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84 THE LOCOMOTIVE. [June, present case the stretch due to the force S l in the outer shell is given by the following equation : 3.1416 A/?, = 3.1416i?j x S x h, or x AR =R S k. 1 1 l In the same way we find that the stretch of the inner sheet is A 7? 2 = B. 2 SJ: Passing now to the stay-bolt, we note that its original length (neglecting the ends that are within the sheets) is li — R , and that the tension to which it is subjected, per 1 2 square inch of its sectional area, is T. Hence, if we assume that the material of which the stay-bolt is made has substantially the same resistance to stretching as the material of the sheets, we shall find that the increase in the length of the stay-bolt is Fig. 5. — ,..,_ Illustrating the Stretch op the Shells. (R 1 — B!l)Tk But since the stay-bolt is secured to the sheets, its stretch must be precisely equal to A if, — A R . 2 as will be seen upon examining the diagram, Fig. 5. Hence (/?! — R ) 2 Tk= Ai?!— Ai? - 2 But we have already obtained the values of AR t and AR 2 , and (i?,-i? 2 ) T=R S - R S . l l 2 2 if we substitute these in the equation last given, we have (./?, — R%) -T k = Z?j Si k— R 2 S 2 kj or, dividing through by k, Substituting in this equation the values of 8 t and S 2 already found, we have {R X -R 2)T t 1 A t., A (PA — Ta) « " (TaR — t : and upon solving this equation for T we have PAR.) R s . PA(t 2 2V + *i -K 2 3 ) T = a ifj (*, i?x + t t Rs ) + ht s A (i? — x R2 ) which is the formula given for T in the earlier article. <strong>The</strong>se formulae may be tested for special cases, if their trustworthiness is doubted. Thus if we make a = , we see that T remains finite, and hence the total pull on the stay-bolt (Ta) vanishes, as it should. Making a — in the formulas for S x and S 2 , we find (since Ta — 0) that PR —PR X 2 Si = — f— and S = s 1— which are correct ,for this case. (It will be observed that S 2 , being negative, is now xomj)ressive stress.) '
1901.] THE LOCOMOTIVE. a 85 Again, if we make the surfaces flatter and flatter by increasing R t and R 2 until they both become infinite, we have PA(U + '^) I t, J{ 2 \ / 1 1 R»\ a v
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