64 PROBLEMS IN THE CONDUCTION OF HEAT We may apply this at once to the case of the plane z — 0 which has be 0 temperature from t — — co to t = 0, and at temperature 1 from t = £=oo. By (64) 1 f00 eipt-Z*J (ip) fl = A + — ------ dp ........................ ( * Mr Jo P By the methods of complex integration this solution may be transforme* Fourier's, viz. dz which are, however, more readily obtained otherwise. In the case of a cylinder (r = c) whose surface has been at 0 up to and afterwards at v = 1, we have from (S3) with n = 0 ci [* f ,($ eipt+«s~r) J ~^ e ___ ,. /aJtx ' ............ v /_ i (* J3 c) P of which only the real par.t is to be retained. This applies to the regioi side the cylinder. It may be observed that when t is negative (87) must vanish for po 2 and (90) for r > c.y great.