328 



Mr. R. F. Muirhead on 



from 0, the centre o£ the spherical surface, and A2 B2 their 

 images, then, in the limit, Ai Bi and A., Bo are both at right 

 angles to Aj Ao and coplanar with it. 



Fi-. 1. 





Fig. 2. 



C2 



2 



1 A, 





r 



B2 



A? 



^ 



B,-^ 



y^° ^ 





C, 





i 





Also if Ai Bi Ci (fig. 2) are three points in a straight line 

 perpendicular to A^ Aq, then in the limit A2 Bo C2 ^^'e in ^ 

 line perpendicular to Ai A2 and coplanar with Ai Bi Oi, 

 and Ave have by similar triangles 



AiB, 

 A^^Bg 



AiO 

 A2O" 

 A^ B^ A2 Bq 



Aid 



ill ^1 -^2 '-^2 



A B 



By symmetry it is clear that the ratio .r -ry i* unaltered 



^2 -t*2 



when the line Aj Bj turns round about Aj 0, keeping at 

 right angles to it, while A^ remains fi.xed. 



Thus i£ we have any number o£ points in the plane through 

 A perpendicular to Aj 0, thej have for their images a set 

 of points forming a similar and similarly situated figure in 

 the plane through Ao perpendicular to Ai A2. 



Now consider an axial system of spherical refracting sur- 

 faces. Let Ai A2 A3 . . . A„ be a point on the axis and its 

 successive images. The points in the plane of A^ (the plane 

 through Ai perpendicular to the axis) form a figure w^hich is 

 similar and similarly placed to its first image in the plane of 

 A2, therefore also to its second image in the plane of A3, 

 and so on. Finally, we deduce that the last image in the 

 plane of A^ is similar and similarly situated to the original 

 object consisting of a plane figure in the plane of A^. Hence: 



Proposition B. — The lines joining the object-points in the 

 plane of Aj to their image-points in the plane of A-„ are all 

 concurrent in a point a on the axis, which we shall call the 

 vertex for the plane of Aj (or for any point in the plane of Ai). 



