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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.