336 Mr. J. W. L. Glaisher on Multiplication 



where p ! denotes 1 . 2 . 3 . . . p, and p, n are supposed to be both 

 even or both uneven. It is easily seen, by considering separately 

 the two cases of p even and p uneven, that the terms in the 

 resulting expression are always positive; so that (3) is equal to 

 2 n-1 j9 ! a l a 2 . .. a n x the terms of the (p — n)th. order in 



( 1+ 3l + ?! +& °-)( 1+ 3l +&C -)-( 1+ g + &c -)*- 

 Thus, for example, let n = 3; tl^en 



(a + b + c) — (b + c—a) — (c + a—b) — (a + b—c) = 0, 

 (a + b + c) 3 -(b + c-ay-(c + a-b) z -(a + b-c) z =Uabc, 

 (a + b + cy-(b + c-a) 5 -(c + a—by-(a + b-cy 



= S0abc(a 2 + b 2 + c 2 ), 

 (a + b + c) 7 -(b + c-a) 7 -(c + a-b) 7 -(a + b-c) 7 

 o2 _. / a ' + b i + c i t b 2 c 2 + e*a 2 + a 2 b 2 \ 



= 2 ' 7!a H 5! + (317 / 



= 56afc(3a 4 + 3b 4 + 3c 4 + 106V + 10c 2 a 2 + 10a 2 b 2 ) 

 &c. &c. 



Let n =4, then 



(a + b + c + d) p 

 -(-a + b + c + d) p -(a-b + c + d) p -(a + b-c + d)P 



~(a + b + c-d) p 

 + (-a-b + c + d) p + (a-b-c + d) p + (a + b-c-d) p 

 = 0ifp = 2, 

 = 192abcd if ^=4, 

 = 960abcd(a 2 + b 2 + c 2 + d 2 ) if p = 6, 

 = 89Qabcd(3a* + 3Z> 4 + 3c 4 + 3d 4 + 10a 2 b 2 + lOaV + I0a 2 d? 

 + 10b 2 c 2 + I0b 2 d 2 + 10c 2 d 2 ) if p = 8, &c. 



§ 5. We can also obtain an expression for the sum of powers 

 when the exponent p is even and all the terms have the posi- 

 tive sign. For 



2 n ~ 1 cos a x cos a 2 . . . cos a n = cos (a 1 + a 2 > . .+ a n ) 

 + 2)i cos a + S 2 cos a + . . . , 



* Since smix= isinhx, cos ix = cosh x, the trigonometrical formulae 

 become, on writing ia v , . . ia n for a v . . . a n '• — 



2 n "~ 1 sinha 1 smha 2 . . .sinha n = sinh (a x +a 2 .. . + a n ) — 2, sinha-f&c. 

 if n=2m+l, and 



= cosh (a 1 + a 2 . . .+a n ) — 2 l cosh. a+&c. 

 if w = 2w, leading at once to the theorem in the above form. 



