760 
ME. J. C. MALET ON A CLASS OE IEVARIAJSTTS. 
we have 
therefore 
To determine y s , let y 3 =y 2 \( x ) an< ^ we ^ ave to determine x 
x'"+3 Q!x"+ 3 QaX = 0 
/'"+3Q 1 /"+3Q,/'=0 
xf'—xy"'=e s $' *>*’ =e 3 J« a ’ 
but also 
from which we easily find 
therefore 
X= 11 / ’\j^e^ ma *dx^dx 
and the complete solution of the cubic is in the case we are considering. 
y = e \m)-w ^| f'j± )$m*dx}aat 
remembering that an arbitrary constant is implied in each integration. 
The quartic. 
Consider now the equation of the fourth order 
g + 4P 1 3 + 6P 3 g + 4P 3 | + P^=0 
we have three invariants H ; G ; and I, and the equation with its second term removed 
becomes 
S+ 6 H S+ 4 G I+( I + 3H %=° 
If we seek the condition that two solutions of the equation should be connected by 
the relation y L =y 2 x, transforming the equation by changing y to yy%, the result must 
be of the form 
Hence we have 
dx 4 + dx Z— 0 
H=Q.-Q 1 *-g 
G=2Q 1 8 --3Q 1 Q a - 
& Q 1 
dx 3 
I=-6Q 1 4 +12Q 1 a Q 3 -BQ g 
d s Q 1 
dx? 
and the condition required is the result of eliminating Q x and Q ? from these equations. 
