MR. J. C. MALET ON A CLASS OF INVARIANTS. 
771 
and therefore 
\vdt=\^ hl ‘ 
_ t f ^ldx 
— 1 2 e 
Hence letting 
S=J*‘ h **, K=j'lH s e^ ,,P **' i ' 
Condition (19) may be written in terms of the coefficients of tbe cubic without its 
second term being removed as follows : 
16ft 3 - 24l 3 It 3 S*— 121^4-15S 3 = 0 
Having expressed Q 3 and v as functions of x as above, the cubic may in this case be 
solved as follows : 
We have from previous results the solutions of 
S+Q,f+Q«=o 
* dw_j sJiiPa 
dt-^ e 
'dx 
U 
where Q L and Q 2 are connected by the relation 
dQ. 2 
dt 
+ fQiQa—9 
=J^ 
2/1=® 
Q s,dt 
y 3 
Q 2 dt 
Hence two solutions of the cubic are 
2/i 
and the complete solution may then be found. 
If we seek the conditions for 
transform so that y l may become e l , and as before we have 
l + 3 Qi + 3Q a +Q 3 =0 
m 3 4-SQim 2 +3Q 3 m+Q 3 = 0 
n 3 4 SQpi 3 4-3Q 2 n 4Q 3 =0 
from which we find Q 1? Q 2 , Q 3 three constants a, /3, y, say. 
conditions are 
and 
2a 2 —3/3 T 
*y! - 1 
2 
and the solution is easily found to be 
Hence the required 
y=AJ(v-)* 3 ' + B/J^ +Ce ”IC) ! 
'dx 
