32G • Mr. R. F. MuirlieuJ on 



5. 



We may finally notice the fact that the property of an 

 analytic function of not being more than K-valued (Cantor, 

 Yivanti, Yolterra, Poincare) has no influence on the cardinal 

 number of the aggregate if the function has K variables. 

 In fact, if the function has v variables (where v is any finite 

 cardinal number), the cardinal number is 



but if the function has ^| variables, the number is 



and this would not be altered if b were equal to C. 

 63 Chesterton Eoad, CamlDridge. 



XLI. T7ie Axial Dioptric System. 

 By R. F. MuiEHEAD, M.A., B,Sc* 



Preliminary. 



THE subject of this article is one which has received 

 much attention from mathematicians both before and 

 after the publication of Gausses Dioptrische UntersucJmngen, 

 in which the idea of characterizing the axial dioptric system 

 by means of four Cardinal Points w^as first stated and applied. 

 Most of the papers on the subject \vhich have come after 

 Gauss's have been based on his fundamental conception, and 

 have aimed at simplification or further development. 



The present essay adds one to the number of these. If it 

 has merit that will be found, I believe, chiefly in the sim- 

 plified geometrical treatment of the fundamental theory given 

 in the first section, which contains in a short space a direct 

 geometrical proof of the existence and main properties of the 

 Cardinal Points. The later Sections which aim at greater 

 generality of treatment may be found interesting from the 

 mathematical point of view. 



By an Awial Dioptric System we mean a set of homo- 

 geneous media w^hose mutual boundaries are spherical surfaces 

 ha^dng all their centres on a straight line called the axis of 

 the system. 



We take the conceptions of Ray and Focus in their usual 



* Communicated by the Author. 



