An Infinite Number Of Curves ∫= Pdx

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.1ConcerningAn Infinite Number Of CurvesOf The Same Kind.OrA Method For FindingEquations For An Infinite Number Of CurvesOf The Same Kind.Leonhard Euler.§1.Here I call curves of the same kinds such curves that are only different from eachother because of the length of a certain constant line, which by assuming some orother values determines these curves. The constant line has been called the modulusby the Celebrated Hermann, and by others the parameter; but since the nameparameter can create ambiguity I will retained the name module [History has nottreated Euler well in this regard, as the name parameter is now used ; in thistranslation we use the term parameter, with which the reader is familiar]. Thus aparameter is a constant invariable line, by which each one of an infinitude of curves isdetermined ; moreover it has different values and thus it is a variable, if it is to refer to2different curves. Thus if in the equation y = ax , a is taken for the parameter, fromthe variability of a innumerable parabolas can arise placed on the same axis andhaving a common vertical axis.§2. Therefore an infinite number of curves of the same kind are all expressed by thesame equation, as the parameter is given successive values, to which we always assignthe letter a. Indeed if all and every successive values are attributed to this parameter,the equation continues to give all the curves that are contained in the one equation.With Hermann, we will call this a parameter equation; in which therefore, in additionto the other constants which have the same value for all the curves, the quantities thatbelong to any curve are the parameter a and two pertaining variables, and either theabscissa and the applied line [the coordinates z and y], the abscissa and the length ofthe curve, or the abscissa and the area of the curve, etc., are of this kind as postulatedby the problem to be solved.§3. Therefore, there are the variable quantities x and z, which enter into the parameterequation with the parameter a. It is evident, if an algebraic equation is given betweenx, z, and a for a single curve in which a is considered as constant, then the same islikewise a parameter equation, or to pertain to all curves, only if a becomes a variablein this way. But if no algebraic equation can be given between x and z, it is withdifficulty that a parameter equation can be found. For if z =∫Pdx , where P is givenin terms of a, z, and x in some manner or dz = Pdx , in which equation a is consideredas a constant; it is understood that a parameter equation is to be had, if the integralequation dz = Pdx is differentiated anew, also with a made variable. But when anintegration cannot be performed, a method of this kind is needed, in which thedifferential equation which is produced, if the integral is differentiated anew, alsowith a made variable, can be found.

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.2§4. Indeed, the equation dz = Pdx suffices for the curves to be constructed andunderstood. For, with the parameter given a certain value a, the equationdz = Pdx can be constructed, from which with this done, one of an infinite number ofthe curves is obtained, and in the same way others can be found with different valuesput in place of a. But if for these curves certain points must be assigned, as demandedby some problem; such an equation as z =∫Pdx is not sufficient, but an equation isrequired free from the sums of the signs in which if it is not algebraic, also belongsthe differentials of a. Therefore for a given differential equation or a single curvedz = Pdx in which a is considered as a constant, it is required to look for adifferential equation, in which a is a variable, and this is an equation of the parameter.Now this parametric differential equation is sometimes of the first order, sometimes ofthe second or higher, sometimes also even deeper cannot be found.§5. Therefore the method which I discuss is one from which the parameter can befound from the differential equation dz = Pdx , in which a is constant ; first I put P tobe a function of a and x only, as∫Pdx may be possible to be established byquadrature [i.e. expanding as a series and integrating term by term]. Therefore wehave z =∫Pdx , in the integration of this Pdx , a has to be taken as a constant. Nowwith the differential of∫Pdx if a is treated as a variable also ; where the equation ofthe parameter can be obtained by putting it equal to dz. Moreover, the differentialof∫Pdx has itself this form: Pdx + Qda , and hence dz = Pdx + Qda is the equation ofthe parameter, if only the value of Q can become known.§6. Moreover the following theorem is of use in finding the value of Q. Themagnitude A composed in some manner from the two variables t and u, if it isdifferentiated with t put constant, and this differential is differentiated anew with u putconstant and t varying, and the same results if in the inverse order, A first isdifferentiated with u put constant and this differential is differentiated anew with t putconstant and u to be varying. As let A =2 2( t + u ) be differentiated with t madeconstant, and there is obtained : nudu . This is differentiated anew with u put2( t2 + nu )constant and there is produced : − ntudtdu23. Now in the inverse order, ( t +2u ) is2 2 2( t + nu )differentiated with u constant, and this is the differential tdt , which2( t2 + nu )differentiated anew with t constant gives − ntudtdu3, that agrees with the first result2 2 2( t + nu )found. And agreement is determined in the comparisons of any other examples. [Wenow know that continuity is involved before coming to this conclusion.]§7. Moreover although the truth of the demonstration of this theorem may be easilycompleted, yet I would like to add the following demonstration arising from thisdifferentiation. Since A is a function of t and u, A goes to B ift + dt is put in place of t ;

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.3but with u + du in place of u A goes to C. Moreover witht + dt in place of tand u + du in place of u at the same time, A is moved to D. From these it is evident, ifin B, u + du is written in place of u to arrive at D; and in a like manner if in C, t + dt isput in place of t, this also leads to the production of D. From these presented, if A isdifferentiated with t held constant, C − A is produced, for with u + du put in place ofu , A goes to C, moreover the differential is C − A. If again in C − A, t + dt is put inplace of t, D − B is produced, whereby the differential is D − B − C + A . Now in theinverse order, t + dt in place of t in A gives B, and thus the differential of A with onlyt variable gives B − A . This differential with u + du put in place of u, becomesD − C , whereby the differential of this is D − B − C + A , in which it agrees with thedifferential found from the first operation. Q.E.D.§8. Moreover this theorem can be put to use to find the value of Q in this manner.Since P and Q are functions of a and x, then dP = Adx + Bda and dQ = Cdx + Dda ,and since z is equal to∫Pdx , it also is a function of a and x, and moreover we can putdz = Pdx + Qda . Now following the theorem, z is differentiated with x put constant,and this differential Qda is again differentiated with a put constant giving Cdadx .From the other operation, the differential of z first with a constant is Pdx , and now thedifferential with x constant is Bdadx . Whereby on the strength of the theorem,Cdadx and Bdadx must be equal, from which C = B . Moreover B is given from P;for the differential of P with x put constant divided by da gives B. Therefore sincedQ = Bdx + Dda , then Q =∫Bdx , if a is considered constant in these equations.§9. We have therefore from these, dz = Pdx + da∫Bdx , from dP = Adx + Bda .Therefore if it is possible to integrate Bdx , the desired equation for the parameter isobtained. But if it is not possible to be integrated, then this equation is as equallyuseless as the first z =∫Pdx , for each involves the integration of a differential inwhich a must be considered as constant, which is contrary to the nature of theparameter equation, clearly in which a must be equal to a variable as well as x and z.§10. Moreover when Bdx does not admit to an integration: yet the equation found asuseless is not to be disregarded. For if the integration of Bdx follows from∫Pdx theequation of the parameter can be shown. For if∫ Bdx = α∫Pdx + K with the functionK of a and x arising algebraically, then on this account∫Pdx = z,∫Bdx = α z + K and dz = Pdx + αzda+ Kda,which actually is the [differential] equation of the parameter. Therefore as often aseither Bdx itself can be integrated, or the integration of Pdx is to be deduced, theequation of the parameter is obtained, which is a differential equation of the firstorder. But if Pdx is integrable, indeed there is no need for this : as z =∫Pdx is at thesame time the equation of the parameter.

