1888. | and Eyurpotential Curves. 197 
The similarity of this equation to the equation 
FF) tog PH OM 4 
(ae + oye) 8 Vaak * Oy?h ~ 
given above, indicates the possibility of deriving solutions suitable 
for free surfaces from known solutions applying to cases in which 
the boundaries are fixed*. 
6. It is always possible to find an indefinite number of systems 
of equipotential lines each having a given curve for one of the 
curves of the system. We have only to express the coordinates 
of a point on the curve in terms of a single parameter, which may 
be done in an indefinite number of ways. Thus, suppose that we 
have «= (a) and y=W(a); then 
z= + iy= $(2) +i). 
2 = (E+ in) + ip (E+ tn). 
This will give us two systems of equipotential curves &=const. 
and »=const., the series 7=const. containing the given curve, 
since that curve corresponds to the value 7 = 0. 
Now write 
Further, if we have two or more systems of equipotential 
curves, each containing a given curve, we may combine them with 
the aid of the theorems contained in Art. 1 and deduce new 
systems containing the said curve. 
In a large number of physical problems, however, we find that, 
as a first step to the solution of the problem, we have to discover 
a system of equipotential curves which contains two given curves; 
and, although we can write down the functions of a complex 
variable corresponding to as many systems as we please that 
contain either of the two curves, yet we shall often find it im- 
possible to hit upon that particular one which corresponds to the 
system containing both the curves. What is greatly to be desired 
is the discovery of a synthetic method that would enable us to 
obtain the equipotential system containing two givencurves. This 
is, of course, a very hard problem, but I am inclined to think that 
much light might be thrown upon it by the study of general forms 
like those given in Articles 3 and 4. The only methods of a 
synthetic character that we now possess are the method of Kirch- 
hoff, alluded to above, and the method of images. The method of 
Kirchhoff, however, is very limited in its application, and the 
method of images becomes in many cases too cumbrous for use. 
* We have to take a value of h, derived from some known case of fluid motion, 
and split it into two parts such that one part and the logarithm of the other part 
each satisfy Laplace’s equation. 
