MR. J. C. MALET ON A CLASS OF INVARIANTS. 
761 
The required result is also the eliminant of the equations 
3+ 6H 3+ 4G 2+( i +3h>=o 
and 
3 + 3 H 2 + g ^=° 
as is evident. 
Having obtained the result in a very cumbersome form only I do not give it, but at 
once go on to cases where two or three conditions subsist between the solutions of the 
quartic. 
Let us first consider the case where 
Vi=y&=y 3* a 
changing y to yy z the resulting equation must be of the form 
Hence we have 
from which 
^+40 *-o 
H =-Q 1 ^f, G =2Q 1 ^ 
3 ^Qi 
G -fr+ 2 ^ H=0 ’ I -§= 6G i 3H 
dx 
one of the required conditions is then given by equation (6), and the second is easily 
found to be 
“fr-swo-fir 
To solve the equation, we find at once as in the case of the equation of the third 
order 
and y 4 is the solution of the equation 
multiplied by y 3 . 
Hence remembering the value of Q 1? we get for the complete solution of the quartic 
5 E 2 
