Dawson — On the Properties of a System of Ternary Quadrics. 147 



To determine the lines hx + miy + tIiZ, &c. : they are the polars of 

 (/iTOiWi), &c., with regard to «^ + ^/^ + s^ so we have only to write down 

 the tangential equations of the points of intersection of the two annihilating 

 conies a Z7 + i3' r + 7 ^, a" U+(5"V+ Y' W. This tangential equation 

 being the product 



(/iX + niifi + Wiv) (/oX, + m^fx + n^v) (hX + m^fx + Tiiv) (k\ + lUifx + Uiv) 



gives, on factorization, the four lines hx + iihy + 7iiS, &c. 



When we form the tangential equation of the points of intersection of 

 the two conies we obtain 



2Si $.2 $13 i? 



^12 222 ^2z q 



4>,3 $23 223 '^ 



0, 



•p q r 



where p-.q-.r:: jS'y'^ - jS'^y' : y'a'^ - ^'a' : a'/3" - a"/3', 



or 2^ '■ q '■ '^"- '■ ^' '• ^' '• ^' 5 ^'' ^\ ^' being the results of substituting 

 x', y, z , the coordinates of a point common to the two conies 



in U, V, W. 



In the above 2i, 22, 23, $12, ^23, ^n are. the tangential and intermediate 

 contravariants of U, V,W. 



The equation of the cubic then becomes 



( {liX + m^y + Uizf QzX + m^y + iizzf {l^x + m/y + tizzy {Ux + iiiiy + 7142;)^ | 



5 j — - — + — - — + — .^- — + — -pr~ y 



where S is the invariant of the fourth degree, and Pi, P2, &c., are the results 

 of substituting {hm{)ii), (kt'ih^h), &c., for A, /j, v in the equation of the 

 Cayleyan, 



When the points (x^yiZi), (Xiy^Zi), {xzy-^^, {x^y^z^ have been determined 

 and the cubic in consequence reduced to the form 



^*ilf,(aa'i + yyx + ««.)', 



we can show that the ten linear equations obtained by equating the coefficients 

 of the various terms of 



4 



%^My{o:x^^yy^^ zz^"" 



to those of the original form of the cubic 



ax? + hy^ + cs^ + "iciiX'y + &c. 



are equivalent to only four independent equations, as follows : — 



R.I. A. PROC, VOL. XXVir., SECT. A. [21] 