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.4§11. But if∫Bdx can neither be shown to be algebraic nor able to be reduced to theform∫ Pdx , it has to be considered, whether ∫Bdx can be reduced to the integral ofanother differential, in which a is not present. For with such an integral in which a isnot present, then it does not disturb the parametric equation if the differential can betaken as it pleases. And by the same rule, if∫Pdx can be reduced to another integral,which does not contain a, nor indeed is this a help in the determination of Q, but z=∫ Pdx gives the parameter equation at once, as if it becomes ∫Pdx = b∫Kdx with bgiven for a and K [is a function of] x only, and the equation of the parameter is givenby z = b Kdx or dz = zdb + Kbdx .b∫§12. But if all this comes to nothing then the position is indicated that a parameterdifferential equation of the first order cannot be given. On account of which an orderin higher differentials must be sought. To this end I differentiate the equationdz = Pdx + da∫Bdx anew. Moreover I place dB = Edx + Fda , with which done thedifferential of∫Bdx is given by Bdx + da∫Fdx . Therefore with the differentiationcarried out and in place of∫Bdx with the value of this from the same equation surelydz − Pdx put in place, there is obtainedda daddz = Pddx + dPdx + dzdda − Pdxdda2+ Bdadx + da Fdx . Thereforeda da∫ 3∫Fdx = ddz dzdda dzddx dPdx Pdxdda Bdx2−3−2−2+ − . But sinceda da da da da dadz∫Bdx = −da, if∫ Fdx can be reduced to the integrals ∫ Bdx and ∫Pdx or if thePdx = zexpression itself can be integrated, the parameter equation is obtained, which is adifferential equation of the second order. As if∫Fdx = α∫Bdx + β∫Pdx + K , with αand β some given constants in terms of a, and also K for constant a and x, thisequation of the parameter is given :daddz −dzdda−Pdaddx+Pdxdda−dPdadx− Bdx = adz−αPdx+ βzKdada da+3. But B and F areeasily found for a given P.§13. And if moreover as rarely happens,∫Fdx either no longer contains a, or it canbe reduced to another form, in which a is not present, the equation found of thesecond order differentials can be obtained for the proper parameter. But if all these donot yet succeed, then there is still this differentiation to be put in place, in which thedifferential of∫Fdx becomes∫∫PdxdaFdx + da Hdx , by putting dF = Gdx + Hda . Withwhich done it is apparent either∫Hdx can be found, or the integration depends on thepreceding∫ Fdx , or ∫ Bdx and ∫Pdx , or it is possible for a to be eliminated fromunder the integration sign. Because of this if it should be concealed still, a parameterequation of the parameter of the third order in the differentials is obtained; but if nowand

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.6n−1also if∫dx(A + X ) can be expressed algebraically, then the equation sought hasbeen found. If neither appertains to the other, then another differentiation is to be putn−1in place. Moreover the differential of∫dx ( A + X ) is :nn−1n−2dz−(A+X ) dxdx( A + X ) + ( n −1)dA∫(A + X ) dx = Diff.ndAnn 1n 2dz ( A X ) dx dx( A X )And thus there arises : dx( A X ) 1 Diff .+ −−− +∫+ =− .( n−1)dAndA ( n−1)dAn−2Whereby it is to be apparent whither or not∫dx(A + X ) can be integrated or to bereduced to an integral earlier.§19. If n is a positive whole number then the equation of the parameter is algebraic.nFor( A + X ) can be resolved into a finite number of terms, of which each multipliedby dx can be integrated, thus as the parameter a does not enter under the sign of thesummation. Moreover, this is the equation of the parameter :n −− −z = A x + n n 1nA∫Xdx + n.n 1 2 2A∫X dx etc. Therefore there remains to be11.2examined for which cases if n is not a positive whole number, that have the abovementioned conditions in place.m§20. In the first case, let X = bx , where b can also depend on a ; hencemnz =∫( A + bx ) dx . [The original has x rather than z as the subject.] In the first placethis formula is integrable , if m = 1 with i designating some positive whole number :iand then also if m = − 1 . Therefore in these cases the equation of the parameter isn+ialgebraic. But if m = − 1 , where b is unable to depend on a, for that indeed thenequation does not admit to integration, but the following :−1 −1n−1n nndz = ( A + bx ) dx + ndA∫dx( A + bx ) avoids the integral, and makes theequation of the parameter a differential of the first kind.§21. Moreover, not only for whatever value of m is given, and a differential equationof the parameter can be found of the first degree, but also if we should havem k nz =∫x dx( A + bx ) . For this becomesm k n m kn−1dz = x dx( A + bx ) + ndA∫ x dx( A + bx ) . Butm+1 k nm k n x ( A+bx )m k n−1x dx( A + bx ) =+ nkA x dx( A + bx ) , orm+nk+1 m+nk+1∫∫m+1 k nm k n−1( m+nk + 1)x ( A+bx )x dx( A + bx ) = −. Consequently this equation of thenkA m+nk + 1k n m m+1parameter is produced : Akdz = ( A + bx ) ( Akx dx − x dA ) + ( m + nk + 1)zdA. Ina similar manner, the equation of the parameter can be found, ifm knz = B∫ x dx( A + bx ) , for in order that the difference can be produced in this way∫

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.8§24. With these put in place dz = Pdx or z =∫Pdx , let P be a function of ndimensions in a and x, therefore z is such a function of of dimension n + 1. Wherebyon putting dz = Pdx + Qda , then we have Px + Qa = ( n + 1)z.From which the valueof Q substituted gives the parameter equation : dz = Pdx + da (( n + 1)z− Px ) oraadz − ( n + 1 )zda = Padx − Pxda . Which is a differential equation of the first orderonly. Moreover since generally Q = Bdx , then in this case∫( n + 1 )∫Pdx = a∫Bdx + Px. From which it is evident that in this case the integral∫ Bdx can always be reduced to ∫ Pdx .§25. Likewise the equation of the parameter comes about by considering P alone. Forby putting dP = Adx + Bda , there becomes nP = Ax + Ba.But if we letdz = Pdx + da∫Bdx , then dz = Pdx + da∫( nPdx − Axdx ) in which integration a isakept constant. Therefore we have nPdx = nz, and Axdx = Px − Pdx as Adx = P[on integrating Adx by parts]. Thus we haveclearly agrees with the preceding.∫ ∫ ∫ ∫dz = Pdx + da ( n + 1 )z − Px , whicha§26. With P retaining it value of n dimensions. Let z =∫APXdx , where A is afunction of a and X of x only. Therefore we have the equation: z =∫PXdx . ByAputting dP = Adx + Bda,(in which the letter A is not to be confused with the otherwhich is a function of a only[Thus, the A in the formula APX is not the same as that indP = Adx + Bda ]) there results [from the theorem in §23] nP = Ax + Ba.Thereforethe differential of PX with x put constant is BXda. Consequently the differential isobtained d . z = PXdx + da BXdx PXdx da ( nPXdx AXxdx ).A ∫= +a ∫− Hence,∫nPXdx = nz and∫AXxdx = PXx −∫PXdx −∫PxdX . [on differentiation by partsAagain, where the integral of A w.r.t x giving P, and subsequently X and x are diff. inturn] Whereby the total derivative becomes. z PXdx PXxda ( n+1)zdad = − + + da PxdX .Aa Aa a ∫Therefore unless∫PxdX is able to bereduced to∫PXdx or clearly to be integrated, a differential equation of the first orderis unable to be given.§27. But if the equation should be given by z = R∫Pdx , with R being some algebraicfunction of a and x with z constant, but P is a function of a and x of dimension n.Since we have z =∫PdxRthen [from the previous sections]1d z da ( n+)z. = Pdx + ( − Px Rdz zdR2R a R) = − orR2 2Radz − zadR − ( n + 1)Rzda= PR adx − PR xda. Moreover it is held in general, that

