316 Mr J. Brill, On the Geometrical Interpretation of the [Nov. 26, 
In some cases a portion of the system of lines we are con- 
sidering may constitute a proper envelope, while the remaining 
portion comes under one or more of the other cases we have 
mentioned. In this case the curves of the system will have only 
a certain number of their foci coincident. 
5. In Article 3, we supposed that the successive curves of 
the family in question did not intersect. If however the curve 
m=a cut the curve 7 =a+ da, then the equation /’ (&E +74) =0 
will be satisfied by real values of & and it is evident that the 
two curves will intersect in branch points. Thus if each curve 
of the family cut the consecutive one, all the curves of the family 
will pass through a set of fixed points, these points being branch 
points of the system to which the family belongs. The set of 
fixed points through which the curves of the family pass need not 
however include all the branch points of the system. 
It is easy to shew that an equipotential family of curves cannot 
have a real envelope which does not consist of a system of discrete 
points. We have shewn that if two consecutive curves of the 
family intersect, they intersect in branch points; and it remains 
to be proved that these branch points are discrete, and do not form 
a locus. Let 
J (E+) =f, (E 0) +7, & 7). 
Then in order that f' (w) = 0, we must have 
Pe.) — 0 andy, |e) 0: 
These equations represent two curves in the w plane which are 
subject to the condition that they should intersect orthogonally. 
Hence their points of intersection must be a system of discrete 
points; as these points could only form a locus in case the two 
curves should coincide, which is impossible. Thus the branch 
points of the family, which are the corresponding points in the 
z plane, also form a discrete set. 
There is however one special case in which the members of an 
equipotential family of curves may have a real envelope which 
is not a point. This occurs when all the members of the family 
touch the line at infinity. It is also conceivable that the members 
of an equipotential family may have the line at infinity for a 
node locus, a cusp locus, or a locus of singularities of a higher 
order. 
Other special peculiarities may arise through infinity turning 
up as a branch point. An instance of this will be given 
further on. 
6. We now proceed to illustrate the above theory by the 
discussion of some particular cases, taking first that of a family 
of straight lines, A straight line has no singular points, and 
two real straight lines always meet in a real point. Thus it is 
