Dawson — On the Properties of a Si/ stem of Ternary Quadrics. 149 



where F\, &c., are, as well as Fx, &c., determinants of the matrix 



a hi Ci m ffg «2 

 «3 h Ci hi 111 hi 

 a-i hs c C-, Ci m 

 and are connected by the relations 



H\G\ - H\G\ = K'Fi, F\G\ - F\a\ = K'H,, F\H\ - F',H\ = K'G^, 

 S\G\ - H\G\ = K'F„ F\G\ - F\G\ ^ K'H„ F\H\ - F',H\ = K'G,, 

 H\G\ - H',G\ = K'F^, F\G\ - F\G\ = K'H„ F\H\ - F\R\ = K'G,. 



There are also the nine relations SzGi - H^Gz = KF^. 

 Further, if 



A = 



then 



Fi Gx Hi 



Fi Gi Hi 



Fz Ga H 



and A' = 



F\ G\ H\ 



F',- G\ H\ 

 F 3 G\ H\ 



A = K{FiF\ + GiG'i + HiH'i), 



A' = K'{FiF\ + GiG'i + HiH'i ; 

 also, {K^yA' = (determinant formed of the minors of A) = A'. Whence, 



A = K'K', A' = K'^X, 

 and FiF\ + GiG\ + HiHi is equal to F,F\ + G,G\_ + H,H\ ; 



.-. FiF\ + GiG'i + HiH\ = F,F, + G,G', + H,H\ = F,F\ + G^G', + H^H, = KK . 



Similarly, 

 FiF'i + F,F\ + F,F\ = GiG'i + G,G', + G^G^ = HiHi + HH\ + HzH\ = KK\ 



We can express the coefficients of the Hessian and Cayleyan in terms of 

 these determinants. 



Let the cubic, its Hessian H, and its Cayleyan P be 

 ax^ + hy'^ + cz^ + ^xiiX'-y + 'dchX'z + ?)hiy'x + 3&3?/-2; + oci'zrx + ocr'^-y + Qmxyz, 

 8iX^ + hy^ + Gz^ + &c., 

 AX' + B^' + CV i . . . ; ■ 

 then we easily find that 



Fi = - n., Gi = 3,2+ C2, Hi = a,3 + B^, 



F, = hi + C\, G, = - b, H = hz + A„ 



Fz = ci + Bi, Gi = c^ + A„ H, = -c, K=m- M. 



Also, 



F'l = A, G'l - 2c2 - A,, Hi = 2b, - A^, 



F\ = 2c: - Bi, G\_ = B, H\ = 2a3 - B^ , 



F\ = 2b. - Ci, G', = 2a2 - C,, H\ = G, IC = 4m + 2M. 



[21*] 



