198 Mr J. Brill, On Conjugate Functions [Feb. 27, 
We may remark, however, that if by any means we can dis- 
cover the distribution of & along one of the bounding curves, 
supposing those curves to correspond to two values of 7, we have 
virtually solved the problem. We shall be able to express the 
coordinates of any point on the bounding curve in terms of the 
value of & at that point, and thus we shall obtain an equation of 
the form z=f(&) which must hold along the boundary. The 
function of a complex variable that we are in search of will then 
be given by the equation z=/(w). 
7. There is one more point to which I wish to draw attention. 
In a paper in the fifty-fifth volume of Crelle entitled “Ueber die 
graphische Darstellung imaginadrer Funktionen”, Siebeck gives 
the following theorem: 
If two diagrams are such that one may be derived from the 
other by means of an isogonal transformation, then to a system of 
straight lines in the one diagram there will correspond a system 
of confocal curves in the other. 
If the transformation be derivable from the equation z= f(w), 
the straight lines lying in the w plane and the curves in the z plane, 
then Siebeck appears to prove that the positions of the foci of the 
system of curves in question are obtained by substituting the roots 
of the equation f’(w) = 0 for w in the expression f(w). Now, by 
choosing the proper transformation, we can make a given system 
of equipotential curves in the z plane correspond to a system of 
parallel straight lines in the w plane. Thus Siebeck’s theorem 
would seem to be equivalent to the statement that all equi- 
potential systems are confocal. This is obviously untrue, since 
the system of coaxal circles discussed above is not a confocal 
system. In fact, if we apply Siebeck’s method of finding foci to 
this case, we find that it gives the limiting points of the system 
which are obviously not foci*. Thus Siebeck’s statement of the 
theorem is not quite correct as it stands. It is true that the 
points obtained by putting the roots of f’(w)=0 for w in the 
expression f(w), stand in the same relationship to all the members 
of the equipotential system, but these points are not always foci, 
though in a great many cases they are. It is, however, highly 
probable that the majority of, perhaps all, confocal systems are 
also equipotential systems, it being understood that by confocal 
systems we mean systems the members of which have all their 
foci fixed including those at infinity. 
In order to illustrate this matter further, consider the trans- 
* In this case, also, Siebeck’s method fails to give the positions of the foci of the 
constituent members of the system. 
