Ill, 112] ATTRACTION OF ELLIPSOIDS. 211 



where X is defined by 



/y2 nft nZ 



_i 



" 



To get the potential of the whole ellipsoid, we must integrate 

 for all the shells, and 



(3) V= 2>rrabc ( I 

 Jo 



o 



For every value of 6 there is one value of X, given by the cubic 

 (2), we may say X = <f> (6). 



Let us now change the variable s to t, where, 6 being constant, 

 s = 0% ds = d*dt ; and put X = fru. 



Then 



(4) V= 27rabc P 0d0 T -p 



Jo Ju v /(a 2 + 



where u is defined by 



* si 



Since 0* is thus given as a uniform function of M, we will now 

 change the variable from to u. 



Differentiating (5) by 0, 



When = 0, u = oo , and when = I,u has a value which we 

 will call cr, defined by 



a? y 2 2 _ 



+ + 



Accordingly, changing the variable, 

 (8) F= TO 6o/J ( ^ 2+( ^ +( ^}^/ --= 



The three double integrals above are of the form 



where /" () = , . 



v/(a 2 + 0(& 2 + 0(c 2 + 



142 



