Dawson — -On the Properties of a System of Ternary Quadrics. 153 



Section III. — Relation hehueen the Original Cubic and the Reciprocal V of D. 



Let 



P,,Q„R,; F,,Q„R,;P,,Q,,R,; &; r„ Q\, R\, &c., S' 



be the same functions of the coefficients of V that 



F„G,,H„&Q., K; F\,G\,H\,r 



are of those of U: then, as in Section I., we have 



AP, = -IKP\, AQ, = -lK'a\, /^R, = -IK'H\, 

 l^P^^-\K\F\, /^Q,^-\K'G\, ' AR, = -IK'H\, 

 AP, = -IK'F\, AQ,--IK'G\, ARf=-lK^H\, 

 AS = ^K'K'; 



and, in the same manner as in that section, we see that 



AF\ = \KP„ AQ\ = ^K'G„ AR\ = \K'Hu 



■ ' AP, = iK'F,, aQ\ = iK'G,, AR\ = ^Xm,, 



AP\ = ^KP„ AQ', = lK'G^, AR\ = \K^E„ 



AS" =-K\ ' 



Let H' and F' be the Hessian and Cayleyan of V; therefore, using the 

 above written values of P,, P',, &c., 



R' = ^- \F\:>f + G\f + H\^ + (2G^3 - G',)xhj + &c.} 



= -7— ^ (reciprocal of F). 

 Again, in the same way, 



P = -^ {- F,X' - G,fx' - E^v' + &c. 4 {K' 4 2K) X^v] 



A A 

 . = - H— X. (reciprocal of ^). 



Again, if T' be the invariant of the sixth degree in the coefficients of V, 



then 



P = &SS' + 8 {Q,Q, 4 P,P3 + R^R■:) - 4 {Q\Q\ + P^P, 4 R\R,) 



- 8(P,F\ + Q,q\ + FhP^) 4 (2.S' 4 8'){S' - 4>S), 



