308 Mr J. Larmor, On Graphical Methods [May 27, 



deduce all other pairs by linear construction without any knowledge 

 of the internal constitution of the optical system. It has in fact 

 been shown by Pascal how, given five points on a conic section, 

 the other point in which any line through one of them meets 

 it again may be determined by linear construction ; and the 

 conjugate or polar theorem, known as Brianchon's, solves the 

 present problem. The lines connecting the three given pairs 

 of conjugate foci, and the two axes of the pencil, form five tangents; 

 and, simply by drawing three lines, the other tangent through any 

 point on one of the axes is constructed, and this meets the other 

 axis in the conjugate focus. 



3. There is an important case in which this general construc- 

 tion reduces to a very simple form. Suppose the problem relates 

 to refraction of the filament of light at a single interface ; or to 

 passage across an optical system whose total thickness can be 

 neglected in comparison with the focal distances of the pencil, for 

 example, eccentric refraction across a thin lens, or across a thin 

 plate of a medium with any kind of curvatures in its two faces. 

 The incident and emergent axes meet on this plate, and if the 

 incident focus is at their point of intersection, the conjugate one is 

 obviously the same point. Thus the two axes of the pencil touch 

 the conic at the same point ; therefore the conic must be a finite 

 line, and all tangents to it pass through one of its extremities. 

 The line connecting conjugate foci in all such cases therefore passes 

 through a fixed point. 



4. Single Refraction in Primary Plane. The position of this 

 point is determined if the positions of two pairs of conjugate foci 

 are known. 



For the case of a single refraction at a curved surface (including 

 of course reflexion) it is desirable to specify precisely its position, 

 as this case is fundamental in the ordinary analytical theory. 

 Now for a spherical interface there are always two exactly 



aplanatic points on each radius: the equation of the spherical 

 surface can in fact be thrown into the form rj — ^ir^ = 0, the origins 



