350 Mr J. Brill, On Solutions of Differential [May 6, 
the transformed problem, we shall have three equations of the 
form 
= da, +5 Ages Aiea ane: 
0 0 CEPR A180) 
tt Wy 5, te, ee 
oe ft Be OY = de, + F dy. 
The second of these equations may be written in the form 
on a . dy, = 5 + le 
and, subtracting this from the first equation, we obtain 
Ou ou of of oF oF 
—— ay —-~— dy. —-— 0a, dy — — At, 
Oy, Oy, Se, gt tenn e OY 
Now it is clear that 0f/d«,, df/dy, and dz,/dy, can be ex- 
pressed in terms of y,, and also that OF'/dx,, oF /oy, and da,/dy, 
can be expressed in terms of y,. Consequently we shall arrive at 
an equation of the form 
Ow ou 
(ay + r) i & + x) dys, 
where 2 is a function of y,, and w a function of y,; and it is plain 
that we shall obtain similar equations connecting du/dx, and 
du/dx,, Oujox, and du/dy,, also du/dx, and du/oy,. It is however 
plain that the finding of the values of du/d# and du/oy, in terms 
of « and y respectively, from these equations, will be a much 
more difficult matter than it was in the case discussed in the 
preceding article. 
The case in which the values of the rate of variation in the 
direction of the normal of the dependent variable are given alon 
each of the bounding curves, is very similar to this. We shall 
have three equations of which the following is a type: » 
Ou ow d 2) ( 
oy, dy, — On, da, =f (%,, 4) dy, 1(5e ) ar (ay Ou, 
=6(y,)dy,, say. ; 
Thus it is easily seen that we shall have equations of exactly 
the same type as before, viz. 
Ou Ou 
a +4 r) dy, = Ge + ts) dy,; &e. 
