Prof. Maiden on Cosmes and Sines of Multiple Arcs. 1 1 5 



sin (n + 1) ^ = 2 sin n ^ . cos ^ — sin (w — 1) ^ : 

 , sin 71 _ sin 7^ 



sin [n + 1) 0~ 2 sin w ^ . cos — sin (?j — 1) 6* 

 1 1 



„ sm (« — 1) ^ ^ n 1 



2C0S^ ^; a^— 2C0£^ r- 



sin n e' n a ^ 



2cos^— &c. 



as before ; but in this series, while 2 cos recurs ?i times as 



the integral part of th^ partial denominators, the last frac- 



. sin 

 tional part is —. — rr-, or 0. 

 ^ sni 



sin 3 ^ 1 _ 4 cos- 0—1 



^^' ^^'' smTd ~ ^ 1 '~8cos36'-4cos^' 



2 cos — ^ — 



2 cos ^ — B 



2 cost/ 



, . ., , sin n 



and similarly, -^—, — —— ^ 

 •' sln(7^ + 1) ^ 



(2cos^)"-^-(w-2).(2cos^)"-H ^'^~^^^'^^~^\ 2cos^)"-^-&c. 



(2 cos 0)" - {n- 1).(2 cos 0)"-' + (^-^)(^'-g) (2cos^)»-^ -&c. 



And it may be shown, as before, that if ^ ^ — ■ be of the 



^ ' sinn0 



Sin 71 u 

 proposed form, -r— r-n will also be of the proposed form. 



^ ^ sin (w + 1 ) C7 



But it is not true that, as in the case of the cosines, sin n 



is equal to the series in the numerator, and sin {n + I) 



equal to the series in the denominator. This arises from the 



neglect of the last member of the continued fraction, viz. 



sin ^ 



. ^ , or -. -^. 

 sin sin 6/ 



To reconstruct series which shall be equal to sinw^, and 

 sin(M +1)^, this fraction must be taken into account. Of 

 course, the numerator will make no difference, but the deno- 

 minator will be a constant multiplier ; and 



sin«^ = sin^. |(2cos^)"-' - [n - 2) . (2 cos ^/ 



-3). 

 1 . 



^ (»-3).(»-4) ^^^^^ ^^„„_ (n-4) (.-5).(.-6) ^^ ^^^ ^^„., 



12 



