188 
MR. A. C. DIXON ON SIMULTANEOUS 
whence follows xdgfydf-\-2dc 0, a simpler relation that may be taken instead of 
the first of the three. 
These equations will define x, y in terms of the single variable a, unless all the 
first minors of 
da dli dg 
dh db df 
dg df 2dc 
vanish. 
We thus have three, ordinary differential equations connecting a, b, c, f g, h ; 
they are connected also by the relation F [a, h, b , g, f c ) = 0, and the fifth relation 
among them may he chosen arbitrarily, so that we may put h = <£ (a), an arbitrary 
function. 
Then we have 
db/da — {r/f (a)} 3 , df/da = f (a) dg/da, 
2dc/da = (dg/da) 2 , 
F (a, <f)(a), b, g, f c) = 0 
as the equations determining b, g, f c in terms of a. These are to be integrated, 
and then a is to be eliminated from the equations 
x + ydh/da + dg/da = 0, 
z = c-\-gx+fy + \ (ax* + 2 kxy + by*). 
The result of elimination will be a solution of the differential equation. Three 
constants are introduced by integration, and thus, if the function </> involves two 
constants, the new solution will generally be a complete primitive. 
§ 41. The new complete primitive gives new solutions of the auxiliary equations 
which we shall now examine. Let oq, a 3 , a 3 , a 4 , a- be the new set of parameters. 
Then a, h, g, b, c, f are connected with these parameters by five equations, one of 
which is the original equation F = 0. These five relations are such, that if 
then 
dh = Xda, dg = yda , 
db = \*da, 2dc = /x*da, df = Xyda ; 
of these five, the first two define X, y in terms of a, a 1 , a,, a 3 , oq, a 5 , and the others 
must then follow from the five equations that give h, g, b, c, f in terms of a and 
the same new constants. Thus, in general, we may suppose a, h, g, b, c, f y, 
expressed in terms of X, cq, a 3 , a 3 , oq, a 5 and the expressions will be such that 
dh — \da, dg — yda, db — \*da, 2 dc — yd da, df — Xyda 
