1888.] Singular Points of an Equipotential System of Curves. 317 
evident that the only possible case is that of a family of straight 
lines passing through a fixed point. The case of a family of 
parallel straight lines is of course included in this. 
7. Passing on to curves of the second order we have the 
well-known case of a system containing two families of confocal 
conics, the one family consisting of ellipses and the other of 
hyperbolas. Under this are included two particular cases. In 
the first of these one of the foci passes off to infinity and we have 
two families of confocal coaxal parabolas. In the second the foci 
coincide and we have a family of concentric circles with the ortho- 
gonal family of straight lines. 
Since proper curves of the second order have no singular 
points, it is evident that in order to arrive at a complete enume- 
ration of the equipotential families of the second order, if we 
leave parabolas out of account, we have only two more cases 
to examine. Firstly, the case of a family of conics having one 
common focus and passing through two fixed points, and secondly 
the case of a family of conics passing through four fixed points. 
It is understood that the points of intersection are not necessarily 
real. There is no known family belonging to the first of these 
eases, but there are several families belonging to the second. 
In the first place we have the system consisting of two families 
of coaxal circles. The circles belonging to one family intersect 
in two real points, viz. the limiting points, and in two imaginary 
points, viz. the circular points at infinity. The circles belonging 
to the other family intersect in four imaginary points, viz. the 
circular points at infinity and two imaginary points lying on the 
radical axis. A special case of this is the system consisting of 
two orthogonal families of circles, all the circles of each family 
touching one another at the origin. 
Secondly, we have the system consisting of two orthogonal 
families of rectangular hyperbolas. In this case all the curves 
of each family have double contact at infinity. There is, however, 
a peculiarity about this case. The points in which the members 
of one of the families intersect are not coincident with those 
in which the members of the other family intersect. This pecu- 
liarity arises from the fact that we have not certain discrete 
points at infinity for singular points, but infinity itself turns up as 
a singular point. 
Thirdly, we have the family of rectangular hyperbolas that 
consists of the orthogonal trajectories of a family of confocal 
Cassinians. These byperbolas intersect in two real points, viz. 
the common foci of the Cassinians, and in two imaginary points 
lying on the line which bisects at right angles the line joming the 
said foci. 
There is one more known family of equipotential curves of 
the second order. It consists of a series of parabolas having a 
VOI, VL PT. V. 23 
