172 
MR. A. C. DIXON OX SIMULTANEOUS 
four may be eliminated by means of the relations (7); so that in the end we shall 
have two relations connecting u x , u 2 , u 2 , u±, and the derivatives ; the problem is of 
the same form as the original one, to solve two simultaneous partial differential 
equations in two dependent and two independent variables. 
Interchange of Variables and Parameters. 
§ 23. A curious thing may be noticed at this point. If in the equations d> = 0, 
xji = 0, we treat x 1 , x. 2 , x 3 , x 4 as arbitrary constants and eliminate them by differentia¬ 
tion, we are led to the same differential equations connecting u x , u. 2 , w 3 , rq as were 
just now given by the variation of parameters. Thus two equations in two sets of 
four quantities will give two pairs of simultaneous partial differential equations by 
taking each set of the quantities in turn as variables and the other as arbitrary con¬ 
stants. The auxiliary equations, if expressed in terms of the eight quantities, will 
be the same in both cases; this gives a meaning to the six solutions of the form 
(a?,-, Xj) which we found the auxiliary equations to have, for any one of the six 
will lead to the primitive == 0, xfj = 0 of the second pair of differential equations, 
just as a solution (u,, Uj) leads to this primitive for the first pair ; any new solution 
of the auxiliary equations will in general lead to a new complete primitive for either 
pair, but an exception to this rule will arise when, for instance, the x differential 
equations have a complete primitive which gives three relations among u } , u 2 , u 3 , tq. 
The array (5), transformed so that the variables are aq, x. 2 , x 3 , x 4 , iq, u. 2 , u 3 , u 4 , 
connected by the equations <fi = 0, ifj = 0, will have six rows of the form 
d(xi, xj), 0, 0, 
six of the form d(u ; , nf, 0 , 0, 
and in the other sixteen there will be 
d(xi, Uj) in the first column, 
in the second the minor of ~ in the determinant : 
OXflUj 
d 2 cf> 
3-</> 
8~(j) 
3 3 0 
8(f) 
3 \fr 
3q3zq ’ 
dXydUy ’ 
3,q3;q ’ 
dxf uy ’ 
dxf 
3 
5 2 <j> 
3'0 
3 2 <j) 
3 2 cf) 
dcf) 
3 \Jr 
dx a dic x ’ 
dx 2 d it a ’ 
8x 3 8u 3 ’ 
3q3;q ’ 
dxf 
0Ao 
8 2 (f> 
8 2 (f> 
Pcf) 
3 2 cf) 
ocf) 
3 1 
8x 3 8u x ’ 
8x 3 8u 2 ’ 
8x 3 8u 3 ’ 
dxfUy ’ 
OX 3 
8x s 
d~(f) 
Pc}) 
3 -cf) 
3 2 cf) 
8cf) 
8\fr 
dxf u 2 ’ 
3q3iq 5 
dxfuy ’ 
3 V 
3q 
8(f> 
du-y ’ 
dcp 
3 u 2 ’ 
3 cf) 
3 u 3 ’ 
3 cf) 
8u 4 ’ 
0, 
0 
8\fr 
3 uf 
8\fr 
3 u 3 5 
df 
du 3 ’ 
3 \fr 
o, 
0 
