AN ELEMENTARY CONSTRUCTION IN OPTICS. 
[From the Messenger of Mathematics, vol. vi. (1877), pp. 81, 82.] 
Consider two lines meeting at a point P, and a point A; through A, draw at 
right angles to AP, a line meeting the two lines in the points U, V respectively; 
and through the same point A draw any other line meeting the two lines in the 
P 
points U', V' respectively; also let the points a\ v' be the feet of the perpendiculars 
let fall from U', V' respectively on the line UV\ then we have 
1 11 1 
Au' + Av'~ AU + AV' 
The theorem can be proved at once without any difficulty. It answers to the optical 
construction, according to which, if UPV represents the path of a ray through 
a convex lens AP, then the thin pencil, axis U'P and centre U', converges after 
refraction to the point V', where U'V' are in lined with A the centre of the lens; 
considering as usual the inclinations to the axis as small, we have approximately 
AV' = Av, AU' = Au', and the theorem is 
1 11 1 1 
AU , + AV'~ AU + AV’ “ AF' 
if AF is the focal length of the lens. 
In the original theorem, the line UV need not be at right angles to AP, but 
may be any line whatever; the projecting lines TJ'ul and FV must then be parallel 
to AP, and the theorem remains true.