348 Mr J. Brill, On Solutions of Differential [May 6, 
In the first instance we will suppose that w is to have a con- 
stant value along each of the curves (A) and (B). Then in addi- 
tion to the above equations we have the three following 
a dx, + - dy, = 0, 
a dx, + 7 dy, = 9, 
Hence it follows that 
At this point a knowledge of the general functional solution 
of our equation will prove of service. This, which is well known, 
is readily deducible from the results of the preceding article. In 
that article we have practically proved that 0w/dv remains con- 
stant so long as # is unaltered, and that du/dy remains constant 
so long as y is unaltered. We have, therefore, 
oY = F(a), nO and u=F (a) +f(y); 
and it is evident that if w is to be real when w+ zy and #—vwy 
are substituted for « and y respectively, then # and f must be 
of the same form. Thus we shall have 
u=f (x) +f(y); 
and it will follow that 0w/dx is the same function of w as du/dy is 
of y. 
Suppose now that 
$(a, y)=0 and (a, y)=0, 
are the respective equations of the bounding curves (A) and (B). 
Then, since , =a, and y,=y,, we have the three equations 
PF Y)=9 Wy Ye) =9  $ (Hs) Ys) = 95 
and from these equations we can obtain the values of the ratios 
he, 8 ChE SGI, le) 
3 
Hence, with the aid of the above relations, we obtain the values of 
the ratios 
1 3° 
Ou ow du. Ou 
Ox, Ox, ; OY, OY, 
