62 PROBLEMS IN THE CONDUCTION OF HEAT. [35! When n — 0, that is in the case of symmetry round the pole, this equation takes the same form as for one dimension; hut we have to distinguisl between the inside and the outside of the sphere. On the inside the constants must be so chosen that v remains finiti at the pole (r = 0). Hence rv = A eW (er*J &» - e~r^ (^>)> ...................... (70) or in real form rv = Aer^w® cos {pt + r J(p/Z)} - Ae~r* W* cos {pt - r VQV2)}. ...(71) Outside the sphere the condition is that rv must vanish at infinity. In thti case rv=B eW &-r^ tfw, ............................. (72) or in real form rv = Be~r^^ cos{pt-r</(p/2)} ................... (73) When n is not zero, the solution of (69) may be obtained as in Stokes treatment of the corresponding acoustical problem (Theory of Sound, ch. xvn) Writing r \/(ip) = z, and assuming rv= Ae3 + Be~z, ............................. (74) where A and B are functions of z, we find for B dB nn + l The solution is B = B0fn(z), ............................... (76) where B0 is independent of z and t tr\ - -\ A- 4. - Jn (?)- A + — r,— - I 9 4 ,2 - + *-l • & *J • TD . X& as may be verified by substitution. Since n is supposed integral, the seriei (77) terminates. For example, ifn=l, it reduces to the first two terms. The solution appropriate to the exterior is thus rv = B0Sn6Werr*/Wfn(£ptr') ...................... (78) For the interior we have rv = A,Sne^ {er**J «*' /„ (i*jp4r) - er^ *>/n (- i*^V)}, ...... (79) which may also be expressed by a Bessel's function of order n + %. In like manner we may treat the problem in two dimensions, where everything may be expressed by the polar coordinates r, 6. It suffices tc consider the terms in cos n6, where n is an integer. The differential equatior analogous to (69) is now d?v I dv n2 ^ --- --- -v = ipv, .......................... (80) cZr2 r dr r- ^ ^ !late may be treated precisely upon the lines of tl paper referred to. The potential energy of bending per unit area has tl expression