PARTIAL DIFFERENTIAL EQUATIONS. 
161 
d(x l , x 
o), d(x 4 , x 3 ), . . 
. d(x i} Xj) . 
• d(x 4 ,x s ) 
A 13 , 
CO 
. A, . 
■ Ko 
A' 
i A J2 
A' 
135 
• A', . 
• A' 45 
There are thus eight bidifferential expressions, and the problem is to be solved by 
finding such multiples of these as, when added together, will form a complete 
bidifferential. 
§ 12. As in the case of Lagrange’s linear equation, this will generally, in practice, 
be done by inspection, and the method will be useful for finding solutions in finite 
terms—when such exist. But in any case,* whether the inspection is successful or 
not, there can be no doubt of the existence of suitable multipliers, in infinite number. 
For it is certain that the equations (3) have—possibly among other solutions--an 
infinity of solutions, each involving two arbitrary constants at least, and any one of 
these may be written u = a, v = b, where a, b are the two constants ; u, v are 
functions of the variables, but may, of course, be implicit functions of great com¬ 
plexity. The functions u, v must satisfy the conditions (4), and it immediately 
follows that d(u, v) must be a linear combination of the determinants of the matrix 
formed from (4) as above ; so that a corresponding system of multipliers must exist. 
If the solution is not in finite terms it is not likely to be found by inspection, and 
it is quite probable that the best way to find it would be by solving the original 
equations (3) in series. By whatever means the solution is found, the corresponding 
system of multipliers is thereby determined. 
If nine solutions of the form u = a, v = b have been found, the nine 
bidifferentials d(u 1} v j), d(u. 2 , v. 2 ) . . . c/(m 9 , v g ) must satisfy identically a linear rela¬ 
tion, since they are all linear combinations of eight expressions only. 
We shall say that one of the nine pairs of functions is a “ bifunction ” of the other 
eight pairs. 
The following is, then, the definition of a bifunction. When the bidifferentials of 
any number of pairs of quantities are connected by an identical linear relation, with 
constant or variable coefficients, any one of these pairs is said to be a bifunction of 
the rest. 
The word bifunction is simply used as an abbreviation—at least for the present. 1 
am not without hope that at a future time it may be found to have some connotation. 
* If one of the dependent variables with its derivatives is altogether absent from the equations (3), or 
if it can be made to disappear by a change of the other dependent variable, the equations (3) will in 
general have no solution. This case will then be excluded; it is the only case in which the method of 
solution in series (as given, for instance, by Frau von Kowalevsky, ‘ Crelle,’ vol. 80) cannot be used 
to prove that solutions actually exist. 
Another case that may fairly be excluded is that in which all the derivatives of one of the dependent 
variables do not occur or may be made to disappear by a change of the other. Such a system is equiva¬ 
lent to a single partial differential equation with one dependent variable, since the one whose derivatives 
are absent may be eliminated. 
VOL. CXCV.-A. 
Y 
