460 Miss Slater on the Emission of Negative 



deduced will serve our purpose equally well. We can for 

 instance assume that the class of types of well-ordered series 

 which can be formed from the terms of a limited class is 

 itself a limited class*. We can then prove that there is no 

 greatest cardinal without making any assumption as to the 

 exponential function. 



I may point out that the proofs which I have given of 

 Schroder and Bernstein's theorem, and of the propositions 

 that 



a+a=a, tf a=a and a 2 =a 



do not depend for their validity on the assumptions which I 

 have made. Those propositions are true for all aggregates, 

 and for ail cardinal numbers which exist. But we cannot 

 determine what cardinal numbers do exist without the aid of 

 some assumptions. 



From the proposition that a 2 = a we can infer that the 

 logical sum of an aggregate of mutually similar aggregates 

 is an aggregate ; but we cannot from this alone infer that the 



oo o ' til 



logical sum is an aggregate in the particular case where the 

 cardinals of the component aggregates form in order of 

 magnitude a series with no last term, unless we know 

 independently that there is a cardinal greater than any of 

 those cardinals. It is this latter case of the proposition which 

 I have actually made use of in proving that the class of 

 cardinals is unlimited. In my later paper I have expressly 

 treated this proposition as an axiom. 

 Burdwan, 17th May, 1905. 



LIII. On the Emission of Negative Electricity by Radium and 

 Thorium Emanations. By Miss J. M. W. Slater, B.Sc; 

 Bathurst Student, Neamham College, Cambridge^. 



IN" a recent communication made to the Cambridge Philo- 

 sophical Society J, Prof. J. J. Thomson has described 

 some experiments which show the existence of a new type 

 of rays in connexion with certain active substances. These 

 are negative rays of the same nature as cathode rays; 

 they differ from the ordinary /3 ray in moving much less 

 rapidly, so that their ability to penetrate aluminium foil 

 and other media is no greater than that of the positive a. 

 particles. This accounts for their not having been detected 

 by the ordinary methods of testing for ft rays. Prof. Thomson 



* This axiom is really assumed in Cantor's argument. 

 t Communicated by Prof. J. J. Thomson, F.K.S. 

 X Proc. Camb. Phil. Sot\ yol. xiii. part 1. 



