ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 405 



x^ — Xi" = u'p COS 6i ; x-i—x" = u'p sin Qx cos 0^ ; Xi—x^'—u'p sin 0, sin 02 cos 03> &c. 

 until we arrive at the two last, viz., 



«,_! -x[^-^ = u'p sin^i sin ^^ sin0,_2 cos0,_i, 



X, — ar," = «'/o sin ^1 sin 02 sin 0,_2 sin 0,_i; 



u' being, as before, a vanishing quantity. 



Then by the ordinary formulas for the transformation of multiple 

 integrals we get 



dxi dx-i dx, = u''f/~^ sin^i'"^ sin 02*"' smO^^.^dp d6i dOi...dd,.i, 



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



f dp d9i de, d9,_,p'-' sm9r'' sin 9,'-' sin e,_2 . (1 - ») r' 



/ »+i (1*)- 



But since u' is evanescent, we shall have p infinite, whenever x^, Xi,...x, 

 differ sensibly from x", x^',,..x"\ and as moreover w — * + l is positive, 

 it is easy to perceive that we may neglect all the parts of the last 

 integral for which these differences are sensible. Hence V may be 

 replaced with the constant value VI in which we have generally 



Jbf ^^ vUf • 



Again, because the integrals in (14) ought to be taken from 0,_, = o 

 to 0r-, = 27r, and afterwards from 0,. = O to 9r = -n-, whatever whole number 

 less than 5—1 may be represented by r, we easily obtain by means of 

 the well known function Gamma: 



» 



/sin^i'-'' sin 02'"' sin 03'"' sin0,_2C?0,</02...c?0,., = ^ZL; 



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



r» — * + l> 



f P'~'dp _ V2/ V 2 ) 



Wi + ,f-^ 2r(^) 



3g2 