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.9just as often as z =∫Pdx can be reduced to a parameter equation, so also canz = R Pdx be reduced to a parameter equation. Indeed no other distinction is present,∫except that it is z in the one case, and it must be z in the other. Whereby if R is eitherRan algebraic quantity or such a transcending function that its differential of thevariable a can also be put in place outside the summation, then the parameter equationcan be found through what has been given already. On account of which in such casesthat follow, even if this rule extends more widely, then it is permitted to omit these,[on the understanding that they still apply.]§28. We can put z =∫( P + Q )dx, or z =∫Pdx +∫Qdx, and P is a function of a and xof dimension n – 1, and also Q is now a function of the same a and x of dimension m– 1. Therefore since the differential of∫ Pdx is P( adx−xda )+ da∫nPdx , [from §24,a awhere it is shown with a change of order, that adz − nzda = P( adx − xda ) ] and theQ( adx−xda )differential of∫Qdx is+ da∫mQdx ; thena a( P+Q )( adx−xda )dzdaadz−(P+Q )( adx−xda )=+ ( n Pdx m Qdx ).aa ∫+∫Putting = u,anddathere becomes u = n∫ Pdx + m∫Qdx.If u again is differentiated, [as we have returnedto a relation of the original kind], then( nP+ mQ )( adx−xda )duda 2 2=+ ( n Pdx m Qdx ).daa ∫+∫Therefore on puttingadu−(nP+mQ )( adx−xda )2 2= t, then t = n Pdx m Qdx.da∫+∫Now with the integrals∫Pdx and∫Qdxeliminated from the these three equations of z, u and t, this equationmnz − ( m + n )u + t = 0 is produced. Which equation is the sought parameter equation,if values to be assumed are substituted in place of u and t .§29. In a similar manner, if z =∫( P + Q + R )dx,and P a function of n – 1, Q afunction of m – 1 and R a function of k – 1 dimensions of a and x. Putadz−(P+Q+R )( adx−xda )u =da2 2 2adu−(nP+mQ+kR )( adx−xda ) adt−(n P+m Q+k R )( adx−xda )and t =, and s =. With whichdadadone, this parameter equation is produced :kmnz − ( km + kn + mn )u + ( k + m + n )t − s = 0.k§30. Again let z =∫( P + Q ) dx,where P is a function of dimension n, and Q is afunction of dimension m, both of a and x. Therefore, whendP = Adx + Bda and dQ = Cdx + Dda , then [from the theorem in §23 applied to P andQ in turn], nP = Ax + Ba and mQ = Cx + Da . Moreover, the differential ofkk −1( P + Q ) with x placed constant and divided by da is k ( B + D )( P + Q ) . On thiskaccount, dz ( P Q ) dx kda k−1= + + ( P + Q ) ( Ba + Da )dx.a ∫[The second term is just

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.11k−1§32. Therefore let V = T( P + Q ) and the differential of this with a placed constantk−1k−1is V = dT( P + Q ) + ( k −1)(TdP+ TdQ )( P + Q ) . Hence the following equationis produced :2 222P dx = αPdx + αPQdx+ βPdx + 2βPQdx+ βQdx + PdT + QdT + ( k −1)TdP+ ( k −1)TdQ,which can be divided by dx. But T should thus be accepted, as the correspondingterms can be made to cancel by taking appropriate values for α and β .§33. If z is not completely determined by∫ Pdx , but by the quantity ∫Qdz , with Qgiven everywhere in terms of a and z, and P by a and x; this equation is obtained :Qdz = Pdx in which the indeterminates x and z are in turn separated from each other.Now the equation of the parameter can be found in this manner : Since Qdx = Pdxcan be differentiated and each member to be put in place also with the variable a withthe aid of dP = Adx + Bda and dQ = Cdz + Dda.Hence there arises :Qdz + da∫ Ddz = Pdx + da∫Bdx or Qdz = Pdx + da(∫Bdx -∫Ddz ). Which equation, ifBdx and Ddz can be eliminated, gives the required parameter equation.∫ ∫§34. Let P be a function of m – 1 dimensions of a and x, and Q a function of n – 1dimensions of a and z. With these put in place :mda∫Pdx+P( adx−xda )nda∫Qdz+Q( adz−zda )Diff.∫Pdx =, and Diff.∫Qdz=. From whichaaQ( adz−zda ) P( adz−zda )this equation arises : ( m-n )∫Pdx = − asdada ∫ Pdx = ∫ Qdz .Whereby if it should be that m = n, then Qadz − Qzda = Padx − Pxda , which is theparameter equation, or da Qdz−Pdx= .a Qz−Px§35. Now if m and n are not equal, the differential equation of the parameter is of theQ( adx−zda ) P( adz−zda )second order. For since ( m-n )∫Pdx = −thendadaQ( adz−zda ) P( adx−xda ) m(m-n)da∫Pdx ( m-n)P( adx−xda )Diff . − =+dadaaamQ( adz−zda ) −nP(adx−xda )=.aWhich is the equation of the parameter sought.§36. If in the proposed equation dz + Pdx = 0 the indeterminates cannot be separatedfrom each other, thus so that P is a function involving x, z and a ; then the equationshould be multiplied by a certain quantity R, in which the terms Rdz + PRdx is to beconsidered as the differential of some integral S. Thus we have :dS = Rdz + PRdx = 0,and thus S = Const. But in order to find the quantity R, letdP = Adx + Bdz and dR = Ddx + Edz, where we take a as a constant for the moment.With these in place, we have d. PR = ( DP + AR )dx + ( EP + BR )dz,on account ofwhich we have D = EP + BR.But as D = dR−Edzthis becomesdx

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.12Edz + EPdx + BRdx = dR. Now since dz + Pdx = 0,it is found thatdR = BRdx, and lR =∫Bdx.Now B is known from the given P, and since B dependson z and x, Bdx should be integrated with the help of the equation dz + Pdx = 0 , ifKindeed it can be done. Thus let Bdx = K , then R = e with le = 1.∫K K§37. Therefore since the equation is dS = e dz + e Pdx = 0,in order to find theequation of the parameter, let dK = Fdx + Gdz + Hda,and it becomesK Kde = e ( Fdx + Gdz + Hda ). Then the integral is taken of e K Hdz with z only madevariable, and with x and a constants, with this done the equation of the parameter isK KKKe dz + e Pdx + da∫eHdz = 0,or dividing by e this equation arises :−KKdz + Pdx + e da∫eHdz = 0.Another equation of the parameter is found, byputting dP = Adx + Bdz + Cda,for the differential of e K P with x and z constant here ise K ( Cda + PHda ). Then e K dx( C + PH ) is integrated with only x variable, with−KKwhich done the equation of the parameter is dz + Pdx + e da∫edx( C + PH ) = 0.But the parameter equations of this kind, unless dz + Pdx = 0 can be determinedwithout R, are hardly of any use.§38. Therefore we can consider special cases, in which the equation is given bydz + Pdx = 0,P is a function of zero dimensions of x and z, and with the parameter agiven by constants and not to be evaluated. Now an integrable formula can always bereturned for dz + Pdx if it is divided by z + Px , on account of whichS = dz Pdx∫ + = Const.z+PxMoreover, it is given by dz Pdx l( z Px ) xdP∫ + = +z+Px− ∫ z+Px.Thereupon on putting z = tx,the function P becomes a function of t only which is T.Whereby the equation becomes S = l( z + Px ) − dT∫, which can be shown byt+Tquadrature [i. e. term by term integration of the series expansion].§39. Therefore nothing else has to be done in order that the parameter equation can befound, unless dz+Pdx∫is differentiated for some variable parameter a put in placez+Pxalso. Therefore the equation dP = Adx + Bdz + Cda is put in place, whereAx + Bz = 0 everywhere [by §23]. Now the coefficient of dx [in the integral] isdifferentiated, surely P with only a variable, the derivative of this is Czda2.z+Px( z+Px )Then Czdz2is integrated with having x only for a variable, with which done the( z+Px )equation of the parameter sought is given : dz + Pdx + ( z + Px )da∫ Czdx+ ) 2= 0 . In a( z Pxsimilar manner, from the coefficient of dz which is 1 , this equation of thez+Pxparameter is found: dz + Pdx − ( z + Px )da∫ Cxdz+ ) 2= 0 , in which integration only z( z Pxis taken as a variable. Or also, even by this equation [from §38]:

