by a Table of Single Entry. 337 



the last term being 2 m cosa if n = 2m + l, and £2™ cos a if 

 n = 2m ; and equating the terms of the 2qth order, we find that 



Ox + a 2 ...+ a n y* + 2i« a » + 2 2 « 2 ? + &c, 



the last term being S m a 2? if n = 2m + l, and ±2,^ ifra = 2wi,is 

 equal to 2 B_1 . 2^! x the terms of ihe 2^th order in 



Thus 



(a + & + c) 4 + (5 + c-a) 4 + (c + a~&) 4 + (a + 6-c) 4 



=4(a 4 + b* + c 4 + 66V + 6cV + 6a 2 6 2 ) ; 



and in the case of n quantities a v a 2 , . . . <2 W , 



(«! + a 2 . . . + a n Y + Xi« 4 + 2 2 <2 4 + &c. 



= 2«-\a\ . . . + a* n +6a,y 2 ...+ Ga^aQ, &c. 



It may be remarked here, that any trigonometrical identity 

 in which the arguments are homogeneous functions of the 

 letters gives rise to a series of algebraical identities by equa- 

 ting the terms of each order ; ex. gr. from 



sin (d — h) sin (a— c) + sin (b — c) sin (a— d) 



+ sin (c—d) sin (a — b) = 0, 

 we have, by equating terms of the fourth order, 

 (d-b)(a-c){(d-b) 2 + (a-c) 2 } 

 + (b-c)(a-d){(b-c) 2 + (a-d) 2 } 

 + (c-d)(a-b){(c-d) 2 + (a-b) 2 }=0. 



There are a great number of trigonometrical identities of 

 this kind, such as 



sin (b — c) + sin (c — a) + sin (a — b) 



+ 4 sin ^(b — c) sin J (e— a) sin -^ {a — b) = 0, 



cos (a + b) sin (a — b)+ cos (b + c) sin (b — c) 



+ cos (c + d) sin (e — a) + cos (d + a) sin (d — a) — 0, &c. ; 



and some of the algebraical identities thus obtained are of 

 interest. An identity of this class is referred to in the ' Mes- 

 senger of Mathematics,' vol. viii. p. 46 (July 1878). 



§ 6. In Laplace's section, " Sur la Reduction dos Fonc- 

 tions en Tables," he considers the question of multiplica- 

 tion by means of a Table of single entry. First, assuming 

 that ^?/ = </)(X + Y), where X is a function of x only, and Y a 



Phil. Mag. S. 5. Vol. 6. No. 38. Nov. 1878. Z 



