1900. No. 5. ON PARTIAL DIFFERENTIAL EQUATIONS, ETC. 21 
2 
Case 1. Suppose £ +0, # +0, G+ 0. Eliminating first = from 
the first and third equations of system (II), p. 16, we find that 
AE [ou Qu|? AF+ BG dudu Qu du Ou\ Ou 
res el +2[2] er Gel® (ele = 
Multiplying this equation by an indeterminate quantity 4, and adding 
the result to the second equation of (II), we obtain 
=e ac +0 1Ê | (EVE. 
megle) åa 
~ (35) (5) = 
Now let us see if it is possible, as above on p. 9, to determine A so as 
to make the first member of the equation resolvable into linear factors, 
We cannot say a priori that such a resolution is possible, as we should 
be able to do if that member were homogencous and of the second degree 
with respect to ¢#ree instead of with respect to the four subject variables, 
Ou Ou 94 Ou 
ot or (32 (5 
3 Ou Ou ; : 
Observing that the squares of bel and lg) are wanting in the first 
7 
member of (A), while those oe and © = # appear, we are led to assume 
as the proposed equivalent of that er an expression of the form 
ou =) Ou , (0% ‚au 
BEST a) + (5 (| [as tm | 
Multiplying the factors of this expression together, and then equating 
the coefficients with those of the first member of (A), we find 
ADP) 
Dan n 
== 
