752 Some Problems in the Theory of Probability. 



5. We may show by a simple extension of the previous 

 method that if z is the chance of a particle having no charge, 

 ac p the chance that it carries a charge of p positive units, y p the 

 chance that it carries a charge of p negative units, then 

 the chance of getting a total charge of r units within a certain 

 volume at a given time is equal to the coefficient of t r in the 

 expansion of 



where v is the average number of particles within the given 

 volume. In the case of an atom or molecule the volume may 

 be taken to be the probable sphere of influence of the atom or 

 molecule. 



The probable value of r is easily found to be 



v[x l -y l + 2 (x 2 -y 2 ) + 3(.r 3 — y z ) +...], 



and the probable value of r 2 is 



+y*[(xi-yi)+2(*2-2/2) + -Y- 



If we call W r the chance of getting a total charge of 

 r units, it is easy to see that W r is a solution of partial 

 differential equations of the type 



B 2 V _B 2 V 



&c. 



B 2 V B 2 V 



~dx 1 2 'da^'dz 



Tn the simple case when ^ 2 = .r 3 = .. . =y 2 z=y 3 = .. . = the 

 value of W r is easily found to be- 



W r =(ffie-'IJ(*,y/*ai)i 



the probable value of r is v(cc l —y l )^ and the probable value 

 of r 2 is 



If the probable value of r 2 is known the value of v may be 

 derived from this formula, or if v be also known and x 1 = y 1 , 

 the formula mav be used to estimate the value of x x . 



