OP OBSERVATIONS OP COMPLEX PERIODICAL PHENOMENA. 
399 
X m 2 smm$= — \ cotf, or=0, 
m=0 Z " 
ot=»- ] re 2 re 1 re 3 re 2 re 
2 m 2 cos mb= — w + o a. or= ¥ — 2 _ +6- 
m=0 sin 2 £ 
Demonstrations.. Second set. 
To find when (sijitf)5, ( sa + tb ), (sijitf)tf, and (sb+ta) are multiples of r if «=13 # 
b=8, r=96, and s and t are positive and integral. 
(1) 8(s+ £) = 96c, c being a positive integer, when 
s+£=12<?, I 
s=12c±:t. 
(2) (13s+8tf)=96c, when 
13s=8(12c±f) 
=8{13c— (c+£)}, 
s= 8c— (<?+ 1\ which can only be integral when (c+ 1) is a multiple of 1 3. 
(3) 13(s+£)=96c 
= 8(13 — l)c, 
s^t=8{l-^)c, 
s= 8 (c— ~^+t, which can only be integral when c is a multiple of 13. 
(4) (8s+13£)=96c, when 
8s=96c±13£, 
s=12c+ J ^£, which can only be integral when t is a multiple of 8. 
