OF OBSERVATIONS OE COMPLEX PERIODICAL PHENOMENA. 365 
+ {the sum of all the values of Q s for which 5 is 2, or such that (s~ 2)f is a multiple of r } ~ 
b 2 = 
— { » 
» Q« 
s is 
» 0+2/) 
uiuiupc VJl # | 
» } 
+ { 
» q. 
s „ 
» Ov-2/) 
} 
&c. 
» q* 
s „ 
» 0^+2/) 
» }_ 
&c. 
&c. 
&c. 
A t = 
r + i - 
» p. 
s is t, or 
» 0+0/ 
11 
1 
L+( 
» Ps 
s is 
1 + 
» }J 
r + { 
-{ 
» Qs 
s is t, or 
» 0-0/ 
B t = 
» Q. 
s is 
» 0+0/ 
} 
+{ 
» q. 
s „ 
» (Sff-tf) 
} 
» q s 
s „ 
» (sff + tf) 
it is easy by means of (a), (b), (c), (d), (e), and other similar formal®, to convert the 
equations (6) into 
whence 
whence 
whence 
whence 
0 =i"' ‘[-a»+ft. + y J = 2 ""[-«„] + re „ 
m-0 m=0 
0 HL [ cos mz ( - «..+ P.,+7,,)] = / [cosm(— aj]+^a„ 
[a m cos mz\ \ 
°=|, [ sinm <-“=»+ft.+y,)]=_5""[sinm2(-«J]+^„ 
2 m=r-l 
bi—~% [a m sin m 2 ] ; 
' m= 0 
°=L 1[c ° S 2m (-“»+/ 3 -+5'.)]=5""'[cos 2mr(-a„)]+^ 2 , 
2 m=r— l 
« 2 =- S [a m cos 2 m 2 :] ; 
° = i„ [ s “ 2 ’®(- a -+&+y.)]=2'" , [sin2mr(- a .)]+55, 
whence 
h=~% [a m sin 2 m 2 ] ; 
& c. &c. 
&c. ; 
°-'i. [ cosl!m (-“.+^+y-)]=r"[costo2(- a „)]+^„ 
2 m=r— 1 
2 [a m cos tmz] ; 
whence 
