OF OBSERVATIONS OF COMPLEX PERIODICAL PHENOMENA. 
363 
each of the quantities p x , q x , &c., P 0 , P„ Q 15 &c. vanish*; or, dividing out the 
factor 2, when 
0=S [— a»+0»+yJ> 
w=0 
m=r— 1 
0=2 [cos mz(— a m +0 OT +y m )], 
0=2 [sin mz(-a m +0 m +y m )], 
m = 0 
0=2 [cos 2mz( — a,„ +3™ +7™)], 
771 — 0 
0=2 [sin 2mz( — a«+3„+y*)], 
w=0 
&c. See., 
0=2 [cos tmz{ — a m + fi m -f- y m )] , 
0=2 [sin a ro +0 m +yj], > 
fltsO 
°=|o ^ C0S f ““»+&» + ?«)]» 
°=£o [sin^m2(-a m +0 m + yj], 
0=2_ o [cos 2~rnz(—a m -i-^ m -\-iy m )], 
0=2 ^ [sin 2 £ mz( - « m +0 m + y m )], 
&c. &c., 
0=2 [cos — a m +/3 TO +y m )], 
m=0 ^ 
0=2 [sintf{wz(-a m +0 m +y m )]. 
w=o y 
(C) 
Representing by s the suffix of a p, q, P, or Q in a type term of (0 m +y m ) in each of 
the equations (6) in turn, and by t the integral numerical factor of the angle in a type 
of the sine or cosine which multiplies ( — a m +0 m -j-y m ), let us note that 
cos s ^ mz cos tmz = £ 
m\rz\st+tl sinj rz{si-t\ , , , 
L4_|cosi(r+l>|«i+il + LL_|cosJ(»-+1)zM-4 
sini5r|s^ + ^j L J sinls’js^— n L J 
l rns If 
* The second differential coefficients being all squares, and therefore positive, there is no ambiguity as to 
whether equations (6) correspond to a maximum or minimum value of (5). 
3 c 2 
