50 Mr. W, Makower on the Molecular 



certain analogy with the relation of the transfinite aggregate 

 of type « to the finite aggregates. 



10. 



I will now sum up the results of my investigations on 

 the transfinite numbers, published in three papers in this 

 Magazine. 



The main result is that any aggregate, the cardinal number 

 (or type) of which is not self-contradictory, can be well- 

 ordered. A closer consideration of the proof given in my 

 first paper led (in the present paper) to a proof of the 

 universal validity of Cantor's third principle, and hence of 

 the non-existence of ordinal numbers and Alephs which 

 transcend all those defined or indicated by Cantor ; and (in 

 the first and second papers) the main theorem led to final 

 forms that are to be given to the results of adding and 

 multiplying any two transfinite cardinal numbers. The 

 results on exponential numbers are not final, but one of the 

 theorems on exponential numbers (§1 of the present paper) 

 has been shown to have an important bearing on the theory 

 of functions. 



The Manor House, Broadwindsor, Dorset. 

 September 6th, 1904. 



IV. The Molecular Weights of Radium and Thorium 

 Emanations. By Walter Makower, B.A., B.Sc* 



R 



Part I. 

 The Molecular Weight of Radium Emanation. 



1. Introduction. 



UTHERFORD and Brooks (Trans. Roy. Soc. Canada. 

 1901 ; Chem. News, 1902) have determined the rate 

 of diffusion of the emanation from radium into air by a 

 method similar to that employed by Loschmidt in his inves- 

 tigations on the coefficient of interdiffusion of gases, and 

 deduce that the molecular weight lies between 40 and 100. 

 It is important to know the molecular weight with greater 

 accuracy, and it was with the object of solving this problem 

 that the present investigation was undertaken. During the 

 course of the work, Curie and Danne (C. R. cxxxvi. p. 1314, 

 1903) have published some observations upon the rate of 

 diffusion of the emanation from radium through capillary 

 tubes of different lengths and diameters, and find for the 

 coefficient of interdiffusion between the emanation and air 



* Communicated by Prof. J. J. Thomson. 



