Problems in the Theory of Elasticity. 33 



independence of the functions n.(s) and of the others m.(s), 

 to the separate conditions 



$f( S ).nJs)ds=0, 1=1,2,3,....,* for v(0 1 

 $t(s) .m£*)di=Q 9 1*1,2,3, *' for u(0J 



which are the usual necessary and sufficient conditions for 

 the existence of a solution to (27) at a singular parameter 

 value. 



If these conditions are not satisfied, we can construct the 

 function 



f 1 (t)r,f(t)-^f(s).'P{st)ds .... (45) 



which satisfies the first of the conditions (43), and similarly 

 the function 



t i (t)=f(t)-$F(ts).t(s)ds . . . (45') 



which satisfies the second. This is easily verified by means 

 of the values of "P{si) and "P' (ts) given by (41). Thus there 

 do exist solutions to the modified problems 



i[ w(f )- w(t- ) ] - 1 [ w(<+) + w(r )] =f x (0 



X S (46) 



i[TV(r)-TV(i+)] - ± [TV(r ) + TV(< + )] = -t s (t) 



which are regular at the characteristic value X in che first 

 case, and k ' in the second. The solutions of these problems 

 are also expressible in the form (31). The poles of the 

 dyadics H(£p) and T'(pt) are all simple ; hence in the 

 neighbourhoods of their respective poles, X and X ', they 

 may be written 



K(tp)=K(tp) + P(*p)/(Ao-X) 1 



r(^)=G'(pO+Va'(^)/(V-^J' ' ' { } 



where K{tp) and Or' (pi) remain finite at the poles X and \ ' 

 respectively. It can then be shown, exactly as I have done 

 in the case of the potential theory *, that the solutions of the 

 modified problems (46) are given by 



W(p) = §f(s).K( S p)d S] 



c L . . . . (4b) 



V(p) = fa'(ps).f(s)d s j 



which are regular at the poles considered. 



* Proc. Eoy. Soc. Victoria, vol. xxvii. pp. 169-170 (1915). 

 Phil Mag. S. 6. Vol. 32. No. 187. July 1916. r> 



