278 
If, therefore, the product y(t) gat) is developed in the way described 
above and after dividing by ¢ the integration indicated, is carried 
out, we find 
(a,8)—45(M)= — HUD) + (MB) 4 + (M—B) 5135 M—2)5(2) 
HHD) HM — 5) JE — 4)8(4) 
+ {( uv Mes +(M— 7)p—1} S(M—6)5(6) 
ny (2) 1 % Oe 1 HEEM — 2). 
The calculation undergoes a little modification for a= 1. In the 
development of the product y,(¢) gat) the term y,(t) ga—\(O) occurs 
on the second side, and before the integration this term must be 
transposed to the other side. 
We have then to determine 
M—1 
x B 
ie nig alt) — “9 (0) j eth ee fz Se sin 2a nt mX cos 2a mt—1 
e t (22) t y=! nt ml me 
0 
Notwithstanding the discontinuity of the function g,(¢) for integer 
values of ¢ the integral remains continuous and the integration by 
terms of the product of the series is still allowed. In this way we 
arrive at the result 
Ml 
4(—1) ? Lee aes ON | 1 i | 1 ai 1 Me 
3 (Qa) a wy mmr 1 as ee ee m=! aap, |= 
ive 
4(—1) ? x | 
ae Fg (p,1) — AM 
The integration of the remaining terms takes place as before, and 
in the special case: «=1, the general formula is to be replaced | 
by the following one 
M a en 
pW 1) — HEM) EU) —5(M — 252) 
— §(M — 4) S(4) 
— S{M — 6) 5 (6) 
—5G) E(M — 3). 
Here, a peculiar notation of Eurer’s may be used. For g(a, 8), 
writes g%, while he expresses at the 
same time the product S(@) &(@) arbitrarily by p* or pf. For &M), 
p™ is always written in this .notation. 
In this notation the formulae proved assume the following form: 
m which 
