346 PEOFESSOR A. R. FORSYTH OX THE DIFFERENTIAL INVARIANTS 
which come from the coefficients of ^n> ^ 02 > '> 7 o 3 respectively ; and 
it + ^ it + a'Jo + afo = °. 
as;; + ® aF,, + ^ a'i(; +-r +-^.0 3^^^ • ■ ■ (hl). 
+ + + + + = » ■ ■ • (ny. 
® a4 + dG,: + a^« ■*■’'''« aiy " .^ 
'^a|; + ®a|, + '^‘»a^ + ^«|£ = «.(UU 
a4 + *'1^;; +alj, +^ ■ • • (™e'. 
"^4^o; + ^aw + ®3y- + 2Gg^^ + ^„.A_ + ^„-|- = o . . . (iiy, 
^ it + it + afe = °. 
which come from the coefficients of ^gQ, tj^,, 77,33 respectively. 
15. Consider the set of equations (III 3 ) to (Illg); all tlie Poissox-Jacobi 
conditions of coexistence are satisfied so that, in so far as the third derivatives of 
and xfj and the second derivatives ot E, F, G are concerned, the set niay be regarded 
as a complete Jacobtax' system. The total number of variables occurring in the 
derivatives of f is 
4, for the derivatives of (f) of the third order, 
+ . xjj . 
.E, F, G of the second order, 
= 17 in all; hence as the total number of equations is 8 , there will be nine algebraic¬ 
ally independent solutions involving these 17 quantities. When we integrate the 
set of equations in the usual manner, we find a set of nine solutions, apparently in 
their simplest form when given by 
