1880.] 



some theorems in trigonom,etry. 



387 



1 /e"' e-^'\ 

 § 19. Since sin {a -/) =— . (-j.^ — ^.j , &c., it is evident that 



lent to the algebraical theorem 



(--i-\ (^-9\ (^_h] (^--) 

 \f a) \g a) \h a) \c h) 



+ 



+ 



+ --T 



1 fe"- e' 

 2i W' ~ ? 

 (p) is equivalent to the algebraical theorem 

 ^a f\ fa g\ fa h\ fb c> 

 </ ^/ \9 <^/ V^ ^J 



'\.J\ i\-t\ (^-^1\ (^- ^' 

 v/ bJ [g bJ [h bj \a cy 



'c /\ (c _g\ ,'c h\ fa 6' 



^7 c/ V^ c) \h c) \b a/ 



■b c\ fc a\ fa b\ fabc fgh\_^ 



,c bJ \a cj \b a) \fgh abcj ' 



which may be readily verified. This process of substitution of 

 algebraical expressions for the sines (which is that employed by 

 Mr Scott in evaluating the determinant) affords perhaps the 

 simplest demonstration of (jj). 



§ 20. By putting a, 6, ... in place of a^ ¥, ... in the equation 

 contained in the last section, and by expanding the sines in (p) 

 and equating the terms of the fourth order, we have the following 

 pair of algebraical identities, 



be (b — c) (a —f) {a — g) (a — h) 

 + ca(c-a){b-f)(b-g){b-h) 

 + ab{a- b) (c -/) (c - g){c- h) 

 + (6 - c) (c - a) {a - b) (abc -fgh) = 0, 



{b — c) (a —f) {a —g) {a — h) 

 + {c-a){b-f){b-g){b-h) 

 + {a-b) (c-f){c-g){c-h) 

 + (6 - c) (c - a) (a-b) {a + b + c -/- g-h) = 0. 



It is interesting to notice the manner in which the trigono- 

 metrical theorem establishes, as it were, a connection between 

 these identities. They can of course be easily verified. 



Addition to Mr Hicks' paper on the problem of two pulsating 

 spheres in a fluid (pp. 276 — 285) : 



On page 279 of the last part of the Proceedings it is stated 

 " we know that V^ has no effect on the resultant force." This is 

 of course only true of the resultant force on the two spheres treated 

 as one body, and not of the resultant force on one only. I hope 

 soon to find time to calculate the extra terms, and to lay them 

 before the Society, but I wish to draw attention at once to the 

 error there made. 



