1889. ] Equations with Boundary Conditions. 351 
5. Asa first example of the application of these methods we 
will take a very simple and well-known problem. We will show 
how to find a function of # and y that satisfies the equation 
Oru ste ore =0Q 
On oy © 
throughout the region between the curves 
e+y=a and #’+y' =D’, 
(b >a), and has a constant value along each of the curves. 
Transforming this problem, we shall have to find a value of wu 
which satisfies the equation 
CU 
Oxoy 
throughout the region between the curves #y=a* and #y=6", and 
has a constant value along each of the said curves. 
Differentiating the equations 
wy, = 0, @y,=d', 2y,= a", 
we obtain xdy, +y,dx, =0, 
ady, + y,di, = 0, 
ZY, oa Yd, a 0; 
from which we deduce 
dy, dy, _ _da,__ da, 
1 aa aly alan 
Therefore we have 
Ou Ou ous Ow 
A oy Is dy, 3 dn, 8 aa, 
Thus we shall obtain a solution of our problem by assuming that 
all along the curve wy = a’. 
We may now complete the solution by applying the third 
formula of Article 2, as simplified in Article 3. We will use (a, ¥) 
for the coordinates of @, taken as any point between the two 
bounding curves, and (w’, y’) for the current coordinates along the 
curve PR. We will also take a as the constant value of u along 
the curve #y=a’*, and @ that along the curve zy=b*. Then 
we have 
Ydy’ a x 
mah [SY =k jog y —log S| = Flog Te. 
a 
