Mr. I. Todhunter on Jacobi's Theorem. 51 



positive or it might be negative : we cannot assert at once, as Ivory does, 

 that ^ must be negative. Suppose, for example, that V stood for p 2 — r 2 — 1 ; 



dV . . , ' '.. ' ' 



then would be negative for positive values of r, and p would continually 



increase with r. 



19. As I have already stated, Ivory accepted the correction made by 

 Liouville ; but in the two pages in which the mistake is acknowledged 

 other untenable assertions are advanced. It was in effect necessary for 

 Ivory's purpose to trace the curve determined by V=0 in the first quadrant ; 

 but instead of demonstration such as we have supplied in Art. 14, Ivory 

 gives unwarranted assertions of the kind already noticed in Art. 18. 



We have 



% = - C + 2P-P^) (1 +K) + 2prV}^ 



. . dV 

 from which it follows that, whatever positive number r 2 stands for, — 



dp 



is negative for all values of p that make 3 + 2p— p 2 positive, that is, for 



dV 



all values of p less than 3. This Ivory gives, and so far he is correct ; — 



is certainly negative, and not zero, so long as p is less than 3. Ivory 



wishes to show that — can never be zero. He takes the differential equa- 

 dp 



tion 



dYj . 6?V 7 n 

 ~—dp+--Tdr=0 ; 

 dp rdr 



he says, "Further, in the differential equation cannot be zero ; because, 

 r 2 increasing without limit, ^—rdr is essentially positive." 



TU/T 



This is quite unsound. Regard^? as the abscissa and r as the corre- 

 sponding ordinate of a curve determined by V==0 ; then assuming that 



is always positive, yet ^lL may vanish ; that is, there may be a point 



or points on the curve at which the tangent is parallel to the axis of abscissae. 

 Suppose, for example, that V stood for 



r 3 — p s -\-4ap 2 — 5a 2 p — b 3 ; 



then is always positive, ^ is negative when p is less than a, and 

 dr dp 



vanishes when p—a or — . 



1 3 



I do not assert that ^ can vanish in the present case ; I only maintain 



that Ivory's argument to show that — . cannot vanish is unsound. I shall 



dp 



E 2 