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.13dz + Pdx = ( z + Px )da Cdt∫ +2, [the original has D rather than C in the integral] in( t T )which C and T are only given in terms of t and a .§40. I am not able to present here that generality of the homogenous equations, socalled by the most celebrated Johan. Bernoulli, which are all contained by thatequation dz + Pdx = 0 , but I can add a solution. In as much as from the equation(§. 38) l ( z + Px ) =∫ dT = l( t + T −∫dtt+T) , where t = z and T P.t+Tx= There is thereforeproduced lx + dt∫= 0 , or by adding a constant l c = dtt+T∫. As if the proposedx t+T2 2( x z )equation is nxdz + dx ( x + z ) = 0 then we have P2 2+= , and onnx( + tt )putting z = tx,it becomesT= 1 , and thus l c = ndtn ∫, puttingx nt+( 1+tt )( 1+tt ) = t + s then t 1−ss−ds(1+ ss )= and dt = . Whereby the equation becomes :2s2ssl c −nds(1+ss )2=n ls n l(( n 1)s23= −n 1).x ∫+2− − −( n+1)s−(n−1)s n+1 n −1§41. Yet a special case in which the use of the calculation of §36 is apparent, shall befor the proposed equation dz + pzdx − qdx = 0,in which p and q are given in someway by a and x. Which equation taken together with that general equationdz + Pdx = 0 gives P = pz − q,from which B = p, and lR =∫pdx , or = ∫ pdxR e .Therefore when∫pdx can be designated by quadrature, the value of R is known, andthus the proposed equation multiplied by ∫ pdxe is integrable : therefore it becomes∫ pdx+ ∫ pdx− ∫ pdxe dz e pzdx e qdx = 0 , and the integral of e∫pdx z∫e ∫ pdx pdx= qdx or z = epdx∫ −∫ e ∫ qdx . Thus the equation must be differentiated with aand x put as variables : e−∫ pdx pdx∫e∫ qdx , and the differential put equal to dz, withwhich done we have the equation of the parameter. Therefore with these in position,dp = fdx + gda et dq = hdx + ida the equation of the parameter is produced :pdxpdxpdx pdxdz = −e−∫ ( pdx + da gdx ) e∫∫ ∫qdx + qdx + e−∫ da∫e∫( idx + qdx∫gdx ),or for the sake of brevity, on putting∫e ∫ pdx qdx = T thenpdxpdx pdx pdxdz = −e−∫ Tpdx + qdx + e−∫ da e∫idx − e−∫ da Tgdx.∫From which exercise it can be understood, how the parameter equation can be foundin the most efficient manner, as the indeterminates in a proposed equation can beseparated from each other.∫

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.14DEINFINITIS CURVISEIUSDEM GENERIS.SEVMETHODUS INVENIENDIAEQUATIONES PRO INFINITIS CURVISEIUSDEM GENERIS.AUCTORELeonh. Eulero.§1.Curvas eiusdem generis hic voco tales curvas, quae a se invicem non differunt nisiratione lineae cuiusdam constantis, quae alios atque alios valores assumens eas curvasdeterminat. Linea haec constans a Cel. Hermanno modulus est vocatus, ab aliisparameter; quia autem parametri nomen ambiguitatem creare potest, modulivocabulum retinebo. Est itaque modulus linea constans et invariabilis, dum unainfinitarum curvarum quaecunque determinatur; varios autem habet valores et ideo2variabilis est, si ad diversas curvas refertur. Sic si in aequatione y = ax sumatur a promodulo, ex variabilitate ipsius a innumerabiles oriuntur parabolae super eodem axepositae et communnem verticem habentes.§2. Infinitae igitur curvae eiusdem generis omnes unica aequatione exprimuntur,quam modulus cui nobis semper litera a indicabitur, ingreditur. Huic enim modulo, sisuccessive alii atque alii valores tribuantur, aequatio continuo alias dabit curvas, quaeomnes in una aequatione continentur. Aequationem hanc modulum continentem cumHermanno modularem vocabimus; in qua igitur praeter alias constantes et eiusdemvaloris in omnibus curvis quantitates insunt modulus a et duae variabiles ad curvamquamlibet pertenentes, cuiusmodi sunt vel abscissa et applicata, vel abscissa et arcuscurvae, vel area curvae et abscissa etc. prout problema solvendum postulat.§3. Sint igitur quantitates variabiles x et z, quae cum modulo a aequationemmodularem ingrediuntur. Perspicuum est, si detur aequatio algebraica inter x et z et a,pro unica curva, in qua a ut constans consideratur, eandem fore simul modularem, seuad omnes curvas pertinere, si modo a fiat variabilis. At si inter x et z non poteritaequatio algebraica dari, difficile erit aequationem modularem invenire. Nam sitz =∫Pdx , ubi P in a, z, et x, quomodocunque detur, seu dz = Pdx , in qua aequationea ut constans consideratur; intelligitur aequationem modularem haberi, si integralisaequationis dz = Pdx denuo differentietur, posito etiam a variabili. Sed cumintegrationem perficere non liceat, eiusmodi methodus desideratur, qua differentialisaequatio, quae prodiret, si integralis denuo differentietur posita etiam a variabili,inveniri possit.§4. Ad construendas quidem et cognoscendas curvas aequatio dz = Pdx sufficit.Nam, dato ipsi modulo a certo valore construentur aequatio dz = Pdx , quo factohabebitur una curvarum infinitarum, eodemque modo aliae reperientur aliis ponendis

