the A.riiil Dioptric Sf/stem, 327 



sense, as abstract oeometrical conceptions snogested by a 

 very narroAV beam, and a small reoion towards or from which 

 rays converoe or diveroe ; and further we shall keep the 

 usual restrictions as to the distances of points from the axis 

 and the inclinations of rays to it being so small that we may 

 reject the squares of these small quantities. In order to 

 avoid infinitesimals in our diagrams we shall suppose these 

 to be distorted by having the measurements at right angles 

 to the axis depicted on an indefinitely larger scale than those 

 parallel to the axis. Thus the whole theory serves as a first 

 approximation to the representation of the actual phenomena. 



Several convenient new terms will be introduced in the 

 course of the article ; for the present w^e may remark that 

 we shall avoid the use of the w^ord conjugate, which is apt to 

 be misleading on account of its use in other branches of 

 mathematics to denote a relationship which is reciprocal in 

 its character. We shall speak of object-points and image- 

 points and their correspondence ; also of in-rays and out- 

 rays when we need brief terms for incident and emergent 

 rays. 



We shall suppose throughout that a strict convention as to 

 the sifins of the various quantities is adhered to. We shall 

 understand that all lines are continued indefinitely in both 

 directions, even in the case of rays belonging to media of 

 limited extent, so that for geometrical purposes we may 

 suppose the w^hole of space occupied by as many different 

 media, each with its points and rays, as there are separate 

 media in the dioptric system. 



Before entering on our special method we shall state a 

 proposition with regard to refraction at a single spherical 

 surface, omitting for brevity the proof, which is easy and 

 well known. 



Proposition A. — When a thin pencil of rays is incident on 

 a sino-le refracting surface, each rav meetino- the surface 

 nearly at right angles, it emerges as another pencil, the focus 

 of the emergent pencil being collinear with that of the in- 

 cident pencil and the centre of the sphere ; and the distances 

 of these two foci from the centre having a relation depending 

 only on the refractivity of the surface and the radius of the 

 sphere. 



Section I. 



From Proposition A it follows that if a pair of points be 

 equally distant from the centre so are their images. 



Hence if Ai Bi (fig. 1, p. 328) be two points equally distant 



