538 Mr. Sylvester's Analytical Development 



The double sign will give rise to two positions of the refracted 

 front 1 G H. 



The propositions which follow are perhaps more curious 

 than immediately useful. 



Proposition 10. 



To determine the portion of a line of vibration in terms of the 

 two velocities of its corresponding front. 



We have here to determine the quantities -^ — -' (of Prop. 1) 



in terms of v n v ll9 or on putting xf + yf + zf — 1, x l y l z t 



are to be found in terms of v n v u 



I m n 



By Prop. (3.) *,: y,:z :: j^—^ : -^—^ : -pr^ 



and by Prop. (5.) I 2 : m* : w 2 : : b 2 - c 2 . a 2 



c ^__ a 2 # 6 9 _ ^9 d ^_^ s 



^2 «2 



7/ 9 ^ 2 



a? — v 



2 7,2 7 , 2 ^» 2 r 



Let a, /3, y be the < es made by the given line of vibration 

 with the elastic axes, then 



= (b 2 - <?) (a 2 - v 2 ) (b 2 - v 2 ) (c 2 - v 2 ) 

 divided by 



f (b 2 - c 2 ) (a 2 - v 2 ) {b 2 - V,*) (c 2 - v 2 ) 



+ (c 2 - a 2 ) {b 2 - v 2 ) (c 2 - v 2 ) (a 2 - t;, 2 ) 

 + (a 2 - 6 2 ) (c 2 - v 2 ) (a 2 - *>,*) {b 2 - v 2 ) 



and therefore 

 __ ( /32 _ c *) (g » - g| « ) (ff - P| t) (c « - ^ ) 

 ~~ K* ~ V) («" - ^) (b 2 -c*)(c*-a*) 

 (where it is to be observed that the reduction of the denomi- 

 nator is simply the effect of a vast heap of terms disap pearing 

 under the influence of contact with the magic circuit a* — b 2 , 



