PARTIAL DIFFERENTIAL EQUATIONS. 
169 
§ 20. Before we can claim in any sense to have found the general solution of the 
auxiliary equations, we must be in possession of thirteen pairs of functions satisfying 
them; we have only eleven when we know one complete primitive, and hence one 
more complete primitive, or even possibly two, must be found. An example (below, 
§ 29) will show that one more is not always enough. 
It is perhaps worth while to remark that any complete primitive defines the 
whole system of solutions, since it defines the differential equations. 
§ 21. The question of finding new solutions when a complete primitive is known 
may be attacked by the method of varying the parameters. Take the equations 
(6) or (7) of § 17. The problem is then to find such variable values for u x , u. 2 , u s , tq 
as will satisfy the equations 
2 TT-diii 
i — i OU[ 
0 , $ l^-du; = 0 . 
i= 1 O ill 
(9). 
Since all variables are supposed functions of x x , x 2l we may make one of two 
suppositions with respect to iq, u. 2 , u 3 , tq; either they are connected by three 
relations and are all functions of the same variable, say t, which is of course a 
function of x x , x 2 , or they are only connected by two relations, so that two of them 
may be taken as functions of the other two. 
Suppose first that they are all functions of the one variable t. Then, generally, 
the four equations (7), (9) will define aq, x 2 , x s , aq. also as functions of t, and hence 
this supposition is not admissible unless it is possible to choose the functions of t in 
such a way # that the four equations (7), (9) will be only equivalent to three. The 
If these conditions are satisfied by an expression 
A ijd((JCi) Xj), 
i, j = 1, 2 . . 6 
it can be put in the form 
^ w A -i dxij ^ w Hi dx% 
and then it must further be possible to express 
6 c 
2 Af dxi and 2 m dxi 
i=l i =1 
as linear combinations of three differentials, dv h dv 2 , dv s . The discussion of the conditions therefore 
belongs to the theory of the reduction of two such expressions, that is, of the extended Pfaff problem. 
* It seems obvious that this will not generally be possible; but it may be well to give an example. 
Suppose the complete primitive to be 
y = ax \ 2 + lx -2 + c, -| 
Z = CX i + eX‘2 2 + lXiX-2 2 J ’ 
so that the differential equations are 
y = irui + P2*2 + ?i - 
z = 21*1 + - P 2 Z 1 & 2 2 , 
then the variations of the parameters a, b, c, e must satisfy the equations, 
X\ 2 da + x 2 db + dc = 0 
Xidc + *2 2 de + XiX 2 2 db = 0; 
VOL. CXCV.—A. Z 
