35] 
ORDER WITH THREE VARIABLES. 
233 
The equation V U — 0, 
combined with that of the curve, determine, as Dr Hesse has demonstrated in the paper 
quoted, the points of inflection of the curve. It may be inferred from this, that if U 
reduce itself to the form 
U = {ax 2 + ¡3y 2 + 7z 2 + 2lyz + 2kxz + 2\xy) P — VP (22), 
P a linear function of x, y, z: then VU takes the form 
VU=P{pV+crP 2 ) (23), 
where p is of the second order in the coefficients of P, and also in the coefficients 
a, /3, 7, t, k, X : and a is equal to the determinant 
a, X, k 
X, ¡3, t 
k, l , 7 
multiplied by a numerical factor. If U is of the form 
■(24), 
U = PQR (25), 
then VU= pPQR = pU (26), 
and this equation is consequently the condition of the function U being resolvable into 
linear factors. The equation in question resolves itself into 
- = B = = K = Il = Jl = K '- 1 } 
a b c i j k i x j 1 p l \ 
a system which must contain three independent equations only. It would be interesting 
to verify this ¿i 'posteriori. 
C. 
30