1888.] Singular Points of an Equipotential System of Curves. 315 
Thus one of the points of intersection will be the point whose 
courdinates are 
E=A+i(p—-4), N=Z, 
and this point will be in general imaginary. 
Suppose that one of the families of the system consists of 
algebraical curves having no real points of intersection. The 
number of points of intersection will be even, and it will be 
possible to arrange the points in pairs similar to that discussed 
in the preceding article. Further we see by that article that 
if we joi the points of intersection with the cireular points at 
infinity, the number of real intersections of the lines so formed 
will be equal to the number of intersections of the curves. And, 
if (&,, 7,), (E,,7,), &¢., be the coordinates of the points of inter- 
section of the curves, then the 2z’s of the said real points will be 
f(E,, 0) + oh (E,, 1) =S (E+ m), 
b(E,, m,) +p (E,, 0.) =f (E+ ™,); 
CRETE EEEOHE HEHE 
Thus the 2s of the above-mentioned points are obtained by 
substituting the roots of the equation f’(w) = 0 in the expression 
f(w). In other words they are the branch points of the system. 
4. In the preceding article we obtained a geometrical in- 
terpretation of the branch points of an equipotential family of 
algebraical curves having no real points of intersection, and we 
now proceed to the discussion of the different cases that may arise. 
We have proved that the locus of ultimate intersections of the 
family of curves consists of the collection of straight lines joining 
the branch points of the system to the circular points at infinity. 
Should this collection of straight lines form a proper envelope 
of the family, then the curves are confocal, the branch points 
being the common foci of the members of the family. There 
are a large number of known instances of confocal systems of 
equipotential curves. 
A second possible case is when all the curves of the family 
pass through a fixed set of imaginary points. An instance of this 
is the above-mentioned case of a set of coaxal circles. It will 
be found in this case that the real intersections of the lines 
joining the imaginary points of intersection to the circular points 
at infinity are the limiting points of the system. ‘This may be 
easily proved directly, or it may be deduced from the harmonic 
properties of the complete quadrilateral obtained by joming the 
four points of intersection of two conics. 
In other cases the straight lines joining the branch points 
tothe circular points at infinity may constitute a node locus or a 
cusp locus, or a locus of singularities of a higher order. In special 
cases these lines may coincide with the tangents at the nodes or 
eusps and we again obtain foci. 
