ME. J. C. MALET ON A CLASS OF INVARIANTS. 
767 
As I afterwards make use of this equation I shall here give a full investigation 
of it. 
The condition J=0, it is easily seen may be written 
<T Pj 
dx 2 
6P >f 
dH 
+4P 1 3 +4P 1 H+- = 0 
(15) 
where H is the invariant of the first kind previously considered. 
If now in the equation 
3+>P,*+P*=o 
we change y to y\/ z we find the equation 
d?y 
dx 2 
^+ 2 Ql |+ Q ^=0 
where 
Substituting this value of Q x for P x in condition (15) and reducing we get 
S+ 6 P 5 + 2 (t+ 4p 1 2 + 2P = 
& + 2(4P 1 P 2 +^ 8 )z=0 
dx 
dx 
(A) 
this differential equation in z is evidently the required equation. 
It is to be remarked that if we remove the second term from this equation it 
becomes 
dh 
dx 5 
. T t d/% , dH 
+ 4H ,&+ 2 ^= 0 
(B) 
The cubic. 
Let us now consider the equation of the third order 
1 dx 3 
^+^2+^1+^=° 
dot? 
Substituting <f>(t) for x we get 
where 
w +3Qi ^ +3Qs ^+ Q ^ =l0 
3Q 1 =3P 1 <^'—Q 3 =P 3< P 
3Q 2 =2P a ^-^"+ 3 J-3P 1 f 
5 F 
MDCCCLXXXII. 
