310 M7- J. Larmo7% On Graphical Methods [May 27, 



The following construction, ascribed to Newton, is given by- 

 Barrow, in his Lectiones Opticae^, and is no doubt the earliest 

 solution of the problem : 



Draw AR at right angles to the incident ray meeting GP^ in 

 R, Pi being the incident focus ; and draw AQ perpendicular to 

 the refracted ray on the side towards C, such that 



AQ : AR = AAi : ^A', 

 then the primarj'^ focus conjugate to Pj lies on CQ. 



5. As a corollary we have a construction for the centre of 

 curvature at any point P of a Cartesian oval of which S and H are 

 foci. Draw any circle, centre 0, touching the oval at P and 

 meeting PS, PH in L, M; find the point R in which the line 

 joining the middle points of these chords meets the bisector of the 

 angle LOM ; let SH meet PR in X ; draw XF parallel to LM 

 meeting PS in F, and draw FG perpendicular to PS meeting the 

 normal in G, which is the centre of curvature required. 



When the oval degenerates into a conic section, this becomes 

 a well-known construction : from G the foot of the normal draw 

 GF perpendicular to it, meeting PS in F, and draw FG perpen- 

 dicular to PS meeting the normal in G, which is the centre of 

 curvature required. 



6. Surface of Double Gurvature: principal incidence. The 

 relation between the positions of conjugate foci in the secondary 

 plane is of the same kind as above ; they are in a line through G. 



These results may be at once extended to the case of any beam 

 incident in a plane of principal curvature of a non-spherical surface, 

 provided the focal lines of the beam are in and at right angles to 

 this plane. The relation between the conjugate primary foci is 

 given by the above construction by aid of the circle of curvature 

 in the plane of incidence ; while the line joining conjugate 

 secondary foci always passes through the centre of the circle of 

 curvature in the perpendicular plane, 



7. General case of single interface. The solution of the 

 problem of the determination of the form of a general pencil 

 after refraction at an interface of double curvature had been 

 achieved long ago by Malus ; but until Maxwell's adaptation ^ of 

 the Hamiltonian method of the characteristic function of the pencil, 

 the theory remained too lengthy and complicated to be of service in 

 optical applications. It is now well known, after Maxwell, that if 

 the rays of the incident pencil are normals to the surface 



- ' 2Ai 2Pi C, +--.-^, 



^ See also Newton's Optical Lectures, read in 1669, published posthumously in 

 1728. 



2 Maxwell, Proc. Lond. Math. Soc, iv. 1872 ; vi. 1874. 



