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 P2 and R, has its other end attached to 

 Pi. 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 Pj 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 Pi and Po. 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 Pi 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. 



