[655 
655] 
A MEMOIR ON DIFFERENTIAL EQUATIONS. 
Ill 
3. 40 to 45. 
where l, m are indeterminate functions of x, y, p, q; and the equation in question 
now becomes 
£ [- (my, a) + (ly, b)] 4- y [(mg, a) - (Ig, 6)] = 0 ; 
that is, 
g [- rn (y, a) - y (m, a) + l (y, b)+y(l, b)] 
+ y[ m(g, a) + g(m, a) -1 (g, b)-g(l, 6)] = 0; 
viz. omitting the terms which destroy each other, this is 
-mg(y, a) + lg(y, b) + my(g, a) - ly (g, b) = 0. 
Substituting for mg, &c., their values, we have 
able) is 
(<r, b) (y, a)-(a, a)(y, b) + (p, b)(g, a)-(p, a) (g, b) = 0; 
and the question is whether this is implied in the equations 
g (p, a) + y (a, a) = 0, 
g (p, b) + y (a-, b) = 0. 
42. Write y = Kg, the equation in question is 
(a, b) ( K g, a)-(a, a) ^g, b) + (p, b)(g, a) - (p, a)(g, b) = 0; 
that is, 
{_ %: 1 g! f ( ( :; % 3+»>«. »> - «> «. *»=°; 
viz. 
not in general 
(g, a ) [(p> b) + k (a, b)]-(g, b)[(p, a) + K (a, a)] + g [(cr, V)(k, a)-(a, o)(k, 6)] = 0 ; 
and we wish to see whether this is implied in 
of p, ar, but 
(p, a) + k (cr, a) = 0, 
(p, b) + k (cr, b) = 0, 
of p, ar (and 
’■stem is really 
which give 
(a, b)(p, a)-(a, a)(p, b) — 0; 
or, what is the same thing, whether these last equations imply 
(cr, b) (k, a) — (a, a) (k, b) — 0. 
Suppose k is a function of p, a, then, as is at once seen, 
. . (Ik , . dK , . 
<*■ a)= dp ( ^ “ )+ ^ <<r - 
(•,»)-*<*») + £(,.»). | 
and thence 
¿k 
(a, b) (k, a) - (a, a) (k, b) = ^ [(a, b) (p, a) - (a, a) (p, 6)]; 
viz. k being a function of p and cr, the two equations imply the third.