172 DR. J- MERCER: STURM-LIOUVILLE SERIES OF NORMAL 



Lastly, it is at once evident that, as n tends to oo, 



a (w, *, Q 



= N 



converges uniformly for == * S ir, == < == ir. 



As a result of the investigations of this paragraph, we now see that G(s, t, n) is 

 limited fcr 0*.*, OSISr, SN; and that, as n tends to oo, ,t converges 

 uniformly in those parts of the square 0*s**,0*t**, which correspond 

 to \t-S\Zr). 



Consider the function 



For values of n S N, it differs from G (s, t, n) by 



1 N - 1 / 2 



/, (s)J, (0 + 2 l^+i( a )^+i(0 cos ms cos? 



7T in = 1 \ "" 



We conclude that H (s, t, n) is limited for < s == IT, S t : ir, n S N ; and that, 

 as n tends to oo, it converges uniformly in those parts of the square S s ^ IT, 

 S * S IT which correspond to | ts \ > tj. We proceed now to show that, when tys, 



lim H (*, t, n) = 0. 



9. The normal functions which satisfy the differential equation 



and the boundary conditions 



d u _ n A n du _ , (11) 



ds ~ ' ds = 



are clearly 



A/-, A/-COSS, /\/-cos2.s, ..., A/-COSWS 



7T 7T 7T 7T 



the corresponding values of p. being 



-X, l'-X, 2 2 -X, ..., ?i a -X, ... 



Thus (III., 3) the GREEN'S function of the equation 



d?u 



(22) 



which corresponds to the boundary conditions (11) is 



1 12 



r I- 2 -5 - . - cos ns cos nt. 

 ATT = i n X TT 



