in the Theory' of Probability. 747 



that the Boyle-Gay Lussac law holds, he finds that when v is 

 large the mean value is given by the equation 



V V7T 



but that when v is not very large 

 __ 2 v k e - v 



where k denotes the largest integer which is not greater 

 than v. 



I£ Boyle's law does not hold, then for large values of v 



V vtt V B n 



where ft is the true compressibility and ft the compressibility 

 derived from Boyle's law. 



These formulae have been applied by The Svedberg* to 

 the study of colloidal solutions. He finds that in great 

 dilution the distribution of particles corresponds very exactly 

 to the theory and that Boyle's law is practically exact for 

 dilute solutions. 



2. Having indicated some of the known applications of the 

 formula, we now proceed to a few developments which may 

 perhaps be useful in the future. Consider first the case of a 

 number of particles which carry either a positive or negative 

 unit charge. If the average number of these particles which 

 are present within the given volume is v, what is the chance 

 that at any given time the volume contains a total charge r 

 on account of the presence of particles of these types ? 



This problem is analogous to one considered by Whetham 

 in an electrical theory of coagulation, ' Theory of Solution,' 

 p. 396. In Whetham's problem, however, the electric charges 

 are supposed to be all of one sign, and the probability is 

 calculated from a different point of view with the result that 



Poisson's law .— e~ v is replaced by the simpler law (Av) n 



where A is a constant. 



If we suppose that positive and negative charges are 

 equally likely to be present, then the chance that a group of 



* "Erne neue Metkode zur Priifimg der Gultigkeit des Boyle- 

 Gay-Lussacschen Gesetz fur kolloide Losimgen." Zeitschr. fiir Phys, 

 Chemie, Bd. lxxiii. Heft 5, p. 5i7 (1910). 



3 02 



