IO CALIFORNIA ACADEMY OF SCIENCES. [Proc. 3D Ser. 



with similar identities for the other coefficients. 



Solving these equations for the quantities m x Hi , m 2 n 2 , 

 etc., we have, denoting again by (p q r) the determinant of 

 the transformation 6*2, and by Pi , P 2 , etc., the co-factors 

 of pi , p 2 , etc., in that determinant: 



(p q r ) mi n x — P x a x + P 2 a 2 -f P 3 a 3 = P a , 

 (pqr) m 2 n 2 = Pi b x + P 2 b 2 + P 3 h = P h , 



(17) (pqr) m 3 n 3 = P x c x -f- P 2 c 2 -\- P 3 c 3 = P c , 

 (pqr) ( m 2 n 3 -\- m 3 n 2 ) =2 P t 



(pqr) ( m 3 n x -J- m x n 3 ) =2 P g 



(pqr) ( mi n 2 -j- m 2 n x ) — 2 P h etc., etc. 



From these identities we readily deduce the following: 



(p q r ) 2 n 2 n 3 ( l 3 m 2 — l 2 m 3 ) = P h Q c — P c Q h = 

 (pqr) (r b c) 



(18) (pqrfn? (l 3 m 2 - l 2 m 3 ) = 2 ( P b Q f -- P t Q h ) = 



2 (pqr) (r b f) 

 (pqr) 2 n 3 * ( l 3 m 2 - l 2 m 3 ) = 2 ( P t Q c - P c Q { ) = 

 2 (p q r ) ( r f c) 



and a comparison of the left-hand members leads immedi- 

 ately to the result: 



(19) (rbcy = ^(rbf) (rfc). 

 In the same way we can show that 



(20) (r c af = $ (r c g) (r ga), 

 and finally that 



(21) (r a b) 2 = 4 (r a h) (r h b). 



But a slight consideration of the symmetry of the identities 

 (16) and (17), shows us that equations (19), (20) and 



(21) are equally true if either p or q be substituted for r, 

 and hence that the three conies 



(y b c) 2 — 4 (y bf) (yfc) = o 



(22) (yea) 2 — 4 (yeg) (yga) =0 

 (y ab) 2 — 4 (y a h ) (y h b) = o 



all pass through the three points p, q and r. 



We can then infer that the conies <£* = o , M/" = o and 



