166 
MR, A. C. DIXON ON SIMULTANEOUS 
§ 18. The method of Charpit for a single partial differential equation of the first 
order shows how all solutions may he deduced from one complete primitive, and it is 
a question of interest and importance whether there is any analogous method for 
simultaneous equations. Now it follows at once from the conditions for a complete 
bidifferential that a bifunction of the pairs that can be formed from m functions, say 
?q, Uo ... . u m , will be a pair of functions of u x .... u m . In the present case a 
bifunction of the six pairs that can be formed with m 15 n 2 , m 3 , tq will be a pair of functions 
of these four, and the complete primitive to which it will lead will be the same as 
that given by ?q, m 3 . For when a solution of the auxiliary equations is known it leads 
directly to one and only one complete primitive by the integration of the equations* 
dy = p x dx ] + p 2 dx 2 
dz = q x dx x + q 2 dx 2 ; 
also the complete primitive to which the equations Fj (u x , m 2 . m 3 , m 4 ) = const., 
Fo (u [} w a , m 3 , iq) = const., will lead can be no other than is given by 
Uy — CLy, U. 2 = a 2 , Mg = Ctg, Uy — CL±. 
It must not, however, be forgotten that the system Fj = const., F 3 = const., 
)\ = 0, f 2 = 0 may have a singular solution. If F 1? F 3 involve two other arbitrary 
constants this singular solution will involve four, and therefore in general be a com¬ 
plete primitive of the equations j\ = 0, f 2 = 0. Moreover, all new complete primitives 
are included among those thus given. 
For every solution implies six equations connecting x x , x 2 , y , z, p x , q±, p*, q» (two 
of these six are of course f x — 0, f 2 = 0), and, therefore, by elimination of 
x v x 2 , y, z, py, q x , p 2 , q 2 , two equations or more connecting ?q, m 3 , m 3 , iq, which are 
known in terms of these eight quantities. If u x , u 2 , m 3 , m 4 are connected by four equa¬ 
tions they are constants, and the solution is therefore included in the old complete 
primitive. Let us, then, suppose that ?q, m 3 , m 3 , u 4 are connected by two or by three 
equations, 
F 0 (m 15 Mo, m 3 , m 4 ) = 0 (a = 1, 2 or 1, 2, 3). 
Now if py, p 2 , q x , q 2 , are all expressed in terms of p, q, two of their number, and 
Xy, x 2 , y, z, by means of the equations f x = 0 , f 2 = 0, the expressions 
dy — pydxy — p 2 dx 2 , dz — q x dx x — q 2 dx 2 
must both l)e expressible in the form 
Ayduy + Ardu 2 + A 3 cZm 3 + A/bq, 
* Otherwise thus—if in the auxiliary equations we suppose A to have the known value u h they become 
a pair of linear equations for / 4 , which must be satisfied by u. 2 , u 3 , ; now two linear equations in six 
independent variables can only have four functionally independent solutions, and one of these is known, 
namely, u x . (In exceptional cases the two linear equations for u 3 , w. 4 may be equivalent; for instance, 
suppose fi=pi + qi, u x = p 2 + q. 2 , f-> having any form.) Hence, except in special cases, the particular 
complete primitive is defined when one of the functions u x , u 2 , u 3 , u t , or more generally a combination of 
them, F (q, u 2 , u 3 , u 4 ) is known. In the case supposed in the text two such combinations are known. 
