MR. J. 0. MALET ON A CLASS OF INVARIANTS. 
773 
Hence referring to equation (A) we see tliat 
^ +3F ^ + ' iV *L +7 *y-' 0 
may be expressed in the form 
i er\ \ o( d ® 
J + 4Q/+ 2Q,)|+2(4Q 1 Q 3 +^) 2 /=0 
therefore 
dQn 
Pi—2Qi, P 3 —2^ -fSQiQ 
3P 3 =2^l+8Q 1 2 +4Q, 
From these equations eliminating Q x , Q 3 we get 
dL 
dx 
+2P X L—2P 3 =0 or K=0 
as the required condition. 
The relation yi—yddz involving only the ratios of the solutions must be also 
expressible in terms of the invariants of the first kind considered in this paper, and in 
fact we find 
dL 
dx 
+2P X L—2P 3 = 
3— 
dx 
-2G 
where H and G have the same meaning as before. 
To arrive directly at this condition in terms of H and G, we see on referring to (B) 
that the cubic with the second term removed by the substitution for y of ye~ J Pl4& , viz., 
can be written in the form 
where 
Therefore 
and eliminating H x we find 
g + . H s +Gy =. 
^+2^=0 
i dx' dx y 
dx 3 
H 1= Q 2 - Ql s-f 
3H=4H 1; G=ag 
3^-2G=0 
dx 
the required result. 
