10G Mr. G. H. Livens on the JS'umber of 



under the most favourable circumstances, is perhaps not so 

 exact as one might desire. 



One of the most fundamental quantities it is for many 

 reasons desired to calculate precisely, is the density of th<* 

 electrons which take part in the ordinary phenomena of 

 conduction, and which are, therefore, more or less freely 

 moveable in the space between the atoms. The simplest and 

 first investigation of the number of these effective free 

 electrons, proceeding from a minimum of assumptions, was 

 made by Schuster*, whose conclusion is that the number of 

 free electrons in a metal at ordinary temperatures is equal 

 to the number of atoms, or exceeds that number not more 

 than three times. 



More recently an elaborate investigation of the whole 

 question concerning the number of: free electrons in the 

 metal has been undertaken by Nicholson f, with a view to- 

 discriminating between the results of the various forms of 

 the theory which have been suggested. His conclusion is- 

 that the particular form of theory proposed by H. A. Wilson! 

 is not only the only theory which provides results consistent 

 with those obtained experimentally, at least, within the limits 

 of experimental error, but also that this theory provides a 

 very exact estimate of the required number of electrons. 

 Unfortunately for this conclusion, it appears that Wilson's- 

 theory is incomplete, and the formulae he obtains and which 

 are used by Nicholson are certainly not those which should 

 follow from his fundamental assumptions. 



In a previous paper § the present writer has attempted to 

 formulate a general form of the electron theory of the 

 optical properties of metals, and results were obtained which, 

 although of: the same form as those obtained by Wilson, are ? 

 nevertheless, fundamentally different from his results. These 

 results possess the advantage of being perfectly consistent 

 with the more usual results for steady currents, and it is 

 suggested that, on the usual basis of these theories, they are 

 the only ones which can consistently be obtained. The object 

 of the present paper is to examine the formulae obtained in 

 the previous paper with a view to their application in the 

 calculation of the number of free electrons. 



On a priori grounds, it would, of course, appear to be 



* Phil. Mag. February 1904.; 



t Phil. Mag. August 1911. It is perhaps fair to add that I hear from 

 Dr. Nicholson that he no longer credits his results with the accuracy at 

 first expected. 



X Phil. Mao-. November 1910. 



§ Phil. Mag. Mav 1915, 



