Introduction to the Mathematical Theory of the Conduction of Heat in Solids

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Then v.= J „ .... e^"' \ But from the expansion of e""""* in a series of Zonal Harmonics,* we have e-"--V(y)?- Thus «o=gi^^;7(;^E (2^^+!) ^* Jn.,i^;)Pnioos y).
f" e *"*' 1 /it'r'\ But J^ ae— V„+i(ar)J„+i(a/)rfa = -^-p^.^J„+j^-2^j^t.
Hence ""= 4^^/) § (2^+l)-P^(cos y)J^ ae— V„+i(ar)J„+i(a/)cZa. ...(1) But, as in § 88, J ae-''''Vni.i{ar)Jn+i{a/)da=-Aae-'"'Vn+i{ar')Hn+i{ar)da, over the path (P) of Fig. 14, when r > r' ; and we interchange r, r' when r r', the integral being taken over the path (P
...), and r, r' being interchanged when r •*•'; while we interchange r and r' when r •

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