394 MESSRS. C. AND F. CHAMBERS ON THE MATHEMATICAL EXPRESSION 
It is also because the decennial period would greatly affect the apparent variation of 
magnetic disturbance following the sidereal period of Jupiter, that no attempt has been 
made to apply these observations, extending over less than three such periods, to the 
determination of the character of that variation. 
Appendix. 
Demonstrations . First set. 
To find the sum of each of the following series : — 
(1) sin 00 + sin 0 + sin 20 -j- + sin(w — 1)0. 
(2) cos 00 +cos0 + cos20 + + cos(w— 1)0. 
(3) 0 sin 00 -(-sin 0+2 sin 20 + + (w— 1) sin (n— 1) 0. 
(4) 0 cosO0 + cos0 + 2 cos 20 + +(w— 1) cos (n— 1) 0. 
(5) 0 sin 00 + sin 0+2 2 sin 20+ + (w— l) 2 sin (n— 1) 0. 
(6) 0 cos O0+cos0 + 2 2 cos 20+ +(%— l) 2 cos (n— 1)0. 
If X =1 — 2#cos0+# a 
g = -2cos<3+2<r, 
^-+2 
dx 2 ‘ ’ ... 
^-=4 (x — cos 0) 2 . 
(a) 
(b) 
(c) 
(d) 
If Y =a’sin(a+0)— #”sin(a+%0)++ l+1 sin{a + (w— 1)0}— # 2 sin«, . . 
dY 
faT = sin (a + 0) — nx n ~ 1 sin (a + w0) + (n + 1) x n sin { a + (n — 1 )0 } — 2x sin k, 
( PY 
d^=~ n (n— 1) x n ~ 2 sin («+w0) + (++ 1) nx n ~ l sin{a+(w— 1)0} — 2 sin a. 
And when x=.l and a= 0, these become respectively 
X =2 (1 — cos0), 
§= 2 ( 1 - 0080 ), 
«_ +2 
dx*~ 
^2=4(1— COS 0) 2 , .... 
Y = sin (3 — sin n(6 + sin 02—1) 0, 
(e) 
(f) 
(g) 
(h) 
(i) 
0) 
(k) 
0) 
