ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 417 



where for simplicity the indices at the foot of the letters ^ and a have 

 been omitted, and their accents transferred to the letters themselves. 

 Moreover all the finite integrals are supposed taken from 1 to «+l. 



By making A = in the last expression we immediately get v'<^, 

 and if for a moment, to prevent ambiguity, we write h, in the place 

 of the original «'r and omit the lower indices as before, we obtain 



V>=(l-2a2^4p+(*-«-l)2|^ (33); 



where to avoid all risk of confusion r has been changed into r" , and 

 the double accent of this index transferred to the letters ^ and h 

 themselves. 



We will now conceive the expression (32) to be written in the 

 abridged form 



the order of the terms remaining unchanged. 



If then we recollect that the accents have no other office to per- 

 form than to keep the various finite integrations quite distinct, and 

 consequently that in the final results they may be permuted in any 

 way at will, we shall readily get 



V'Va^ — VaV'0 = 



(l-Sf)(4S2 ,^„,„ .-T^>-' +22-^, X 2^-^ I 

 + 42^'x22-^.-,p^^, +22^x2^x2^-^ 



