ceeds is intelligible, but is unwarranted. For, granting that — is always 



52 Mr. I. Todhunter on Jacob? $ Theorem. 



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presently demonstrate that -j— cannot vanish. 



20. Ivory concludes the two pages in the Philosophical Transactions 

 for 1839 thus:— 



<f If the sign of be changed, the result will be positive ; and hence, 



observing that do is contained between and 1, we obtain a condition between 

 any two values of p and t 2 that satisfy the equation (7), namely, the ex- 

 pression 



(3+p-p 2 )(l+p) + 2pr 2 

 must be a positive quantity, or, which is the same thing, 



This is obscure in its commencement ; but the statement to which it pro- 



dp 



to be negative, this will be secured if 



(3 + 2p-|>V) (1 +px z ) + 2p r y 

 is always positive ; that is, if 



3 + 2p + (3p +p 2 + 2pr 2 K -p\v 4 

 is always positive ; that is, if 



cc 



is always positive. And as at lies between and 1, this condition is cer- 

 tainly secured if 



3 + 2p -f 3p -\-p 2 + 2pr 2 is greater than p* . 

 According to Ivory's statement it would be necessary that 



3 + 4p + 2pr 2 should be greater than^> 3 ; 

 whereas we see that it would be sufficient that 



3 -f- 5p +p 2 + 2pr 2 should be greater than p 3 . 

 Ivory's second statement is inconsistent with his first, but becomes con- 

 sistent with it if we change + 3 to — 3 ; if, however, his second statement 

 is to be taken as what he intended, and his first statement corrected to 

 agree with it, his error is aggravated. 



In fact, however, I doubt whether any such necessary criterion as Ivory 



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proposes can be easily deduced from the value of ~. For, granting that 

 ^ is always negative and never zero, it will not follow that every ele- 

 ment in the integral which expresses ^ must be negative, but only that the 



