[From the Cambridge Philosophical Proceedings, Vol. II.] 



LXV. On the Relation of Geometrical Optics to other parts of Mathematics 



and Physics. 



THE study of geometrical optics may be made more interesting to the 

 mathematician by treating the relation between the object and the image by 

 the methods used in the geometry of homographic figures. The whole theory 

 of images formed by simple or compound instruments when aberration is not 

 considered is thus reduced to simple proportion, and- this is found very con- 

 venient in the practical work of arranging lenses for an experiment, in order 

 to produce a given effect. 



As a preparation for physical optics the same elementary problems may be 

 treated by Hamilton's method of the Characteristic Function. This function 

 expresses, in terms of the co-ordinates of two points, the time taken by light 

 in travelling from the one to the other, or more accurately the distance through 

 which light would travel in a vacuum during this time, which we may call 

 the reduced path of the light between the two points. The relation between 

 this reduced path and the quantity which occurs in Cotes' celebrated but little 

 known theorem, is called by Dr Smith the "apparent distance." The relations 

 between the " apparent distance " and the positions of the foci conjugate to the 

 two points, the principal foci and the principal focal lengths, were explained * ; 

 and the general form of the characteristic function for a narrow pencil in the 

 plane of xr was shewn to be 



V= 7 i + Ml r I + / v. + * - -' ~ + & c., 



* [i.e. to the Cambridge Philosophical Society at a meeting when the results of the foregoing 

 paper LXIV. were communicated.] 



