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
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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
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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.