ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 427 



p = „i") f„2. _ 2a>x2i'"+2^ + v?-2 o„_2 

 t' -(H 2 ^ 2i''» + 4w + »-3 ^ 



, 2«) . 2a)-2 X 2i'~' + 2ai + *-2 .2i'^' + 2w + *-4 , - „ , 

 2.4 X 2«<-' + 4to + «-3,2« + 4w + w-5 ^ ' 



where w represents any whole positive number. 



Having thus determined all the factors of (f>, it now only remains 

 to deduce the corresponding value of H. But Ho the particular value 

 satisfying the differential equation in H, will be had from by simply 

 making therein 



since in the present case we have generally «/ = «'. 



Hence, it is clear that the proper values of 0,, di, 9^, &c. to be here 

 employed are all constant, and consequently the factor 



0102 ©3 ©s-l 



entering into (f> is likewise constant. Neglecting therefore this factor 

 as superfluous, we get for the particular value of H, 



a' 



since ,0^ = ?.= + ?/ + + ?/ = ^^ = fj , 



ga a 



and Pa represents what P becomes when p is clianged into — . 



a' ^ 



Substituting this value of Ho in the equation (25), No 10, there 

 results since a' = a'^ + h^ 



H=K.P„ f ^'"""^^ , (46) 



a' 



K being an arbitrary constant quantity. 



Thus the complete value of V for the particular case considered in 

 the present number is 



Vol. V. Part III. 3 K 



