1889.] Equations with Bowndary Conditions. 347 
-(% ap ma dy) i" (= ay Ou a i" 
P 
oy ay! On 
Ou Ou Ou Ou 
; Ou _ (dw 
1.@. (= de) = & de) 
Finally, we will apply our theorem to the region PQR in 
figure (3). It then takes the form 
20u Rou Ou Ou 
Leal, Sadie | (5, An dus) = 0; 
and therefore we have 
Ou Ou i) 
R 
Quy (tp tu) =| (a, Ya dx 
3. We now proceed to show how the results of the preceding 
article may be utilized to obtain the solutions of particular problems. 
\ 
3 2, 
(B) 
1 
Hatt oe ria 
0 L 
Suppose the curves (A) and (B) in the accompanying figure to 
be the curves along which the specified boundary conditions are 
to be satisfied. On (A) take a point 1; through 1 draw 1 2 parallel 
to Oy to meet (B) in 2; and through 2 draw 2 3 parallel to Ox 
to meet (A) in 3. Let (x,, y,), (,, Y,), (#, Y,) denote the co- 
ordinates of the points 1, 2, 3. Then by the preceding article we 
have 
Ou Ou ou Ou 
dx, = -—dx, and ~— dy, = ay, 
Ga, 1 Oe, OY, “Yo 
