178 C. E. Weatherburn : Parameter Valines. 



1^10. — Further relations. — By means of " the preceding results we 

 may obtain relations corresponding to those of §6 for the ordinary 

 potential. From the first equation (7) we find on multiplying by 

 dt and integrating over the boundary, 



/F{tp)dt=Xj[l-Ic'X{6)]F(dp)dO 

 that is, 



(46) (1 -Ao yFitp)dt= -k-'\JX{d)V{dv)dd 



which reduces to (•■32) when /.-^ is put equal to zero. Similarly from 

 the first of (8) we find on integration with respect to t 

 )^/[\-k'X{d)]H(ej7)d$=/B(tp)dt-c-^lc'X{p) 

 -l/F{tp)dt 

 -or 



(47) {i-xyn{tp)d(=c-k:'X(p)-xkyx(0)iJ(ep)d6 



^J^---/x{e)Fi0p)dd 



1— Afl 



where c has the value 2, 1, or 0, according as p is Avithin the inner 

 region, on the boundary, or in the outer region. This relation 

 reduces to (33) when k is zero and p on the boundary. 



Tliese might have been derived from (5), the first of wliicli be- 

 comes on integration 



(48) {l-X)/H(f.p)df=c-PX{p)-X/rrX(0)Ji{ep)de 



Substituting from (6), multiplying by (A.,, — A) and proceeding to 

 the limit A=Ao "^^'e arrive at (46). Then substituting from this 

 in (48) we find (47). 



The preceding investigation deals with the singular parameter 

 values of the first two boundary problems only. In another paper ^ 

 the author considers the third boundary problem for the equation 

 (2), requiring the determination of a solution satisfying the rela- 

 tion 



^(^+)=A/?(ov(^^)-^(OU(o 



The singular parameter values for this problem are there discussed. 



1 Weatherburn. "The mixed boiuuiai-y problem for the geiieraUsed potential correspond in;.;- 

 to the equation y -i m- A:-2it = 0." Qniirterly .Journal, vol. 46, pp. 83-04. 



