1889. ] Equations with Boundary Conditions. B45 
boundary condition to satisfy along each of two given curves. In 
what follows I have endeavoured to adapt Riemann’s method to 
obtain solutions of problems im which the boundary conditions are 
of this character. I have not succeeded in making the process 
purely synthetical, and the portion of the work that is of a ten- 
tative character will in general be found to amount to a very 
difficult piece of algebra. I think, however, that this will be 
found to be a simpler matter than the guessing of the transcen- 
dental function that gives rise to a solution suitable to a given 
form of boundary. Also I am of opinion that by studying Rie- 
mann’s methods of working we shall eventually be enabled to 
extend and to adapt them so as to render the process purely 
synthetical. 
2. The majority of the equations that turn up are not of a 
form suitable for treatment by means of Riemann’s method, but 
this difficulty can be surmounted in a large number of cases with 
the aid of transformation. Thus consider the equation 
Ow dU 
Oa?” Oy" 
If we write €=x2+ wy and »=a#—wWy, the transformed equation 
becomes 
Cia 
O£0n 
which is of a form suitable for treatment by Riemann’s method. 
If, therefore, the equations of the bounding curves when expressed 
in terms of & and 7 are real, and the values at the boundaries of 
the dependent variable, or the values of its normal variation, as 
the case may be, are also real; then the transformed problem will 
be capable of treatment by the method in question. We proceed 
to show how the method may be adapted to suit this particular 
equation. 
We have the theorem 
Ou Ou Ou 
lee fo 1 |(5 tse): 
the line integral being taken round the boundary of the region 
throughout which the surface integral is taken. Hence, if we have 
Fu 
Ody — 
throughout the said region, we have also 
Ou ay 
| (ay meas »)=0, 
