by a Table of Single Entry 335 



indicated in § 1, viz. by equating the terms of the nth order in 

 the equation giving sin a x sin a 2 . . . sin a n as a sum of sines or 

 cosines. 



Denoting by 2 r sina the sum of the sines of the angles 

 ±«i ±«2- • • ± a n, in which r signs are negative and n — r 

 positive, Adz. denoting by 2 y sin a the expression which would 

 be written in Sylvester's notation, 



Z sin (-a 6i -ae 2 . . ■— %+«e r+1 + « 0H _ 2 . . .+«©„), 



and attaching a similar meaning to X r cos a, we have 



( _) m 2 n_1 sin % sin <2 2 . . . sin a n = sin (a 1 + a 2 . . . + a n ) 



— 2i sin a + 2 2 sin a . . . + ( — )™2 TO sin a 

 if w = 2m + 1, and 



( — ) m 2 n ~ 1 sin a x sin a 2 . . . sin a n = cos (a x + a r . . . + a n ) 



— Si cos a + 22 cos a . . . + J(— ) m 2 m cosa 



if 7i= 2m. The factor J has here the same explanation as 

 before ; viz. each term under the sign 2 TO occurs twice over, 

 and the \ merely causes it to be counted once instead of 

 twice. The truth of these formulas is readily seen by starting 

 with sin a sin b = J {cos (a— &) — cos (a + b)} } multiplying by 

 sin c and obtaining the expression as a sum of sines, then mul- 

 tiplying by sin d and so on ; the general law then soon becomes 

 apparent. There is a simple and direct investigation of the 

 formulas by Mr. R.Verdon in the * Messenger of Mathematics,' 

 vol.vii. pp. 122-124(1877). 



The formulas of § 3 in the form (2) follow at once by equa- 

 ting terms of the nth order in the expansion of the terms in 

 these trigonometrical formulas, or, what is the same thing, by 

 writing a Y x, . . . a n x for %, . . . a n and equating coefficients of x n . 



We also see that if in the formulas in § 3, the exponent, in- 

 stead of being equal to n, the number of quantities a ly . . . an, 

 be less than n, and differ from it by an even number, the ex- 

 pressions on the right-hand side of the equations are equal to 

 zero, and that in general, if we denote 



2(-a 01 . . .-a 0r + a 0r+J . . . + a 6n )P 

 by 2) r a p , then 



(a 1 + a 2 ... + a n )P-2 1 a^ + S 2 aP-&c., . . (3) 



the last term being (-) m 2 m o?if ra = 2m + l, and i(-) m 2, m a p 

 if n = 2m, is equal to ±2 n-1 p! a 1 a 2 ...a n x the terms of the 

 (p— n)th order in 



