162 
MR. A. C. DIXON ON SIMULTANEOUS 
It is, of course, evident that if u, v are functions of variables x x , x. 2 . . . then the 
pair u, v is a bifunction of all the pairs that can be formed from x 1 , x 2 . . . Other 
examples will be found later on in the paper. 
§ 13. Sometimes solutions exist for systems of partial differential equations in 
which the number of dependent variables is less than the number of equations. 
If, for instance, with the system just considered we take a third equation of the 
same form, the coefficients being distinguished by two dashes, there may he solutions 
common to the three equations. If u — a, v = b give such a solution, then it 
follows in like manner that d(u , v) is a linear combination of the determinants of the 
following matrix : — 
d(x x , x L ) . 
. . d(x h xfj . . . 
Aj2, . . 
• Ay, . . . 
A' la , . . 
A'-- 
. JX y, . . . 
A" 12 , . . 
It. 
xl. y, . . . 
Similarly for a greater number of equations. 
Application to other Differential Equations. 
§ 14. There are two classes of equations whose solution depends on that of a pair 
of such linear homogeneous equations as we have just been considering ; they are, 
firstly, systems of two equations in two dependent and two independent variables, 
and, secondly, equations of the second order with one dependent variable and two 
independent. We shall consider them in order. 
Firstly, let y, z be the dependent variables and x x , x 2 the independent; sometimes 
we shall write x z for y and aq for z. Let p x , p 2 be the partial derivatives of y and 
q x , q 2 those of z, and let the equations be 
/j('^ i, y> p i> Po,> yfi (/•>) ffi 
M X V ® 2 » V, Z,Pl,Pi* $0 &) = °- 
A complete primitive will consist of two equations connecting x x , x 2 , y, z and 
involving four arbitrary constants. By differentiation these equations yield four 
more involving p x , p 2 , q x , q. 2 . As the two equations are supposed to be a complete 
primitive it must be possible to find expressions for the four arbitrary constants in 
terms of x x , x 2 , y , z, p x , q x , p 2 , q 2 ; the elimination of the four constants must give 
/, = 0 ,/ ; = 0 . 
Let a x , a 2 , « 3 , a. x be the constants, and u 2 , u 4 the expressions for them in 
terms of x x , x 2 , y, z, p x , q x , p 2 , q 2 . Suppose / 3 , f x , / 5 , / 6 to stand for u x , u 2 , v 3 , u x 
respectively. Then by differentiation we have for any value of the suffix i from 
1 to 6, 
