318 Mr J. Brill, On the Geometrical Interpretation of the [Nov. 26, 
common focus and passing through a fixed point. And, since 
it is possible to draw two parabolas having a given focus and 
passing through two fixed points, it is evident that we shall 
obtain two sets of parabolas satisfying the given conditions. 
These two sets must however be considered as two distinct 
families of equipotential curves*. 
8. We may remark that it is theoretically possible to discover 
all the families of equipotential curves that belong to the cases 
enumerated in the preceding article. For example, take the 
case of a family of conics passing through four fixed points. The 
most general form of the equation representing such a family is 
an’ + 2hay + by’ + 29a + 2fy+eo+nr (Aa? + 2Hxry + By’ 
+2Ga+2Fy+C)=0, 
where A is the parameter of the family, and the other coefficients 
are constants. The condition that this may represent an equi- 
potential family of curves is that the expression 
Sone: 
Ox oy ox? = Oy” 
may be a function of X. If the value of this expression be calcu- 
lated, it will be found that the numerator and denominator are of 
the sixth degree in # and y, and consequently the said condition 
will be of the form 
2) =) 
(a = _ pvr+qV+rr+s 
FON DBS Oher iy oe 
On Oye 
Substituting for X, reducing, equating coefficients, and eliminating 
p,q 7, 8, P, Q, R, S from the resulting equations, we should at 
length obtain the necessary relations that must exist among the 
constants a, b,c, f, 9, h, A, B, C, F, G, H in order that the family 
of conics may be equipotential. } 
Similarly it would be theoretically possible to discover if there 
were any equipotential families consisting of conics having a~ 
common focus and passing through two fixed points. We could 
also test the remaining case of parabolas, viz. a family of parabolas 
passing through three fixed points. 
It is to be observed that the work for discovering the equi- 
potential systems among the higher plane curves is also theoreti- 
* This case may be derived from that of a family of straight lines passing 
through a point by means of the transformation w?=az, the straight lines lying 
in the w plane and the parabolas in the z plane. If we look at the case from 
Riemann’s point of view, we must consider the two sets of parabolas as traced 
upon different sheets of the Riemann’s surface belonging to the given trans- 
formation, 
