Ternary Quintic to Hubert's Canonical Form. 297 



where % , 2 1} 2 2 are curves of class two; further, we may assume 

 without loss of generality that % Q , 2 l5 2 2 lack respectively terms 

 in fo 2 > £i 2 , fa 2 - Suppose 



then the conditions of apolarity to C s impose the following condi- 

 tions upon the 15 coefficients of 2 , % and 2 2 : 



{«! (a y ) 2 — a 2 (a/3) 2 } a x 2 = 0, {a 2 (« a ) 2 — a (a v ) 2 } a^ 2 = 0, 

 {a (a^) 2 - «j (a a ) 2 } a^ 2 = 0, 



for all values of x , x 1 , # 2 ; eighteen conditions, homogeneous in 

 fifteen unknown coefficients, have to be satisfied. But clearly the 

 conditions obtained by equating to zero the coefficients of x Q x s in 

 the first, of x x x s in the second, and of x 2 x s in the third are not 

 independent, s standing for either 0, 1, or 2: thus the eighteen 

 conditions are brought down to fifteen. The possibility of satisfy- 

 ing these depends on the vanishing of a determinant of 15 rows ; 

 but this determinant is obviously skew-symmetrical and so must 

 vanish. 



