174 C. E. Weatherbiirn : 



This is an identity in A, and P+^(<s) does not contain X. We may 

 therefore put X— 1 giving 



/P+Hts)dt=l 

 which may also be deduced from (31) in virtue of the properties of 

 the distribution <^r(0- This last relation combined with (33') 

 shows that 



(34) /H+Hts)d(=0 



while from (32) and (33) it follows that 



(34') {1-Xyil[ts)dt = l A,±l. 



This may also be proved from the first of (8), multiplying by df 

 and integrating over the boundary. 



^7. — Expansions. — The second member of the equation (19a) 

 assumes, when X,, =1. the form 



E(0 = f (0 -./■£ {0)<l>riO)de = f (0 - Gr 

 r= I, 2, . . . . , m 

 according as f is on the 1st, 2nd . . . ?»th surface. 



The series (22a) now becomes, by (29) 



(35) v{t) = [i{t)-Gr] + X[f,{t)-Gr]+X\Ut)-Cr]+ .... 



and since v(^) noAv possesses no pole at X= +1, while X= — 1 is not 

 a singular value, this series is convergent for |Aj = l. The terms 

 therefore decrciise indefinitely, and we have for the constant Or 

 the value! 



n= cc 



=Lt j'i{6)hn{et)de 



n=cc 



whei-e f is on tlie rth surface. The constant C,- assumes m different 

 constant values, one on eacli of the surfaces. 



In (35) we may put A= + 1 and tluis obtain tlie moments of the 

 strata, which satisfy i-espectively the boundary problems. 

 W(<-)=-[f(0-C,] 



W{t+)=i{t)-Or 



The singular value A — I also corresponds to the second problem 

 for the inner region. The second member of (19b) for this pole 

 takes the form 



V{t)=i{t)-/'P{te)i{e)d6 



^i{f)-MnfHO)dO = i(t) 

 provided the usual condition for the inner region, viz., 



/i{e)dO = 

 be satisfied. The function /u(^) represented by (22b) now becomes 



(36) fL{t) = i{t) + Xi,'{t)-i-X\'{t)+ 



1 Cf. Plemelj. Potentialtheorctische lliitersuch\iiigeii, S. 60. 



