ME. J. C. MALET ON A CLASS OF INVAEIANTS. 
769 
I 3 , however, is not a distinct invariant, since it is evident from the method of forming 
it that it can be expressed in terms of I x and I 2 . 
It is to he remarked that if I is any invariant of the cubic, of the kind we are 
considering, then 
A f and [P.*ldx 
P 3 S dx ) 6 
are also invariants of the same kind, as follows at once from the relation 
P ^(f>' 3 =v or P ^dx=v i dt. 
Let us now seek the condition that two solutions of the cubic, y lt y 2 should be 
connected by the relation y$) 2 — 1. 
Transform the equation so that d shall be a solution and let it become 
f+BQg+SQ,|+^=o 
and we have 
1 + 3Q 1 +3Q 2 +Q 3 =0 
1 —BQj+SQg—Q 3 =0 
from which 
1 + 3Q 2 =0, 3Q 1 +Q 3 =0 
Substituting for Q 2 and Q x in the equations 
dQ 
dt 
^-■+3QiQ 3 
=1, 
q 3 * 
§+ 2 Q I »- 8 Q, 
Q.s 
=L 
then letting Q 3 = 2 3 and reducing we get 
3|=I/ +Z * 
9^|=2s 6 -9I 2 2 3 +9 
dz 
from which equations we are to eliminate z. Eliminating^-, we get 
Ctl 
* e +3lj#+ 9l s z a —9=0. 
(16) 
(17) 
dz 
Now differentiating with regard to t and then substituting for — from (16) we 
dt 
find 
2#H-4l 1 ^ + ^(4V-6I 2 +| i § N )-6lA+|lf=0 • ■ • (18) 
P, dx 
5 F 2 
