DIFFERENTIAL INVARIANTS ()F SRACF. 
201 
The last three equations replace A- {a) = per, Ag (cr) = per, A,, (cr) = per, l:)einn; 
A^ (cr) — Ag (er) = 0, A^ (o') — Ag (cr) = 0, A- (a) + Ag (cr) + Ag {cr) = Sfxcr. 
Inspection of the first eight equations shows that they are the difihrential equations 
of the invariants and the contravariants of the ternary quadratic 
{a, h, c, f p, /^X, Y, Z)^ 
the contragradient variables being ^oiO’ ^ooi- ^ ^^6 such an invariant or the 
leading coefficient of such a contravariant, that is, the coefficient of the higliest power 
of ; in the latter case, the contravariant is uniquely determinate when t is known. 
Now i satisfies tlie five equations 
2/; q_ ^ 
M ^ dh ^ dfj 
= 0 
d I 3^ I 3^ 
s;. + / M + '■ d„ 
Bt Jr C 
^ 37; + ^ 3/' 
dll 
= n 
h~ + h = 0 
rip dj rc 
dt . 1 dt dt „ 
+ * a/7 - •" a7; =" 
This is a complete Jacobian system. It apparently contains five members : but the 
fifth equation is a linear combination of the first four, and therefore the system really 
consists of four equations. It involves six variables, viz., a, h, c,f, g, h ; and therefore 
it jDossesses two independent solutions. One of these is Ifi; the other is easily seen 
to be J)c — f^, = A. 
lieturning now to the system of nine equations, and Ijearlng in mind the fact that 
the first eight possess two solutions, one being and the other being a contravariant 
of the ternary quadratic which has A for its leading coefficient, we have the 
contravariant in the well-known form 
0 = (A, B, C, F, G, HX<^joo> <^oiu> ^oo\f- 
The ninth ecpiation is satisfied for any solution which is homogeneous in a, h, c, f, g, h 
of deg ree m and is also homogeneous in (^jqq, </)qoi of degree 7i, provided 
2m n = SjjL. 
Hence the value of p for @ is 2. 
There is thus a single absolute differential invariant of the type specified; its 
value is 
@L-2. 
2 P 2 
