Mr. I. Todhunter on Jacobi's Theorem. 



47 



therefore 



% 21" 2l 2 



_dg_ 3(9 + l* .-ij 3(94-70 

 dl 21' 2(l+/ 2 )/ H ' 



r 



Thus vanishes when 

 dl 



-^WWfei « 



Ivory gives these formulae with some misprints. 



Now it has been shown by Laplace that the equation (6) has one positive 

 root, and only one, namely, when I— 2*529 . ... See the ' Mecanique Celeste/ 

 Livre III. No. 20. 



11. "We now return to Jacobi's theorem. We take the integral in (3), 

 and put x for u; we put X— ii=r, and we may suppose X greater than [i, so 

 that t is positive ; and we put Xfx=p. Thus (3) becomes 



*i a :\l-a?) (l-/>V)<to _ Q ,~ 

 {(l+^ 2 ) 2 + rV}i • 



Denote the left-hand member of (7) by V ; then we propose to examine 

 the range of values of p and r which make Y vanish, supposing both p 

 and t positive. If we regard p as an abscissa, and r as an ordinate corre- 

 sponding to p, we in effect propose to trace in the first quadrant the curve 

 determined by the equation V^=0. 



12. If we put r = 0, the equation (7) becomes 



Jo — iT+^y (8) 



Ivory says : " It is obvious that there is only one value of p that will 

 verify the equation just found ; for the integral can pass only once from 

 being positive to be negative while p increases from 1 to be infinitely 

 great." 



I cannot admit that this assertion is obvious ; the result, however, may 

 be established by the following investigation : — 



Denote the integral by u ; as long as p is less than unity, u is positive, 

 and when p is infinite u is negative. Thus u must vanish once as^? changes 

 from unity to infinity : we have to show that u can vanish only once. 



If the equation u=0 could have more than one root, it must have three 



roots at least ; and then the equation ^==0 must have two roots at least : 



this we shall show to be impossible. 

 "We have 



du^_ Ci x\\ -ce 2 ) (3 + 2p-p*x 2 )dx . 



dp Jo (i+p»-y 5 



