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



the case in which both p is greater than 4i and T 2 + '—^L+ K 1 } 



7 2 d~V 

 is less than : and we have to show that — — is negative in this case. 



22 dp 



Put p=~ ; thus Y becomes a function of q and r ; and we have to show 

 that cannot vanish when q lies between and ^ , and r 2 + 

 + 9gg+ 2 > is less than ? so that r ¥ + (3g + Dg + 9(3g + 2)g» |g lesg 



10 22q~ 2 10 



22* 



(g 2 — a? 2 )^ 



7 



than — . We have 



where D 2 stands for <f + 2^ 2 + a? 4 + § VW* ; therefore 



c? g 



Separate the integral into two parts, one. extending between the limits 

 # = and # = and the other between the limits x — q and #=1. It is 

 obvions that the second part is positive, so that we have only the first part 

 to examine ; this is 



and we shall show that this is positive. 

 Let Q stand for 



- <f + (3<Z + <f - r¥K + (3 + 2q + 3qr 2 X ; 

 then we shall first show that 



'2 



{ - 2 3 + (3ff + <f - r¥K + (3 + 2? + 3<zr> 4 J «fe?, 



J: 



(1— OQtftf 



is positive ; and next that 



jo D 5 

 is positive. 



Now I ? (1 — a?)Q,dt6 will certainly be positive if 



(l-^){-?H(3? + ? 2 -rYK + 3 2 r¥}^ 

 is positive. The last integral is 



-4-|) + (3 2 + f /- rY )(|-i) + v(£-i). 



