416 Proceedings of the Royal Society of Edinburgh. [Sess. 
“ ce qui parait etre tres difficile a efiectuer.” He then announces the 
analogous theorem : “ Soit U une fonction homogene et de Fordre v des 
deux variables x, y, et VU la determinante 
02U 0HJ /0 2 U\ 2 
dx 2 dy 2 \dxdy ) » 
Ion a 
(v - 2)(v - 3) ■ V(U + aVU) = { -v{v- \){v-zyaj+v(v- l)(2v - 5) 2 a 2 I }u 
+ { (v - 2)(v -3f + (v- 2)(v - 3)(2v - 5)a 2 J } VU . 
En representant par i, j, h, l , m les coefficients difierentiels du quatrieme 
ordre de U, on a 
I = ikm - il 2 — mj 2 - k s + 2jkl , 
J = 4jl - 3 k 2 - mi , 
de maniere que I, J sont des fonctions de x[ y des ordres 3(y — 4) et 2(V — 4) 
respectivement.” To this he adds the remarkable fact, that if the binary 
quartic 
i + 4 + 6 k£ 2 rj 2 + 4 + myf , 
where i, j, k, l , m are now any quantities independent of y, be trans- 
formed by the substitution 
£ = Xg + fiy ) 
y = X'g + fx'y j 
and I and J thus become I' and J', then 
F = (A .y! — X' />t) 6 I , J ' = (Xy! — X' . 
In other words, he makes known for the first time the two “ invariants ” of 
a binary quartic, and notes the curious fact that expressions of exactly 
the same form occur in his equivalent for V(U v + <xVU 2i! ,). 
Another remark is equally suggestive, namely, that simpler results 
might be reached if U were taken a homogeneous function in x', y' as 
well as in x, y , and VU were defined as 
_ 0 2 u 0 2 u 
dxdx' dxdy' tydy' dydx 
For example, U being a quadric in both sets of variables, namely 
U = x^Ax 2 + 2B xy + C y 2 ) 
+ 2£c 1 ?/ 1 (A , £c 2 + 2B ’xy + Q>'y 2 ) 
+ y 1 2 ( A"x 2 + 2B "xy + C ,f y 2 ), 
