770 
ME. J. C. MALET ON A CLASS OF INVARIANTS. 
Hence the required result is the eliminant of the two cubic equations (17) and (18). 
I do not give here the expanded result thus obtained, as I have arrived at it in a 
more compact form, as follows : 
Remove the second term, from the cubic we are considering, by change of the 
independent variable, and let the result be 
d s y , dy , 
Suppose now that y^y<^— 1, and let 
^ + 2 Qi f ,+Q22 ' =0 
be the linear equation of which the solutions are y 1 and y%, m it is evident then that 
the cubic may be written in the form 
J{f+ 2 <+«^}- 2 42+<+M=» 
Since this equation is evidently satisfied by y l and y 3 , and we can determine X so 
that any other function of t shall satisfy it. 
Comparing coefficients we find 
2Q 1 -2\=0, 2^ J -2XQ 1 +Q a =tt 
dt v 
we have also in consequence of the given condition 
From these we have 
6Q 2 X+'y=0 3Q 3 +2jwfo=0 
2--2XM-Q *=« 
Hence we easily derive 
lfijjwftfj + jj'rcfo j — 12^j'y^+15 / y 3 =0.(19) 
which is the required result expressed in terms of the invariants u and v. 
To write the result in terms of L and I 2 , we have from which we get 
v a dt 
