216] 



General Dynamical Theory 



183 



If r is the length of a principal axis of the section, the plane (444) must 

 be a tangent plane to this cone, and the condition for this is the vanishing 

 of the bordered determinant 



I 



fa* 



A' = 



r- 



' fa 



' 8/V + 



r* 



, ..., 



The equation A' = accordingly gives the lengths of the principal semi- 

 axes. 



In the expansion of A', the coefficient of the highest power of is 



where the summation extends to all the variables. The term independent of 

 is of course obtained from A' by omitting terms containing - a , leading to 



a new determinant, say A". Hence if we denote the product of the semiaxes 

 by II (r), we have 



The volume of the 2n 1 dimensional ellipsoid is now found (as in 46) 

 to be 



If we denote this volume by H, and substitute in the value just obtained 

 for H the value of II (r) given by equation (448), we obtain an equation of 

 the form 



where C does not involve $ t , being in fact given by 



_ (27T) 



.(449). 



