402 
Proceedings of the Royal Society 
A 
And again writing na — A, a = — , this takes the form 
n 
• a . A / n 2 - 1 1 / , A\ 2 n 2 - 1 n 2 - 4 1 / , A \ 2 ) 
sm A = n sin — J 1 - i ( cho - ) + 7-7- -5-7- *( cho ) - & c .f 
n\ 1.2 3 \ n ) 1.2 3.4 5\ n ) > 
Now when n becomes indefinitely great, n sin - becomes A, so also 
n 
cho - , wherefore sin A = ^ = — 3 + — — 
n 1 1 . 23 ^ 1 . 2 . 3 ./ 
&c. 
1 1.2.3 1.2.3. 4.5 
7J* 
In order to obtain the series for the cosine we must put pa = ^ 
and therefore sin (p + l)a = cos a , which gives 
cos na = - 1 
( n - 1 n - 
(T“l 
2 n 1 n -i p n 
— — - cho a- + &c. 
A o 
+ cos a 
f n w-llw+1, 9 ,£> ) 
It — r 1 ~r ch0<l + &a [ 
and if, in this, we substitute for cos a , its value 1 - \ cho a 2 , 
-1 71 n- 2 , 5 l n-l?i» + lfl-4 , 4 „ 
cos = 1 - _ — - — cho a 2 + • — - — - — — cho a 4 &c. 
i A 1 A O 4 
whence, proceeding as before, 
cos A = 1 - + 
A 4 
A 6 
i-2 1 .... 4 r 
6 
+ &c. 
4. Note on the Bifilar Magnetometer. By J. A. Broun, 
F.B.S. Communicated by Professor Tait. 
5. Addition to the paper “ On the establishment of the 
Elementary Principles of Quaternions,” by G. Plarr, — 
published in Vol. XXVII. of the Transactions of the 
Society. Communicated by Professor Tait. 
6. Note on Mr Muir’s Solution of a “ Problem of Arrange- 
ment.” By Professor Cayley. 
This note has been printed along with Mr Muir’s paper, ante , 
p. 382. 
