Or OBSERVATIONS OE COMPLEX PERIODICAL PHENOMENA. 
395 
^ = sin0 — n sin w0 + (?z+ 1) sin (n— 1) 0, . 
d*Y 
-j~Y = — n (n — 1 ) sin n(B + (n + 1 ) n sin ( n — 1) 0. 
(m) 
(n) 
When x=l and ot=^ (a) to (d) have the same values as in (h) to (k) respective! y, 
and (e) to (g) become as follows : — 
Y =cos0 — cosw0 + cos(w— 1)0— 1, (o) 
cos 0 — n cos %0+(?i+l) cos (n — 1) 0 — 2, (p) 
cPY 
-^=—n{n—V) cosw0+(w+l)wcos(w— 1)0 — 2 (q) 
Let S=,£sin(a+0)+,!r 2 sin(a+20)+# 3 sin(a+30) + . . . . x n ~ l sin{a+(w — 1)0} . (1) 
sin («+w0+0)+sin (a+w0— 0)= 2 sin (a+%0) cos 0, 
2# ,l+1 sin (a-| -%0) cos0=^ m+I sin{(a+%0)+0} +#" +1 sin{ (a + w0)— 0}. 
Hence, by giving n the values 1, 2, 3 .... . (n— 1), 
2a’ 2 sin { a + 0 } cos 0 = # 2 sin { a + 20 } + x 1 sin a, 
2$ 3 sin{a+20}cos 0— x 3 sin{a + 30} -\-x z sin{a+0}, 
2x i sin{a-f- 30}cos 0=# 4 sin{a+40} -\-x 4 sin{a+20}, 
&c. = &c. &c., 
2x n sin{ a + (n— 1) 0 }cos 0 =x n sin{ a n(3 [ +# re sin{ a +(%— 2) 0 j- . 
Now adding 
2#S cos0 = S — x sin (a +0) + #™sin{ a + w0 } +# 2 S — # w+1 sin { a + (w — 1 ) 0 } -j- # 2 sin«, (2) 
S ( 1 — 2# cos 0 + .ir 2 ) = , 2 ? sin (a + 0 ) — sin (a + w0) + # n+ 1 sin { a + (« — 1)0}— sin a, (3) 
which, when x=l and a=0, becomes 
2S(1— cos0)=sin0— sinM0+sin(^— 1)0, (3 a) 
which, when w0 = 2<?7t, 
=0, whether or not 0 is 0 or a multiple of 2tt ~) 
)■ (ob) 
= sin 0-f sin 20 + sin 30+ + sin(w— 1)0. J 
If in (3) x be made=l and «=-, 
2S(1 — cos0)=cos 0— cos%0+cos(w— 1) 0 — 1, (3c) 
which, when W0 = 2c7r, 
== — 2 (1 — cos 0) (3f7) 
S= — l=cos 04-cos 20+cos 30+ +cos ( n — 1)0, 
3 g 2 