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.15valoribus loco a. Sed si in his curvis certa puncta debeant assignari prout problemaaliquod postulat; talis aequatio z =∫Pdx non sufficit sed requirtur aequatio a signissummatoriis libera in qua si non est algebraica, etiam differentialia ipsius a insint. Exdata igitur aequatione differentiali pro unica curva dz = Pdx in qua a ut constansconsideratur, quaeri oportet aequationem differentialem, in qua et a sit variabilis,haecque erit modularis. Haec vero modularis interdum erit differentialis primi gradus,interdum secundi et altioris, interdum etiam penitus non poterit inveniri.§5. Quo igitur methodum tradam, qua ex aequatione differentiali dz = Pdx , in qua aest constans, modularis possit invenire, quae a ut variabilem contineat; pono primo Pesse functionem ipsarum a et x tantum, ut∫Pdx saltem per quadraturus exhiberipossit. Erit igitur z =∫Pdx , in integratione ipsius Pdx , a pro constanti habita.Quaeritur nunc differentiale ipsius∫Pdx si etiam a ut variabilis tractetur; quoinvento ipsique dz aequali posito habebitur aequatio modularis. Differentiale autemipsius∫Pdx habebit hanc formam Pdx + Qda , eritque dz = Pdx + Qda aequatiomodularis, si modo valor ipsius Q esset cognitus.§6. Ad inveniendum autem valorem ipsius Q sequens inservit theorema. Quantitas Aex duabus variabilibus t et u utcunque composita, si differentietur posito t constante,hocque differentiale denuo differentietur positu u constans et t variabili, idem resultatac si inverso ordine A primo differentietur positu u constante hocque differentialedenuo differentietur posito t constante et u variabili. Ut sit A =2 2( t + u ) ,differentietur posito t constante, habibitur nudu . Hoc denuo differentietur posito2( t2 + nu )u constante et prodibit − ntudtdu2. Iam ordine inverso differentietur ( t3+2u ) posito2 2 2( t + nu )u constante, eritque differentialetdt( t2 + nu2posito u constante, eritque differentiale)tdt , quod denuo differentiatum posito t constante dabit − ntudtdu2( t2 3, id quod+ nu )2 2 2( t + nu )congruit cum prius invento. Atque similis convenientia in quibusque aliis exempliscernetur.§7. Quamvis autem huius theorematis veritatem exercitati facile perficiant,demonstrationem tamen sequentem adiiciam ex ipsius differentiationis natura petitam.Cum A sit functio ipsarum t et u, abeat A in B si loco t ponatur t + dt ; at positou + du loco u abeat A in C. Posito autem simul t + dt loco t et u + du loco u mutetur Ain D. Ex his perspicuum est, si in B scribatur u + du loco u provenire D; similquemodo si in C ponatur t + dt loco t proditurum quoque D. His praemissis, sidifferentietur A positio t constante prodibit C − A, nam posito u + du loco u abit A inC, differentiale autem est C − A. Si porro in C − A ponatur t + dt loco t prodibitD − B , quare differentiale erit D − B − C + A . Inverso nunc ordine posito t + dt loco tin A habebitur B, eritque differentiale ipsius A posito tantum t variabili B − A . Hoc

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.16differentiale posito u + du loco u abit in D − C , quare eius differentiale eritD − B − C + A , in quod congruit cum differentiali priori operatione invento. Q.E.D.§8. Istud autem theorema hoc modo inservit ad valorem ipsius Q inveniendum. Cum Pet Q sint functiones ipsarum a et x, sit dP = Adx + Bda et dQ = Cdx + Dda , atque zcum sit =∫Pdx , erit quoque functio ipsarum a et x, postitum autem estdz = Pdx + Qda . Iam secundum Theorema differentietur z posito x constante, eritquedifferentiale Qda hoc porro differentiatum posito a constante dabit Cdadx . Alteraoperatione differentiale ipsius z posito primo a constante est Pdx , huius verodifferentiale posito x constante est Bdadx . Quare vi theorematis aequalia esse debentCdadx et Bdadx , ex quo sit C = B . Datur autem B ex P; differentiale enim ipsius Pposito x constante divisum per da dat B. Cum igitur sit dQ = Bdx + Dda , erit∫Q = Bdx , si in hac integratione a ut constans consideretur.§9. Ex his ergo habebitur dz = Pdx + da∫Bdx , existente dP = Adx + Bda . Siigitur Bdx integrari poterit, desiderata habitur aequatio modularis. At si integrari nonpotest aeque inutilis est haec aequatio ac prima z =∫Pdx , utraque enim involvitintegrationem differentialis, in qua a ut constans debet considerari, id quod est contranaturam aequationis modularis, quippe in qua a aeque variabile esse debet ac x et z.§10. Quando autem Bdx integrationem non admittit : non tamen aequatio inventa utinutilis omnino est negligenda. Nam si integratio ipsius Bdx pendeat a∫Pdx aequatiomodularis poterit exhiberi. Si enim fuerit∫ Bdx = α∫Pdx + K existente K functioneipsarum a et x algebraica, erit ob∫Pdx = z,∫Bdx = αz+ K et dz = Pdx + αzda+ Kda,quae aequatio revera erit modularis. Quoties igitur Bdx vel reipsa poterit integrari, velad integrationem ipsius Pdx deduci, aequatio habebitur modularis, quae eritdifferentialis primi gradus. At si Pdx est integrabile, ne hoc quidem opus est: sedz = Pdx erit simul aequatio modularis.∫§11. Si autem∫ Bdx neque algebraice exhiberi neque ad ∫Pdx reduci potest,dispiciendum est, num∫Bdx ad integrationem alius differentialis, in qua a non inest,possit reduci. Tale enim integrale in qua a non inest non turbat aequationemmodularem, cum si libuerit per differentiationem tolli possit. Atque eodem iure, si∫Pdx reduci poterit ad aliud integrale, quod a non continet, nequidem hac ipsius Qdeterminatione opus est, sed z =∫Pdx statim dat aequationem modularem, ut si sit∫ Pdx = b∫Kdx data b per a et K per x tantum, erit aequatio modularis z = b∫ Kdx seudz = zdb + Kbdx .b

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.17§12. Si autem haec omnia nullum inveniant locum indicio est, aequationemmodularem primi gradus differentialem non dari. Quamobrem in altioris gradusdifferentialibus quaeri debebit. Ad hoc differentio denuo aequationem∫∫Bdx differentiale∫dz = Pdx + da Bdx . Pono autem dB = Edx + Fda , quo facto erit ipsiusBdx + da Fdx. Differentiatione igitur peracta et loco∫Bdx eiusvalore ex eadem aequatione nemne dz − Pdx posito, habebiturda daddz = Pddx + dPdx + dzdda − Pdxdda2+ Bdadx + da∫Fdx . Erit igiturda daddz dzdda dzddx dPdx Pdxdda Bdx∫Fdx =2−3−2−2+3− . Cum autem sitda da da da da da ∫∫Bdx =, si∫ Fdx reduci poterit ad integralia ∫ Bdx et ∫Pdx vel si reipsa duciPdx = zpoterit ad integrari, habebitur aequatio modularis, quae erit differentialis secundigradus. Ut si fuerit∫Fdx = α∫Bdx + β∫Pdx + K , datis α et β utcunque per a etconstantes, et K per a et x constantes, erit aequatione modularis haecdaddz −dzdda−Pdaddx+Pdxdda−dPdadx− Bdx = adz−αPdx+ βzKdada da+3. At B et F ex dato Pfacile reperiuntur.§13. Si∫Fdx quod autem rarissime evenit vel non amplius in se contineat a, vel adaliud possit reduci, in quo a non insit, aequatio inventa differentialis secundi graduspro legitima modulari poterit haberi. Sed si haec omnia nondum succedant, adhucdifferentiatio est instituenda, in qua differentiale ipsius∫Fdx erit∫positodzda−PdxdaFdx + da Hdx= Hda . Quo facto videndum est vel an∫Hdx re ipsa possit exhiberi,dF Gdx +vel an pendeat a praecedentibus∫ Fdx , ∫ Bdx et ∫Pdx , vel an possit ex signosummatorio a eliminari. Quorum si quod obtiget, habebitur aequatio modularisdifferentialis tertii gradus; sin vero nullum locum habuerit, continuanda estdifferentiatio simili modo donec signa summatoria potuerint eliminari.§14. His generalibus praemissis ad specialia accedo, casus evoluturus, quibus functioP quodammodo determinatur. Sit igitur P functio ipsius x tantum, a prorsus noninvoluens, quam littera X designabo, erit ergo dz = Xdx , quae quidem aequatio quianon continet a, ad unicam videtur curvam pertinere, neque integratione constantemaddere liceat, poterit esse z =∫Xdx + na seu differentiando dz = Xdx + nda , quae estvera aequatio modularis. Eadem aequatio prodiisset, si iuxta regulam Xdifferentiassem posito x constante, unde prodit B = 0 et∫Bdx = n constanti, orta igituresset aequatio modularis dz = Xdx + nda cuius loco potius integralisz =∫Xdx + na usurpatur.§15. Sit nunc P = AX , existente A functione ipsius a, et X ipsius x tantum. Cum igitursit z =∫Pdx erit z =∫AXdx seu quia in integratione a ut constans debet considerari,et

