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360 23 Inhomogeneous perfect fluid solutions with symmetrywhere a, b, c, m and n are constants. Here the four-velocity obeys u 1 =−bu 4 sinh 2u/(a+b cosh 2u), so this solution is tilted if b ̸= 0. The casen =b =0,m 2 +2a 2 = 1 is a Petrov type D solution found by Allnutt (1980),while n = m = 0 gives (15.82). Baillie and Madsen (1985) obtained thespecial case m = √ 3/F , n =1/F − 1, a 2 =3b 2 =3(F 2 − 1)/F 2 ,byapplying Theorem 10.1 to (13.59).Similarly, generalizing the G 3 VI 0 solution (13.58), which is a limit of(13.57), led to the stiff fluid solution (Wainwright et al. 1979)ds 2 = A 2 (−dt 2 +dx 2 )+t(B dy 2 + B −1 dz 2 ),κ 0 µ =(a 2 − b 2 t 2 )/2A 2 t 2 ,[(](23.3)A 2 = t a2 + 1 2 (m2−1) exp n 2 + 1 2 b2) t 2 +2(nm − ab)x ,B = t m e 2nx ,where a, b, m and n are constants, which is a corresponding limit of (23.2).It is tilted if b ̸= 0 (in the obvious tetrad, the velocity obeysu 1 = btu 4 /a).There is also a p = µ solution found from the Bianchi type VII h vacuumsolution (13.62), which is given by replacing G 2 (ξ) in (13.62), ɛ = 1, withG 2 = (sinh 2ξ) a2 +b 2 −3/8 (tanh ξ) 2ab e m2 ξ−(m 2 +2a 2 −2b 2 −11/4)w , (23.4)andΦbymξ where mA =1soh =1/m 2 (Wainwright et al. 1979). Theenergy density and the velocity in the obvious tetrad are given byκ 0 µ =2(a 2 + b 2 +2ab cosh 2ξ)/k 2 G 2 (sinh 2ξ) 2 (23.5)and u 1 = −bu 4 sinh 2ξ/(a + b cosh 2ξ). If b ̸= 0, this solution is tilted.Finally, among the p = µ solutions given by Wainwright et al. (1979),there are two with an H 3 contained in a family of comoving solutionswhich in general have only a G 2 ; they were pointed out by McIntosh(1978a) and Collins (1991). The relevant metrics take the formds 2 = t 2(q−1) F (u)(−dt 2 +dx 2 )+ √ t[dy + W (u)dz] 2 + t 3/2 dz 2 , (23.6)where u = t − x and q is constant. One has F = e −u and W = u.The other has F = u −2p and W = 2 √ 2pu, where p is constant andpu > 0; the homothety is an isometry if p = q. For both solutions, κ 0 µ =(q − 13/16)/t 2 g xx .Among the comoving separable solutions with a G 2 in Wainwright classA(ii) (see §23.3) studied by Wils (1991) are three stiff fluid solutions withan H 3 on S 3 . The first two can be given as follows:[ds 2 = sinh t e −6x dy 2 +e 4x (dz + √ 10e −x cosh t dx) 2]+e 2x sinh 2 t(−dt 2 +6dx 2 ) , κ 0 µ = 7 4 e−2x / sinh 5 t ; (23.7)