56 PKOBLEMS IN THE CONDUCTION OF HEAT Jn being, as usual, the symbol of Bezel's function of order n. For, if be even, cos n cos = COS ?i^|r COS l COS o and, if ?i be odd, cos n = - i cos 9i-v^ sin (^'p cos o In either case Thus and (24) becomes (28) cos 7i = 2?r cos nO In (of), ............... (29) This gives the temperature at time t and place (p, z) due to an init instantaneous source distributed over the circle a. The solution (30) may now be used to find the effect of the initial soui expressed by (22). For this purpose we replace q by a da, and introdi the • additional factor Jn (lea), subsequently integrating with respect k between the limits 0 and GO . Comparing the result with that expressed (20), (23), we see that is a common factor which divides out, and that there remains the identity ™ ......... (31 This agrees with the formula given by Weber, which thus receives interesting interpretation. Keverting to (30), we recognize that it must satisfy the fundamei equation (1), now taking the form _ ____. ~d# dp*- p dp + pn- dfr ~ dt' .(32 and that when t = 0 v must vanish, unless also z = 0, p — a. r