148 



Proceedings of the Boyal Irish Academy. 



Let - ■ ■ 



4 



^ J/i {xvy 4 yi/i + zziY = 2^^"^ + 9.y^ + ^'^^ + 2>'ptX^y + &c. + 6sa;?/« ; 



then it is clear, if - . ^ . 



{A,B,C,F,G\H,){xyz)\ {A,BAF,G,H,) {xyzf 



are the two annihilating conies which pass through the four points (a^i^iS,), &c. 



{Xiy^z^, that we have 



{A.B.CF.G^ffr) {pq,r„s,2hp2) -0, (a) 



' (A,B,C\F,G,R,) {2hqr,q,sq,) =0/ 



{A,B,C,F,G,H,) {2M-^Tr,r,s) = 0, 



M. . . . ){P • . ■) = 0, 



[A, . . . ){lh . . .) = 0, 



[A,. . . )(Ps . . .) =0; 



and also the same six equations again, with a, h, c, &c,, instead of p, q, r, &e. 

 Hence the ten equations of identification 



a - 2^> ^ - q> ^ =" ^^ ''^■2 = 2^i> *^c., 



are connected by the six linear relations (a), and are therefore equivalent to 

 four independent equations, which four serve to determine the values of 

 ifi, M„ M„ M,. . - \ :^ \ ^ 



As the system of annihilating conies is the system of polar conies of a 

 certain cubic, it follows that any two sets of four poles lie on a conic. 



If {A,B,C,F,G,H) (x, y, zy = be an annihilating conic, then replacing 



x,y,zhj 



d d d ' 



dx dy dz' \ 



and operating on the cubic taken in the general form ax^ + &c. (Salmon's 

 ISTotation), we obtain three relations between the coefficients A, B, &c,, if we 

 solve for F, G, H, and substitute, we obtain a result of the form 



A {F{yz + Giza + H^xy - Kx?) + B{F{yz + G^zx + Hzxy - Ky-) 



. + CiF^yz + G^zx + S^xy - Kz') 



as the general type of annihilating conic, the coefficients F, &c., being 

 determinants of the third degree in the coefficients of the cubic. 



In a similar manner, by the elimination of A, B, C, we can reduce the 

 system to the form 



A {K'yz - F\x' - F\y' - F\z-) + B {K'zx - G\x' - G\y' - G'^z") 



+ CI{K'xy - H\x?- E\f' - H\z% 



