1888. ] and Equipotential Curves. 189 
If for the two first-mentioned families we take the two con- 
jugate families derived from the same function of a complex 
variable, this reduces to the well-known theorem: 
The isogonal trajectories of a family of equipotential curves 
themselves form an equipotential system. 
It is evident that our theorem may easily be extended to suit 
the case where we have n families of equipotential curves traced 
on the plane. In this case we should have n curves intersecting 
mea clven point; and 3), 3,,...:.. ,S, would be the angles made 
by the tangents to these curves, at their point of intersection, 
with the positive direction of the axis of 2 The angle made with 
this same line by the tangent at the given point to the particular 
curve of the derived system that would pass through it, would be 
g 
Mb an Wig acaeee tM 
M,+M, +... PHD, 9 * 
WEE M1)3.M2, 202. , M, are given constants. And, if we wish 
to generalize still further, we may make this last angle equal to 
i 
ig aes ora snd oae + 770. S.)\, 
where p is another given constant. 
2. With the aid of the theorems of the preceding article, 
we may obtain several of the known systems of equipotential 
curves by combining systems of concentric circles. Thus, suppose 
that we have two systems of concentric circles, S and S’ being the 
centres of the two systems. Then, if two circles, one belonging to 
each system, intersect in the point P, the bisectors of the angles 
between the tangents at P to the two circles will bisect externally 
and internally the angle between SP and S’'P. Thus we should 
have two families of confocal conics; and, since these two families 
are orthogonal, it is evident that they are both derivable from 
the same function of a complex variable. Now a series of con- 
centric circles, having their centre at the point (a, 0), is given by 
the equation w = log (¢—a), from which we deduce 
log ow — log (z —a). 
Thus, in order to discover the function of a complex variable 
which gives rise to the two systems of confocal conics, we have 
to solve the equation 
dw 
log ge 5 log (2 —a)— : log (z+ a). 
14—2 