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.18∫2z = A Xdx . Quae aequatione seu eius differentialis Adz − zdA = A Xdx erit aequatiomodularis quaesita. Loco A quidem cum sit functio ipsius a tantum, poni potest ipsemodulus a : nam loco moduli eius functio quaecunque eodem iure pro modulo haberipotest.§16. Sit P = A + X litteris A et X eosdem ut ante retinentibus valores. Erit ergodz = Adx + Xdx atque z = Ax +∫Xdx , quae aequatio iam est modularis; quiamodulus A non est in signo summatorio involutus. Si quem autem∫Xdx offendat,differentialem aequationem dz = Adx + xdA + Xdx pro modulari habere potest.§17. Simili ratione modularem aequationem invenire licet, si fueritP = AX + BY + CZ etc. ubi A, B, C sunt functiones quaecunque ipsius moduli a, et X,Y, Z functiones quaecunque ipsius x et constantium excepta a. Namque obdz = AXdx + BYdx + CZdx erit z = A∫ Xdx + B∫Ydx+ C∫ Zdx , quae simul estmodularis, cum modulus a nusquam post signum summatorium reperiatur.§18. SitnnP = ( A + X ) seu z =∫dz( A + X ). Differentiale ipsius P posito x constanten−1est n( A + X ) dA id quod per da divisum dat superiorem valorem B vid. §8. Eritnnn−1n dz−(A+X ) dxigitur dz = ( A + X ) dx + ndA∫(A + X ) dx seu dx( A +− 1∫X ) =.ndACum igitur sit dx( A +nX ) = z , si haec duo integralia a se invicem pendeant, vel∫∫n−1dx(A + X ) algebraice etiam exprimi poterit, habebitur quod quaeritur. Sineutrum contingat denuo differentiatio est instituenda. Est autem differerntiale ipsiusnn−1n−1n−2dz−(A+X ) dx∫dx( A + X ) = dx( A + X ) + ( n −1)dA∫(A + X ) dx = Diff.ndAnn−1n−2dz ( A X ) dx dx( A X )Erit itaque dx( A X ) 1− ++∫+ = Diff .− . Quare( n−1)dAndA ( n−1)dAn−2videndum est an∫dx(A + X ) possit vel integrari vel ad priora integralis reduci.§19. Si n fuerit numerus integer affirmatius aequatio modularis erit algebraica. Namn( A + X ) potest in terminos numero finitos resolvi, quorum quisque in dx ductusintegrari potest, ita ut modulus a in signum summatorium non ingrediatur. Erit autemn −−aequatio modularis haec z = A x + n n 1A− n∫Xdx + n.n 1 2 2A∫X dx etc.11.2Examinandum igitur restat quibus casibus si n non fuerit numerus integeraffirmatiuus, supra memoratae conditiones locum habeant.m§20. Sit primo X = bx , ubi b etiam ab a pendere potest; erit ergomnx =∫( A + bx ) dx . Haec formula primo ipsa est integrabilis, si m = 1 , designante iinumerum quemcunque affirmatiuum integrum : deinde etiam si m = − 1 . His igiturn+i

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.19casibus aequatio modularis fit algebraica. At si m = − 1 , ubi b ab a non pendere potestnilla quidem aequatio integrationem non admittit sed sequens−1 −1n−1n nndz = ( A + bx ) dx + ndA∫dx( A + bx ) evadit integrabilis, fitque aequatiomodularis differentialis primi casus.§21. Non solum autem, quicunque valor ipsi m tribuatur aequatio modularism k ndifferentialis primi gradus haberi potest, sed etiam si fuerit z =∫x dx( A + bx ) .m k n m kn−1Fiet enim dz = x dx( A + bx ) + ndA∫ x dx( A + bx ) . Sed estm+1 k nm k n x ( A+bx )m k n−1x dx( A + bx ) =+ nkA x dx( A + bx ) , seum+nk+1 m+nk+1∫∫m+1 k nm k n−1( m+nk + 1)x ( A+bx )x dx( A + bx ) = −. Consequenter habebitur aequationkA m+nk + 1k n m m+1modularis haec Akdz = ( A + bx ) ( Akx dx − x dA ) + ( m + nk + 1)zdA. Similim k nmodo modularis esset inventa, si fuisset z = B∫ x dx( A + bx ) alia enim nonprodisset differentia nisi quod loco z scribi debuisset z B, et loco dz, Bdz−zdB2siBquidem B ab a etiam pendeat.§22. Missis autem huiusmodi litterae P determinationibus, quippe quae minus latepatent, ad alias accedo, quae multo saepius usui esse possunt. Continentur haedeterminationes ea functionis cuiusdam propositae proprietate, qua functio eundemubique tenet dimensionum quantitatum variabilium numerum. Tales enim functionespeculiari modo differentiationem admittunt. Ut sit u functio nullius dimensionisipsarum a et x, cuiusmodi funt a ( a x ),2 2−aliaeque similes, in quibus ipsarum a et xx adimensionum numerus in denominatore aequalis est numerus dimensionumnumeratoris. Det autem talis functio u differentiata Rdx + Sda ; dico fore Rx + Sa = 0 .Nam si in functione u ponatur x = ay , omnia a sese destruent et in ea praeter y etconstantes nulla alia littera remanebit. Hancobrem in differentiali post substitutionemaliud differentiale praeter dy non reperietur. Cum autem sit x = ay eritdx = ady + yda , ideoque du = Rady + Ryda + Sda . Debebit ergo esse Ry + S = 0 .§23. Sin vero fuerit u functio m dimensionum ipsarum a et x, atquedu = Rdx + Sda; erit umfunctio ipsarum a et x nullius dimensionis. Differentieturxigitur u met prodibit xdu −mudx+1seu Rxdx−mudx+Sxda.mm+1Quod cum sit differentialexxx2functionis nullius dimensionis erit Rx − mux + Sax = 0,seu Rx + Sa = mu.Quare sifuerit u functio m dimensionum ipsarum a et x; atque ponatur du = Rdx + Sda;eritRx + Sa = mu ideoque du = Rdx + da ( mu − Rx ) seu adu = Radx − Rxda + muda.a∫

