OF ELLIPSOIDS OF VARIABLE DENSITIES. 197 



Then by the ordinary formulas for the transformation of 

 multiple integrals we get 



sin 6'* sin df... sin 0\^dpd0^dO z ...dO t 



'a-V 



and the second number of the equation (13) by substitution will 

 become 



dp de i d0 2 ... d6 s _, p'- 1 sin 6>/- 2 sin Of . . . sin 0, 2 . (1 -*) V ., 4 

 , . t m 4) 



But since u is evanescent, we shall have p infinite, when- 

 ever x^ x 2 ,...x 8 differ sensibly from x", x^...x a " ; and as more- 

 over n s + 1 is positive, it is easy to perceive that we may 

 neglect all the parts of the last integral for which these dif- 

 ferences are sensible. Hence V may be replaced with the con- 

 stant value V ' in which we have generally 



Again, because the integrals in (14) ought to be taken from 

 0,_j = to 0,_j = 2?r, and afterwards from 6 r = to 6 r = TT, what- 

 ever whole number less than 5 1 may be represented by r, we 

 easily obtain by means of the well known function Gamma : 







/sin Of sin Of sin Of ...... sin 0. 2 dO, dO % ... dd, 1 = 



and as by the aid of the same function we readily get 



r (*} r 



- L 



the integral (14) will in consequence become 



r 



and thus the equation (13) will take the form 



