774 
MR. J. C. MALET ON A CLASS OE INVARIANTS. 
Hence the condition y^—y^j 3 may be expressed in either of the forms 
we have in fact the relation 
K=0 or 3~—2G=0 
dx 
K 2 =L 3 ^-2G^ 
We see also that in this case the solution of the cubic is reduced to that of the 
quadratic 
S+r.S+ 4 -^" 
the solutions of which are *fy\ and \Zy%- 
The quartic. 
To find the invariants of the second kind of the equation of the fourth order 
3+‘ p .S+« r S+<+^=» 
let us suppose the second term removed by the substitution #==</>(£), then writing the 
result in the form 
we 
find 
dh/ dh/ . dy , 
i+“i+^+ 4(, =° 
2P,f-^=0 to=P 4 <^ 
“=T-T~ l2P ^ +6P#2 .« 
:1 ^-^-9+ 12 L?- 4 Lr-6P,f'f+4P 8 f 3 • . . (/9) 
From the first of these equations we have 
dx~~ <f>'* <£' 3 <£' 3 3 1 
therefore by substitution in (a) we find 
9«=f 2 ('54P,-44P 1 2 -24^) 
