Mr. I. Todhunter on Jacobi's Theorem. 



53 



aggregate of the positive elements, if such there be, should fall short of the 

 aggregate of the negative elements. However, be this as it ma}', there 

 can be no doubt that the specific criterion which Ivory proposes is quite 

 unsupported by demonstration. 



When p and r are very large, the relation between them is approxi- 

 mately that given in Art. 15. 



21 . In Arts. 14 and 1 6 it is shown that for every given value of p greater 

 than p , there is one, and only one, value of r which will make V vanish. 

 We shall now show that corresponding to every value of r there is one, 

 and only one, value ofjj which will make V vanish. 



Whatever be the given value of r, it is obvious that V is positive when 

 p = 0, and negative when p is large enough : thus there must be some in- 

 termediate value of p which makes V vanish. 



We have then to show that there is only one such value ; to show this 

 we shall demonstrate the proposition which Ivory asserted on insufficient 



grounds, namely, that ^ can never vanish simultaneously with V. 



g = _ j*i { 3 + 2p + (3p +f- + 2pr> 2 "i> V } 5 

 dV 



and we know that — cannot vanish so long as p is less than 3. See 

 Art. 19. 



We shall first show that the limit p = 3 may be changed to a larger 

 limit. 



If the integral 



1 d? 4 (l -* 2 ){3 + 2p + (3p -f-p 2 + 2pr)x 2 -pKv l }d.v 



is positive, the integral will also be positive when A 5 is introduced as a 

 divisor under the integral sign ; for by introducing this denominator we 

 diminish all the elements of the integral, but we diminish every negative 

 element in a higher degree than any positive element. 

 Now the value of the last integral is 



2(3 + 2j>) 2(3p+p 2 +2pr*) _ 2p 3 . 

 5x7 7 X9 9xli ; 



we are certain that this is positive if 



2pr >+ p * +3p 3 + 2g [ess than 



7X9 5x7 9x 11 



that is, if 



?+%B+ ?&+M k not less than %. 

 2 10p 22 



On trial it will be found that this condition is satisfied, even when r is 

 zero, provided p be not greater than A\, Thus we have only to consider 



