1889.] Equations with Boundary Conditions. 349 
Now the value of du/dy at any point of one of the bounding 
curves can be expressed in terms of the y of that point, as also 
the value of du/d# at the said point can be expressed in terms 
of the w of the point. And it is obvious that du/oy, is the same 
function of y, as du/dy, is of y,, and also that du/dw, is the same 
function of w, as du/d#, 1s of w,. And we have already established 
that du/dy is the same function of y as du/dx is of « What 
we have to do, therefore, is to express the ratios 
dy, : dy, ; dz, :dz,, 
found as above, in the form 
x (Y:) > X Ys) + x) + x (4), 
with the aid of the three equations given above. We are then 
at liberty to assume that 
oO i, Oe ou 
oy x (y) Ou x (x) 
along the curve (A). It will be found that these assumptions 
enable us to obtain a solution. And we know that with a given 
form of boundary only one solution is possible; and although 
there are an indefinite number of solutions corresponding to 
boundaries including portions of the two curves as parts, yet it 1s 
reasonable to suppose that the simplest result that can be arrived 
at will in general apply to the case in which the boundary consists 
of the two complete curves, and does not involve any portion of 
any other curve. 
Since along the curve (A) we have the relation 
Ou Ow 
the third formula of Article 2 becomes 
poh evou 
—_— — ———s d = 
Ug — & | a Yy da, 
where a is the constant value of uw along the curve (A). This 
formula will enable us to complete the solution as may be seen 
from the examples worked out below. 
4. Our theorems are also very easily adapted to other forms 
of the boundary conditions, besides that discussed in the pre- 
ceding article. Thus suppose that we have certain expressions 
given for the values of the dependent variable along the bounding 
curves, which are not constant all along those curves. Then, 
supposing that the functions expressing them remain real in 
