64- PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



and we may determine X so that qtf 4- 2X V + \ 2 pp' = shall be a pair of lines. 

 Writing V = Hfy —Lm — L'£z 4- Mz 2 , and substituting for pp' and qq their 

 values (£ — az) (y — a'z) and (£ — &) (>? — /3'z), the equation in question is 



fl + 2XZ7 + X 2 ) In - '/3 + 2XZ + X 2 a) ^-(/3' + 2x7/ + XV) £:4- OS/S' + 2X.1/+ X 2 aa ;: 2 = 0. 



and the required condition is 



(1 + 2x7/ + X 2 ) (BB + 2xJ/ + x'-W, = (B + 2x7 + X 2 a; QS' + 2x7' + xV) ; 



or reducing, this is 



(2,1/ + 2H&B 1 - 27'/3 - 27/3) 

 4- X (fa - j8; >' - /3') + 477J7 - 477) 



+ X 2 2M + -Ufa* - 27a - 27a') = 0, 



viz., X is determined by a quadric equation. Calling its roots \, and A„ the 

 foregoing equation, substituting therein successively these values, becomes 

 (£ ~~ 7 Z ) ('J — 7 Z ) = °> and (£ — &)(i/ — S'z) = respectively, say ?•?•' = and s/= 0. 

 153. We have to show that the four foci (p = 0, p' = 0), {q = 0, q f = 0), 

 (r = 0, ?•' = 0), (s = 0, s' = 0) are a set of concyclic foci ; that is, that the lines 

 p = 0, q = 0, r = 0, s — correspond homographically to the lines 7/ = 0, 

 q = 0, r' = 0, s' = ; or, what is the same thing, that we have 



1 , a, a', aa' = , 



1, y, y . yy 

 1, 3, 3, 33 J 



or, as it will be convenient to write this equation, 



a — By — 3 _ a — 3/3 — y 



a' — B y — 3 a — 3' ft — y 



154. We have 



B + 2X,7 4- X.'q ,_ ff + 2x,Z' + x>' 

 7 1 + 2i/X 2 + V ' 1 + 2 ^ x i + V ' 



3 _ | 8 + 2X 2 7 + X 2 2 a _ /3- 4- 2X 2 7' + X 2 2 a' 



1 + 277X 2 + X 2 - ' 1 + 277x 2 + X 2 2 • 



The expressions of a — S, &c, are severally fractions, the denominators of which 

 disappear from the equation ; the numerators are 



for a - 3, = «(1 + 2X 2 77 + X 2 2 ) — (8 + 2X.7 + aX, 2 ), 



= a - B + 2X 2 (a77 - L) ; 



for $ - 7 , = /3 (1 + 2X.77 + X x 2 ) - (0 + 2x,7 + aX x 2 ), 



= X 1 {2(/377-7)(a-/3;}; 



for y - S, = (|S + 27X, + aX, 2 ) (1 + 277x 2 + x/) 

 -{B + 27x 2 + ax 2 2 ) (1 + 2H\ 1 + x x 2 ) , 



= (a' - /3) {277 2 a/3 - 2777 (a + j8) + 27 2 + J (« - /3) 2 } 



