Mr. I. Todbunter on Jacobi's Theorem. 



45 



so that 



2 B-C . B\ 2 -<y 



Put in the values of B and C, and this reduces to 

 o_3Mfi u 2 (l-u 2 )du 



so that w 2 is positive. 



8. Before considering Jacobi's theorem, we will advert to the case in 

 which ju 2 =X 2 . 



We have from (1) 



let p be the density of the ellipsoid, then 



m _4tt^ 3 (1+X 2 ) 

 3 



2 



Put q for 4xp ; thus 



?=3 1 oW (4) 



Here we have changed u into cc, and \ into I, in order to have the same 

 notation as Ivory has. Ivory says : 



" From the equation (4) we learn that q will be known when I is given, 

 or that every spheroid of a determinate form requires an appropriate 

 velocity of rotation. 



" The inspection of the same equation is sufficient to show that q is 

 positive for all values of I 2 ; and as it vanishes both when I 2 is zero and 

 infinitely great, it must pass at least once from increasing to decreasing, 

 or it will admit of at least one maximum value. By differentiating with 

 regard to I we obtain 



2ldl Jo (1+/V) 3 {!>) 



from which formula we learn that is positive between the limits l 2 =0 



and / 2 = 1 ; that it will consist of a positive and a negative part when I 2 is 

 greater than 1 ; and the positive part decreasing while the negative part 

 increases, that it will ultimately be negative when I 2 is infinitely great. 



It follows therefore that can be only once equal to zero, and conse- 



quently that q can have only one maximum value, while I 2 increases from 

 to oc ." 



This is quite unsound, because the words which I have put in italics are 

 untrue. There are two ways of separating the integral into a positive part 

 and a negative part. We may take for the positive part the integral 



