ME. J. C. MALET ON A CLASS OF INVARIANTS. 
Hence we have 
or 
rj p 
54Po—44P, 2 — 24 n 
3 J m 
v 
W 
r/V 
54P 3 -44P 1 3 -24-^ 
VP, 
=Ji say 
is an invariant of the quartic of the kind we are considering. 
Again by differentiating the equation 
4 * _ 2 ^Pi | sp a 
we get 
<f / 4 ~ 3 ^ 3 " r 9 1 d® "P <f >' 5 
<j>'± a dx* 1 * ~ L dx 1 $ 
= 3 1 9 1 & 
^72p r7P 
_ 2 . X 1 j 2 8 p j 1 6"D 3 
“ 3 dx* “P 9 1 ~P 91 
_2 d^P| , i 6 p ^ij_9p /2 ^Py i sp 2 
,3 + 9 Pi .7., H" J Pl 3 ^ i- 9-Pl 
substituting in (/3) we find 
27«=^{40P/-36P 1 §-18 ^-108P 1 P s +108P 3 } 
hence we see that 
/7P Mp 
40P^ - 36P, -18 - 108P* + 108P 3 
P/ 
is an invariant, which I shall call J 2 . 
We easily find a third invariant from the equations 
2P 1= ^ and w=P 1 Q' 1 , 
namely, 
sf + SPA 
Pp 
which may be called J 3 . 
By aid of the invariants J l5 J 3 , J 3 we can solve problems with respect to the quartic 
which are analogous to those already treated of in the case of the cubic. 
MDCCCLXXXII. 5 G 
