772 
MR. J. 0. MALET OR A CLASS OE INVARIANTS. 
More generally let ns seek the conditions that y 2 and y z should he any given 
functions of y v 
"Write for convenience y 2 =(j>(log yj, y 3 =xjf(\og y±) then transforming as before we 
have 
1 +3Q 1 + 3Q 2 + Q 3 =0 
(j )"'+ 3 Qi<l>" + 3 Q 2 ^+ Qb<^> =0 
^ , 4-3Q 1 ^ ,, +3Q 2 i/f / d-Q 3 ^=0 
Now from these equations we find 
111 
<r r v 
-yjr'" Tjr" yjr' 
1 1 T 
<}> r v 
■yj/'" ‘\Jr / 
Hence from the relation JP 3 *efcc=JQ 3 *cfa we find JP 3 ’c&c=F(tf) where F is a known 
function, and the complete solution of the cubic is 
y=Ae F -{K*}+B<;,{F- 1 j[p s W*}}+C^|F- 1 jj‘p 3 W*}} 
To find the required conditions we have only to substitute F^fJP^da;} for t in the 
left-hand sides of equations 
^+3QA T ^ ! +2Q 1 3 -3Q 2 
Q? =Tl ’ Q? =Ia 
Qi, Q 2 , Q 3 being found in terms of t from equations previously given. 
I proceed now to consider an invariant of the cubic which is particularly worth 
noticing. 
Eeferring to the values of I 2 and I 3 given before, we find 
S + 2P.L-2P3 
L* 
=K 
(say) 
and K=0 is the condition that the solutions of the equation of the third order should 
be connected by the relation y}=y^y ^; as follows. 
Let yi=Zi, 2/a=% 2 , and let the equation of which the solutions are z 1 and z 2 be 
