, 70 UK. J. MERCER: STUBM-LIOUV1LLE SEMES OF NORMAL 



5 7 The formula (18) is true for all values of which are greater than, or e,,ual to, 

 . certain positive integer, y N. Let G (, t, .) denote the sum 



i (+..,() +..,()- 1 co w CM mi); 



= !f \ 



then we see from (18) that 



siu u cos mt 2 COBWW sin_rol 



i , m 



2 f cos m cos art [A. (m. s) + A.(m, Q] + siu m* cos mtA. (m. *) 



1T..NI m 



cos mg sin mtA a (m, t) | (m, s, t)} 

 m m a \ 



Now the sum 



^ sm n 



, = i m 



is known* to be limited for all values of 2 and all positive integral values of n ; 

 further, as n tends to oo, this sum converges uniformly for values of z lying m the 

 closed intervals complementary to the set (2mir .7, 2mir + r)) (m = 0, 1, 2, ...), 

 where i) is any assigned positive number. It follows at once that the sum 



" sin ms cos mt 



i , 



being equal to 



, " sin m(s + t) +1 sm m(s-t) 



ny* * _ ffl 



is limited for SSSTT, ^t ^TT, i N ; and that, as n tends to oo, it converges 

 uniformly in those parts of the square O^S^TT, O^t^ir, which correspond 

 to 1 1 s 1 2: TJ. It is easily seen that 



i cos ms sin m 

 m = s m 



has the same property. 



Again, 



i cos nts cos mt A t (m, g) /19\ 



= N m 



may be written 



I ds.j <i (s. t, i. n) q, ds it (20) 



2irJo Jf, 



where 



x 4 cos ms cos m t sm 

 / ^s, t, j, n) = 2, 



m = N Wl 



* For a full ducussion, see HOBKON, ' The Theory of Functions of a Real Variable,' pp. 648-643. 



