48 



Mr. I. Todhunter on Jacobi's Theorem. 



it is obvious then that ~ cannot vanish until 3 ^_ 2 p is S reater tnan unit y* 

 that is, until p is greater than 3. Thus — is negative when p=3 ; and we 



can see that ~ is positive when p=cc . Hence ^ changes sign once as^> 



increases from 3 to infinity. 



Put^ 2 = tan 2 0, and let p = tan 2 /3 ; thus 



du I ft sin 4 . 2 _ .: 2fl J, A, 3\ o n ) Ja v 

 -7-=—. 1 — tttKP c °s — Sirrfl) 4 1 — ( 3 + - ) cos"0 I dd = — , say. 

 ^ 1* Jo cos'0 Vi 1 \ jp/ J pi 



Then we know that v is negative when p=3, and positive when £>=x . 

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



roots at least, and then the equation ~ = must have two roots at least. 



dp 



Now 



dv 



«{7, 



C p • 4 fl /i o 2fl 3sin 2 0\ 

 1 sm 01 1 — 3 cos ^ — V 



Jo V i> / 



* i?Jo 2(1 +pf 



d^v " • 



Thus — - 2 is always positive when ^> is greater than 3, and therefore — 

 efyr dp 



continually increases, and so cannot vanish more than once. 



Hence v cannot vanish more than once, and therefore u cannot vanish 

 more than once. Thus u is always negative when p has any value greater 

 than that which makes u vanish. 



13. There is also another way in which the result may be established. 

 It will be found that 



'1^(1— a? 2 ) (1— pW)dx_3 + l3p _3+l4p + 3p^ tan ~i^~ 



jo u+i^r >v Si 



Then it may be shown that the last expression will vanish once, and only 

 once, as p changes from 1 to oc . This method is adopted by Liouville 

 in an article in his 'Journal de Mathematiques ' for April 1839: the 

 article consists of observations on the memoir by Ivory, which we are 

 discussing. 



Liouville says that the value of p, which makes the last expression 

 vanish, is a little less than 2. Ivory has a formula which is equivalent to 

 this, but he does not employ it to show that there is only one value of p ; 

 for he had already, as we have seen, stated this to be obvious. From the 

 circumstance that Liouville gives a strict demonstration, it is plain that he 

 agrees with me in thinking that Ivory's statement is not obvious. 



According to Ivory, the value of p which satisfies (8) is 1*94 14. . . . 

 "We will denote this by p . 



14. I shall now show that if we ascribe to p any value greater than^? , 

 a corresponding value of r exists, which will make V vanish. 



