160 
MR. A. C. DIXON ON SIMULTANEOUS 
dx x , dx^^ . . . j d*)C'fi_^ jj 
Xi, x 2 , . . . , x, i+1 , 
Y ' ¥ ■' Y' 
-A-l J a 2 > • • • ) -A- M+l? 1 
but in general, of course, it will not be possible to combine them so as to form a 
perfect differential. 
§ 11. An analogous process of integration may be given for two simultaneous 
equations 
— P/li) } + SB (pi + SC iCji + E = 0 
i, j i i 
t{A'ij(piqj — pjqi)} + SB '(Pi + SC'® + E' = o 
h j i i 
in which the coefficients A, B, C, E, A', B', C', E' are functions of n 
variables, x x , x 2 . . . x m and two dependent y, z, and 
cz 
ffi -2 
independent 
To fix the ideas, take n — 3 and let x p x 5 stand for y, z respectively, A / 4 for C„ 
A;j for —B,, A 45 for E, and make similar changes in the accented letters. Then, if 
u = a, v — b are two equations constituting a solution, # a, b being arbitrary con¬ 
stants, we must have 
* dpt, v) 
ij ^i, *,) 
SA'* 
d(u, v) 
3(&ij Xj) 
lor 
d(u, v ) 
b(y,z) 
d(u, v) 
d(x v z) 
0 , &c., if u — a, v — b, 
and the values thus given for p x , q x , p 2 , q 2 , y> 3 , q s must satisfy the equations (3) identi¬ 
cally, since a, b are supposed arbitrary. The equations to be solved are thus reduced 
to others which are linear and homogeneous in the Jacobians, and which do not 
contain the dependent variables. 
The equations (4) give two of the Jacobians of u, v linearly in terms of the 
others ; if we substitute for these two in the identity 
d(u, v ) = 
—« 
£j j 
b{u, v ) . 
Xj) ^ ^ 
we find that d(u, v) is a linear combination of the determinants of the matrix of tent 
columns. 
* This solution will not be a complete primitive unless a certain number of other arbitrary constants are 
involved as well as a, b, a supposition which is neither made nor excluded. 
It may he well to point out that the solution here assumed consists of two equations, and not of one 
equation involving an arbitrary function • in fact, any solution whatever necessarily consists of two 
equations, and one point of the present method is that these are to be sought together, not successively. 
f For n independent variables the number of columns in the matrix will be £(n + 1) (it + 2), the 
number of rows being still three. 
