756 
MR. J. C. MALET ON A CLASS OP INVARIANTS. 
Substituting the value of Q x derived from this equation in H and reducing we find 
for our required condition 
/ 7TT\ O / J79.TT Jin \ 
...... ( 6 ) 
*+«.-(”)■+, h(S_ S ) = , 
This condition, as is seen, is the eliminant of the equations 
S+3PS+3P.® + P a =» 
and 
dx* 
g+2 p^+p* 
= 0 
To solve the equation when condition (6) is true. First consider the equation witl 
the second term removed, and we have to find y 2 the equations 
<&y 
do? 
+ 8 H l+^=° 
dhj 
+H y 
=0 
of which equations y 2 must be a common solution, as follows at once from the con¬ 
ditions Q 3 =0, Q 3 =0. Thence we easily find 
and therefore 
y 1= zzx\/He~^ az 
These values of y 1 and y 2 are to be multiplied by e~l Fldx to get the corresponding 
solutions of the cubic when the second term is not removed. 
To get the third solution we have 
d?y_ 
dx z 
+3Q.S- 
which gives, remembering the relation 
g— fr+ 2H Qi=° 
thence we easily find the complete solution of the original cubic 
y- v/He~K Fl+ ®)* { A+B»+C f | 
where A, B, and C are arbitrary constants. 
