114 Prof. Maiden on Cosines and Sines of Multiple Arcs. 

 cos 3 ^ 1 



2 cos 6 — 



2 cosO — 



2 cos 6 — 



cos 6 



(4 cos ef — 3 cos e 



8 (cos 6)'* — 8 (cos 6)- + 1 ' 

 and similarly, 



cos n e _ cos e. {{2 cos ^)»- ^ —n (2 cos ( 9)"-^+ &c . } 

 cos(w+ 1)^ ~ cor^~{(2cos^)" — (?«+l).(2cos6')"-2 + &c.}* 



And as the fraction on the right-hand side of the equation is 

 formed from the fraction on the left-hand side by taking it to 

 pieces, and then reconstructing it in the reverse order, with- 

 out the omission or cancelling of any part, not only are the 

 fractions equal, but the numerators are equal, and the denomi- 

 nators are equal ; and 



cos ne = cos6 A (2 cos 0) "-^ - n . [2 cos 9) ""^ 

 n . « — 3 . - /i\ „ . ?? . ?« — 4 . ?z — 5 .^ „,„ - 



w.w — 5.« — 6.«— 7 ,^ 



+ — t:273:^ ("^^^ 



ey-^ - &cA. 



which is the usual series in a form slightly varied. 



It will not be difficult to make the induction strict, by show- 

 ing, that, if the form of series be true for cos (?j — \) 6 and 



a .u . cos in— \)6 , . . . „ 



cos n a, so that j^ — may be expressed as the ratio ot 



cos no ■' ' 



two series of the proposeil form, since 



cos nd _ 1 



cos (w -f 1 ) ^ " ^ -, cus [n — 1)6 



^ ' 2 cos d — — ~— 



con n 



cos(w-f 1) 6 will come out to be of the proposed form likewise. 

 The general form of the coefficient, after the second term, 

 or the coefficient of the ?wth term, is 



n . {ii — m) .{n — m — \) {n — 2m + 3) 



1 .2.3 (?« - 1 ) ' 



as Mr. Booth has shown in another shape. 



Sin 6 may be expanded in a similar manner, and I am not 

 aware that this series has ever been given. 



