xlvi Trans. Acad. Sci. of St. Louis. 
Consider an apparatus consisting of a smooth wire and two 
small rings, P, and P,, not on the wire, all rigidly attached to a 
frame. The wire is in the form of any curve (plane or twisted), and 
on it a small weightless ring R is capable of sliding without friction. 
A weightless and perfectly flexible string with a weight at one end, 
after passing through the rings P, and R, has its other end attached to 
P,. The string slides through the rings P, and R without friction. The 
ring R, under the action of the forces which act upon it (i. e., the 
tension of the string and the reaction of the wire) will be in equil- 
ibrium at certain points P of the wire, If now the rings P, and P, be 
replaced by a point source of light and the eye of an observer respect- 
ively, the observer will see images of the light (i. e., actual brilliant — 
points”) at certain points Q of the wire. The first theorem of -this 
paper is that the points P and Q are identical. 
Let us now think of a weightless and perfectly flexible string 
with both of its ends attached to fixed points P, and P, On the 
string a small heavy ring R is capable of motion without friction. 
Consider this apparatus as being situated in any field of force for 
which a force function exists. The ring R under the action of the 
forces which act upon it (i. e., the weight of the ring in this field 
of force and the tension of the strings) will be in equilibrium at 
certain points P of the field. If now the points P, and P, be replaced 
by a point source of light and the eye of an observer respectively, the 
observer will see at P an image of the light (i. e., an actual brilliant 
point”) in the equipotential surface which passes through P. If no 
force function exists, the point P will be a virtual extra brilliant 
point” of the line of force which passes through P. If, in particular, 
For the definitions of the different kinds of brilliant points see 
Transactions of the American Mathematical Society, Vol. IX, No. 2, 
pp. 245-279. 