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.20§24. His praemissis in dz = Pdx seu z =∫Pdx sit P functio n dimensionum ipsarum aet x, erit igitur z talis functio dimensionum n + 1. Quare si ponatur dz = Pdx + Qda ,erit Px + Qa = ( n + 1)z.Ex quo valor ipsius Q substitutus dabit aequationemmodularem dz = Pdx + da (( n + 1)z− Px ) seu adz − ( n + 1 )zda = Padx − Pxda . Quaeatantum est differentialis primi gradus. Cum autem generaliter erat Q =∫Bdx , erit hoccasu ( n + 1 )∫Pdx = a∫Bdx + Px.Ex quo perspicitur hoc casu integrale∫Bdx semperreduci ad∫ Pdx .§25. Eadem aequatio modularis proveniet ex consideratione solius P. Posito enimdP = Adx + Bda , erit nP = Ax + Ba.Cum autem sit dz = Pdx + da∫Bdx , eritdz = Pdx + da∫( nPdx − Axdx )a∫= nz, et∫Axdx= Px −∫Pdxob∫in qua integratione a constans habetur. Erit igiturnPdx Adx = P. Habebitur itaquedz = Pdx + da ( n + 1 )z − Px , id quod prorsus congruit cum praecedentibus.a§26. Retinente P suum valorem n dimensionum. Sit z =∫APXdx , ubi A sit functioipsius a et X ipsius x tantum. Erit igitur z =∫PXdxA. Posito dP = Adx + Bda , (in quolittera A cum altera quae est functio ipsius a tantum non est confundenda) eritnP = Ax + Ba. Ipsius PX differentiale igitur posito x constante erit BXda.Consequenter habebitur d . z = PXdx + da BXdx PXdx da ( nPXdx AXxdx ).A ∫= +a ∫− Estvero∫nPXdx = nz et∫AXxdx = PXx −∫PXdx −∫PxdX . Quare fietA. z PXdx PXxda ( n+1)zdad = − + + da PxdX .Aa Aa a ∫Nisi igitur∫PxdX reduci poterit ad∫PXdx vel prorsus integrari, aequatio modularis differentialis primi gradus darinequit.§27. At si fuerit z = R∫Pdx , existente R functione quacunque algebraica ex a, x etetiam z constante, at P functione ipsarum a et x dimensionum n. Quia est z =∫PdxR1erit d z da ( n+)z. = Pdx + ( − Px Rdz zdR2R a R) = − seuR2 2Radz − zadR − ( n + 1)Rzda= PR adx − PR xda. In universum autem teneatur,quoties z =∫Pdx ad aequationem modularem reduci possit, toties etiamz = R Pdx ad aequationem modularem reduci posse. Nullum aliud enim discrimen∫aderit, nisi quod in illo casu erat z, hoc casu debeat esse z R. Quare si R fuerit velquantitas algebraica, vel talis transcendens, ut eius differentiale posito etiam avariabili possit sine summatione exhiberi, aequatio modularis per praecepta data

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.21reperietur. Quamobrem in posterum tales casus, etiamsi latius pateant praetermitterelicebit.§28. Ponamus esse z =∫( P + Q )dx, seu z =∫Pdx +∫Qdxet P esse functionemipsarum a et x dimensionum n – 1, Q vero functionem earundem a et x dimensionumP( adx−xda )m – 1. Cum igitur differentiale ipsius∫Pdx sit+ da∫nPdx et differentialea aQ( adx−xda )ipsius∫Qdx sit+ da∫mQdx ; erita a( P+Q )( adx−xda )dzdaadz−(P+Q )( adx−xda )=+ ( n Pdx m Qdx ).aa ∫+∫Ponatur = u,daeritque u = n∫ Pdx + m∫Qdx.Si igitur porro differentietur erit( nP+ mQ )( adx−xda )duda 2 2=+ ( n Pdx m Qdx ).daa ∫+∫Posito igituradu−(nP+mQ )( adx−xda )2 2= t, erit t = n Pdx m Qdx.da∫+∫Eliminatis nunc ex his tribusaequationibus ipsarum z, u et t integralibus Pdx et Qdx , prodibit haec aequatiomnz − ( m + n )u + t = 0.Quae aequatio, si loco u et t valores assumti substituantur, eritmodularis quaesita.§29. Simili modo si fuerit z =∫( P + Q + R )dx,et P functio n – 1, Q functio m – 1 et Radz−(P+Q+R )( adx−xda )functio k – 1 dimensionum ipsarum a et x. Ponatur u = etda2 2 2adu−(nP+mQ+kR )( adx−xda ) adt−(n P+m Q+k R )( adx−xda )t =, et s =. Quo facto eritdadaaequatio modularis haec : kmnz − ( km + kn + mn )u + ( k + m + n )t − s = 0.k§30. Sit porro z =∫( P + Q ) dx,ubi P sit functio n dimensionum, Q vero functio mdimensionum ipsarum a et x. Quando igitur est dP = Adx + Bda et dQ = Cdx + Dda ,kerit nP = Ax + Ba et mQ = Cx + Da . Differentiale autem ipsius ( P + Q ) posito xk−1constante divisum per da est k ( B + D )( P + Q ) . Hanc ob rem eritkdz ( P Q ) dx kda k−1= + + ( P + Q ) ( Ba + Da )dx.a ∫Cum autem sitBa = nP − Ax et Da = mQ − Cx, et Adx = dP et Cdx = dQ ob a in hac integrationekconstans, erit dz ( P Q ) dx da k −1= + + ( P + Q ) ( nPdx + mQdx − xdP − xdQ ),a ∫seuk( P+Q ) ( adx−xda )kdzda−1= + ( P + Q ) (( nk + 1)Pdx+ ( mk + 1)Qdx ).aa ∫Ponaturkadz−(P+Q ) ( adx−xda ) −zdak−1= u erit u (nPdx + mQdx )( P + Q ) .kda=∫Quare sik−1kintegrale ( nPdx + mQdx )( P + Q ) pendet ab integrali∫( P + Q ) dx habebitur∫aequatio modularis differentialis gradus primi; sin minus differentiatio estcontinuanda. Fit autem∫∫

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.22k−1−du = ( nPdx + mQdx )( P + Q ) + uda − dak 1( nP + mQ )( P + Q ) x +a ada 2 2222k−2∫(kn P dx + ( 2kmn+ n − 2mn+ m )PQdx + km Qdx )∫(P + Q ) .aEt positok−1adu−uda−(nP+mQ )( P+Q ) ( adx−xda )t =da=2 2222k−2t∫( kn P dx + ( 2kmn+ n − 2mn+ m )PQdx + km Qdx )( P + Q )§31. Cum igitur habeantur tria integralia videndum est, num ea a se invicem pendeant,hoc enim si fuerit, habebitur aequatio algebraica inter t, u et z, quae dabit loco t et usubstitutis assumtis valoribus aequationem modularem differentialem secundi gradus.Quo autem facilius in casibus particularibus perspici possit, an pendeant a se invicem,kad alias formas eas reduci convenit. Cum igitur sit z =∫( P + Q ) dx,eritk−1u = mz + ( n − m ) ( P + Q ) Pdx, et∫2 22k−22t = ( 2 km + n − m )u − ( km − m + mn )z + ( n − m ) ( k −1)( P + Q ) P dx.Quaerendum itaque est an∫k−22( P + Q ) P dx reduci possit ad haeck−1k∫( P + Q ) Pdx et∫(P + Q ) dx . Vel an sitk−22k −1k∫(P + Q ) P dx = α∫(P + Q ) Pdx + β∫(P + Q ) Pdx + V designante Vquantatem algebraicam quamcumque per a et x datam, et α ac β sunt coefficientes exconstantissimis et a compositae.k−1§32. Fiat igitur V = T( P + Q ) huis differentiale posito a constante sitk−1k−1V = dT( P + Q ) + ( k −1)(TdP+ TdQ )( P + Q ) . Prodibit ergo sequens aequatio2 222P dx = αPdx + αPQdx+ βPdx + 2βPQdx+ βQdx + PdT + QdT + ( k −1)TdP+ ( k −1)TdQ,quae per dx dividi poterit. At T ita debet accipi, ut termini respondentes sesedestruant, sumtis ad hoc idoneis pro α et β valoribus.§33. At si per∫ Pdx non absolute determinetur z sed quantitas ∫Qdz , data Q utcunqueper a et z, atque P per a et x; habebitur ista aequatio Qdz = Pdx in qua indeterminataex et z sunt a se invicem separatae. Modularis vero aequatio hoc modo invenietur :Quia est Qdx = Pdx differentietur utrumque membrum ponendo etiam a variabili opedP = Adx + Bda et dQ = Cdz + Dda. Erit ergoQdz + da∫ Ddz = Pdx + da∫Bdx seu Qdz = Pdx + da(∫Bdx -∫Ddz ). Quae aequatio, sipoterunt eliminari, dabit modularem quaesitam.∫Bdxet∫Ddz∫

