OF OBSERVATIONS OF COMPLEX PERIODICAL PHENOMENA. 
397 
and as 0 cos 0/3=0, 
?»=> i — 1 yi 
2 tocosto 3 = — x , ( 11 a) 
m = 0 " 
except when 3 is 0 or a multiple of 27r, in which case 
X mcosra3={0+l+2+3 + +(w— 1)}=— — 
m = 0 w 
Again, multiplying equation (6) by x , 
sin (a+3) + 2.r 2 sin (a+23) + 3^ 3 sin (a+33) + 
+ (w— 1) #" -1 sin{a+(w + l)3} 
=^ X - 1 - Y fx- 2 }, 
(IB) 
( 12 ) 
d_( dS 
dx 
x^\ = sin (a + 3) + 4# sin (a + 23) + 9# 2 sin (a + 33) + 
+ (w— l) 2 af" 2 sin{a+(?i— 1)3} 
■- Y f x i+*{ 
=^i x - 
dY rfX 2 dY dX 
d* 2 X “d* ' d* X — dx ' dx A - 
■ Y fx-H 2 Yfx- 
(13) 
= X -'{S + ^}- X 1( Y + ^g)f + ,Y«} + 2X-.y,§ a , . . (14) 
which, when x=\ and a = 0, 
= 2 -1 (1 — cos 3)- 1 [sin 3— w 2 sin w3 + (^+l) 2 sin (n— 1)3] 
— 2 _2 (1— cos3) _2 [{3 sin3— (2w+l)sin w3 + (2w + 3)sin (n— 1) (3\2 (1 — cos3) 
+ 2{sin3 — sinw3 + sin n— 1)3}] 
+ 2 _3 (1 — cos3) -3 [8(l — cos3) 2 {sin3 — sinw3 + sin(w -1)3}] } • • • (15) 
which, when n(3=2c7r, 
= — 2 _1 (1 — cos 3)" 1 [{w 2 +2w}sin 3] 
— 2 -2 (1 — cos 3)" 2 [2(1 — cos 3){ — 2w sin 3 }] 
— 2 _i n — 
(1 — cos 3)- 1 % 2 sin 3 = 
• 3 3 
2 sm 2 COS 2 , n 2 B 
^ = — 2 " COt 2 
4 sin 2 ? ^ 
=sin 3+4 sin 23+9 sin 33+ + (n— l) 2 sin (n— 1) 3 ; • . 
and as 0 2 sin 03=0, 
X m 2 sin m(5= — o cot 5, when 3 is not 0 or a multiple of 27 t. 
m = n " & . ± 
(16) 
(17) 
(17a) 
