466 Mr. Sylvesters Analytical Development 



' (a 2 - & 2 ) (« 2 - c 2 ) 

 Similarly, 



(cos * ) - (F - « 2 ) (6 2 - c 2 ) 

 (cos*j - (c2 _ fl9) (fi9 _ £ 2) . 



Proposition 6. 

 To find i? y v„ in terms of eo, <$>, v(/. 

 By the last proposition 

 (cos oo) 2 a 2 



a*-v? " (a 2 - 6 2 ) (a 9 - c 2 ) ~ V[ 



(cos4>) 2 __ # 



& _ V/ 3 - ( a 2 _ £2) (a 2 _ c «) % 



(cos vl/) 2 c* 



c 2 - v? - (a 2 - 6 2 ) (a* - c 8 ) *'" ; (c 2 - a 2 ) (c* - £ 2 ) 



(cos co) 2 (cos <$) 2 (cos vj/) 2 _ 

 •'• a 2 _ v 9 + j«— T^ + C 2 _ ^2 - °> 



Just in the same way 



(cos eo) 2 t (cos $) 9 (cos \[/) 2 

 6 2 - » 2 + c"^^ 



2 ' 2.2 „ 2 "I* -o „ 2 — "• 



so that v* 9 v^ are the two roots of the = n 



(cos a?) 2 (cos $) 2 (cos \|/) 2 _ 

 a 2 — u 2 + 6* - u 2 + c 2 — d 2 " — 



Cor. — Hence the = n to the wave surface may be obtained 

 by making 



(cos 00) x + (cos <p) y -f ( cos 4/) s = v, 



or if we please to apply Prop. (5), we may make 



/ (^-«ft(a«-V ) A 2 - Q (fl» - Q 



V (^-/3 2 )(a 2 -c 2 ) ' + V (6 2 - a 2 ) (6 2 - c 2 ) * y 



^ x /E^s3EEM,M^v 9 



