Mr. I. Todhunter on Jacobi's Theorem. 49 



Let such a value be ascribed to p ; then from Art. 12 it follows that, with 

 this value of p and with r=0, the value of V will be negative : we shall 

 now show that by taking t large enough V will be positive, 



Y-C 1 ^ 2 (1-^ 2 )(1-JQ 2 ^ 2 )^ . 

 Jo {l+(2 i? + r> 2 + i? V}i ; 



thus the sign of V is the same as that of 



r' 1 x\\-x 2 ) {\-p 2 x 2 )dx 

 o (c 2 + a? 2 +p 2 cV)* 



1 



where c 2 stands for 



2p+r 



"When c is made small enough, the term p 2 c 2 x l may be neglected in com- 

 parison with x 2 ; and so the sign of the integral will become the same as 

 that of 



f 1 x\\-x 2 )dx C l pV(] -x 2 )dx 



Jo (c 2 +Of Jo ( c 2 +OI 



The first term is infinite when c=0, and the second term is finite ; thus 

 the sign of the integral is positive when c is small enough. Since V is 

 negative when r=0, and is positive when r is large enough, it must vanish 

 for some intermediate value of r. 



15. Moreover, when c is very small, we have approximately 



J*. 



Kv\l-x 2 )dx ,23, 2 



_ 1 1 = log _— - = log — 



o ( c 2 + t r 2 )f & c 2 b cesje 



' x p 2 x\l — x 2 )dx_ p 2 

 fo (c 2 + x 2 )% 



Thus to make V vanish when r is very large, we have approximately 



therefore 



so that approximately 



ce s/ e 



2(2p + r 2 ) = e^2; 



16. We shall next show that corresponding to a given value of p there 

 is only one value of r which will make V vanish. 

 Put 



A»=(l+JMO a + rV, 

 P 2 =(l+_p) 2 + r 2 ; 



then 

 Thus we have 



VOL. XIX. 



(l-O (1-2jV)=A 2 -=PV. 



Y C l x 2 dx p2 fi^V.r 

 V ~Jo ~A~" r Jo ^ ' 



