with a concentric core of a different material 401 



Now introduce the path (P) of Fig. 1*. In this standard path 

 the argument of a lies between and j at infinity on the right, 



and between -^ and it on the left. The condition (2') at r = 6 

 suggests the suitable value iov A^. Then we are led to the solutions 



6^0 f sin ar e-'i^-' 



ITT j F (a) a 



_ Bvq [ [sin [jia (r — a) sin aasin fia (6 — r)| e~*'""^ , ,_. 

 ITT J (sm fia {0 —a) f (a) smfia [b — a)] a 

 where 



F {a) = a cos aa sin ^xa {h — a) + sin aa cos ^a (6 — a) 



H ' — sm aa sm u.a (o — a), 



and the integrals are taken over the path (P) of Fig. 1 in the a-plane. 





.+ 00 



Fig. 1. 



The value of u^ given by (7) reduces to 

 6^0 [G («) e-"''^'^ 



^, , (?a (8), 



F{a) a ^ " 



where 



G{a) = a cos aa sin ixa [r — a) + sin aa cos fxa {r — a) 



1 — LLCr .... 



H — sm aa sm ixa (r — a), 



fxaa ^ 



and the integrals are taken over the path (P). 



The values of u^ and u^ given in (6) and (8), from the way in 

 which they have been obtained, satisfy the differential equations 

 (1) and (1'), and the conditions (2), (i) and (5), which hold when 

 r = and r ^ a. 



Putting r = 6 in (8), we have -^ da, over the path (P). 



L7T J a 



Introduce the path (Q) of Fig. 2 formed by the path (P), the image 

 of this path in the real axis, and the circular arcs dotted in the 



* A similar path was used by me in Chapter xvin of my book on Fourier's 

 Series and Integrals and the Mathematical Theory of the Conduction of Heat, 1906. 

 See also, for the method of this paper, Phil. Mag., London (Ser. 6), 39, p. 603, 1920. 



26—2 



