Dawson — On the Properties of a System of Ternary Quadrics. 155 



we obtain the identity 



<:2) -- 



F\, 

 K' + 2K, 



2(R\ + ff,), 



8 



Or 1, 



H\, \K' 0, 



H\, 0, 0, i^' 



■2{H', + E,\ 2{rr,+ G,\ -SFu F\ + F, r, + F, 



K' + -IK, 2 (F\ + Fo), Cf, + G,, - 36^3, a\ + G, 



2 {F\ + #3), ^' + 2^, H', + iJ,, ^'1 + H,. - -^H, 



If, now, we write down the equation (1) for the cubic F, and then replace 

 Pi, Qt. i?i. &c., by their equivalents in terms of F'l, G\, H\, &c,, and write 



we obtain the equation 



since it is found that the determinant in (1) becomes the determinant in (2) 

 when the change from Pi, Qi, Pi, &c., to F'^ G-\, H\, &c., is effected. 



Again, writing down the equation (2) for V, and changing P\, Q\, R\, &c., 

 into their equivalents in terms of F^, Gi, H^, &c., we find 



8 



K" 



or, replacing ;S" 1 )y , we get 



>S? 



64A' X 8A' 



A'.A^ 



K' 



Now, when /S' = 0, the annihilating conies of V pass through three common 

 points ; that is, the polar conies of the points 



he - &3C2 : &3C1 - &iC : hxCt - hci; 

 c^a^ - ca-i : etc - Cjffa : cizCi - acz ', 

 cujbz - a^h : «3&i - abs : ah - a^x ; 



or, rt'i : i/i : <;,, &c., pass through three common points. 

 Hence, either 



«i 2/1 ?-i 



Ol-i, y.2 Zr. 



= 0, 



^\0.?H. 0,^-5 = 



?'• ?;y vz 



''-'3 Vz ^'3 

 R.I. A. PROC, VOL. XXVn., SECT. A, 



[22] 



