44-47J The Method of General Dynamics 43 



this sphere satisfies equation (67), so that points of which the coordinates 

 satisfy equations (67) and (78) simultaneously lie on a sphere of (n 1) 

 dimensions and of radius equal to that of the former sphere. 



We see then that the region of the generalised space in which equation 

 (67) is satisfied and in which K < K , is a sphere of (n 1) dimensions, and of 



radius \/\- -)> provided that the e's are finite everywhere inside this 



V \ n ' 

 sphere. 



46. In space of s dimensions (, 2 ...|:) the volume of a sphere of 

 radius R is 



where the integration extends over all values of the variables which are 

 such that 



The multiple integral (79) is of the type associated with the name of Dirich- 

 let, and its value is known to be* 



s 



R s (80). 



47. It follows that the volume of our (n l)-dimensional sphere of 



radius A /( -^ -) will be 

 V V n J 



n ] 



2 



By differentiation with respect to K , it follows that the volume for which K 

 lies between K and K + dK is 



n - 1 n - 1 



and in virtue of the general property of the Gamma-function which is 

 expressed by the equation 



this may be written in the simpler form 



-i i 



" ' l } ' K 9 ~*~dK (82). 



n / 



* Williamson, Integral Calculus, p. 320. 