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.23§34. Sit P functio m – 1 dimensionum ipsarum a et x, et Q functio n – 1 dimensionemipsarum a et z. His positis eritmda∫Pdx+P( adx−xda )nda∫Qdz+Q( adz−zda )Diff.∫Pdx =, and Diff.∫Qdz=. Ex quoaaeruitur ista aequatioQ( adz−zda ) P( adz−zda )( m-n )∫Pdx = − obdada ∫ Pdx = ∫Qdz . Quaresi fuerit m = n, erit Qadz − Qzda = Padx − Pxda , quae est aequatio modularis, seuda Qdz−Pdx= .a Qz−Px§35. Sin vero m et n non sint aequales, aequatio modularis erit differentiales secundiQ( adx−zda ) P( adz−zda )gradus. Nam cum sit ( m-n )∫Pdx =−eritdadaQ( adz−zda ) P( adx−xda ) m(m-n)da∫Pdx ( m-n)P( adx−xda )Diff . − =+dadaaamQ( adz−zda ) −nP(adx−xda )=.aQuae aequatio est modularis quaesita.§36. Si in aequatione proposita dz + Pdx = 0 indeterminatae non fuerint a se invicemseparatae, ita ut P sit functio involvens x et z et a; debebit per quantitatem quandam Rmultiplicari, quo formula Rdz + PRdx ut differentiale integralis cuiusdam S possitconsiderari. Erit itaque dS = Rdz + PRdx = 0,ideoque S = Const. Sed ad quantitatemR inveniendam, sit dP = Adx + Bdz et dR = Ddx + Edz,ubi a tantisper pro constanthabemus. His positis erit d. PR = ( DP + AR )dx + ( EP + BR )dz,quocirca debet esseD = EP + BR. At ob D = dR−Edzfiet Edz + EPdx + BRdx = dR.Cum vero sitdxdz + Pdx = 0,habebitur dR = BRdx, et lR =∫Bdx.Cognita vero est B ex dato P, etquia B et z et x involuit, Bdx integrari debet ope aequationis dz + Pdx = 0 , si quidemKfieri potest. Sit itaque Bdx = K , erit R = e posito le = 1.∫K K§37. Cum igitur sit dS = e dz + e Pdx = 0,ad aequationem modularem inveniendamK Ksit dK = Fdx + Gdz + Hda,eritque de = e ( Fdx + Gdz + Hda ). Sumatur deindeintegrale ipsius e K Hdz posito tantum z variabili, x vero et a constantibus, quo factoK KKKerit aequatio modularis e dz + e Pdx + da∫eHdz = 0,seu diviso per e haec−KKdz + Pdx + e da∫eHdz = 0.Alia aequatio modularis invenitur, positodP = Adx + Bdz + Cda, erit enim ipsius e K P differentiale posito x et z constante hoce K ( Cda + PHda ). Integretur e K dx( C + PH ) posito tantum x variabili, quo facto erit−KKaequatio modularis dz + Pdx + e da∫edx( C + PH ) = 0.Sed huiusmodiaequationes nisi R possit sine aequatione proposita dz + Pdx = 0 determinari, nulliusfere sunt usus.§38. Consideremus igitur casus particulares, sitque in aequatione dz + Pdx = 0,Pfunctio nullius dimensionis ipsarum x et z, non computatis constantibus et modulo a.Formula vero dz + Pdx integrabilis semper redditur si dividatur per z + Px ,

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.24quamobrem erit S = dz Pdx∫ + = Const.z+PxFit autem dz Pdx l( z Px ) xdP∫ + = +z+Px− ∫ z+Px.Deinde posito z = tx,fiet P functio ipsius t tantum quae sit T. Quare eritS = l( z + Px ) − dT∫quod per quadraturus potest exhiberi.t+T§39. Ad aequationem modularem igitur inveniendam nil aliud est agendum, nisi utdz+Pdx∫differentietur posito quoque modulo a variabili. Ponatur igiturz+PxdP = Adx + Bdz + Cda , ubi erit Ax + Bz = 0.Differentietur nunc coefficiens ipsius dx,nempe P posito tantum a variabili, erit eius differentiale Czda2. Deindez+Px( z+Px )integretur Czdz2tantum x pro variabili habia, quo facto erit aequatio modularis( z+Px )quaesita dz + Pdx + ( z + Px )da∫ Czdx+ ) 2= 0 . Simili modo ex coefficiente ipsius dz( z Pxqui est 1 z+Pxprodit haec aequatio modularis dz + Pdx − ( z + Px )da∫ Cxdz+ ) 0 2= , in( z Pxqua integratione z tantum pro variabili habetur. Sive etiam haecdz + Pdx = ( z + Px )da Cdt∫ +2in qua C et T per solum t et a dantur.( t T )§40. Praemittere hic non possum, quin generalem aequationum homogenearum, uti aCel. Ioh. Bernoulli vocantur, quae omnes hac aequatione dz + Pdx = 0 continentur,resolutionem adiiciam. Namque reperitur ex (§. 38)l(z + Px ) =∫ dT = l( t + T − dtt+T) ∫ubi = z et T P.t+T x= Prodibit igitur lx + dt∫= 0 seut+Tadiecta constante l c = dt∫. Ut si proposita sit aequatiox t+T22 2( x z )( + tt )nxdz + dx ( x + z ) = 0 fiet P = , positque z = tx,erit T = 1 ideoquenxnl c = ndt∫fiat ( 1+tt ) = t + s erit t 1−ss−ds(1+ ss )= et dt = . Quare eritx nt+( 1+tt )2s2ss∫l c −nds(1+ss )2=n ls n l(( n 1)s23= − +2− − n −1).x ( n+1)s−(n−1)s n+1 n −12 +§41. Quo tamen usus calculi §36 in casu speciali appareat, sit aequatio propositadz + pzdx − qdx = 0,in qua p et q utcunque in a et x dantur. Quae aequatio cum illagenerali dz + Pdx = 0 collata dat P = pz − q,ex quo fiet B = p, et lR =∫pdx , seu= ∫ pdxR e . Cum igitur∫pdx per quadraturas possit assignari, cognitus est valor ipsiusR, ideoque aequatio proposita per ∫ pdxe multiplicata sit integrabilis : erit igitur∫ pdx+ ∫ pdx− ∫ pdxe dz e pzdx e qdx = 0 huius integralis e∫pdx z∫e ∫ pdx pdx= qdx seu z = e−∫ pdx∫e∫ qdx . Differentiari itaque debete−∫ pdx pdx∫e∫ qdx positis et a et x variabilibus, et differentiale ipsi dz aequale poni,quo facto habebitur aequatio modularis. Positis igitur

Commentarii academiae scientiarum Petropolitanae 7 (1740)pp. 174 - 189. Translated and annotated by Ian Bruce.25dp = fdx + gda et dq = hdx + ida prodibit ista aequatio modularispdxpdxpdx pdxdz = −e−∫ ( pdx + da gdx ) e∫∫ ∫qdx + qdx + e−∫ da∫e∫( idx + qdxseu posito brevitatis gratia∫e ∫ pdx qdx = T eritpdxpdx pdx pdxdz = −e−∫ Tpdx + qdx + e−∫ da e∫idx − e−∫ da Tgdx.∫Ex qua operatione intelligi potest, ad aequationem modularem inveniendam idmaxime esse efficiendum, ut in aequatione proposita indeterminatae a se invicemseparentur.∫∫gdx ),