Mr. I. Todhunter on Jacobi's Theorem. /. . ' 55 



that is, 



3 



and this is positive, for q does not exceed — . 



1 o 



Now the coefficient of x 2 in Q is positive, since r 2 q 5 is less than and 



3 



thns Q cannot change sign more than once ; it changes sign once, for it is 

 negative when x=0, and positive when x — q. The factor ~ will be 

 found to increase continually as x increases from to q. Hence in 

 passing from f 9 (1 — x 2 )Qdx to ( ^C 1 ^)Qdx ^ a i arger f rac ti on i s taken 



of every positive element than of any negative element ; so that since the 

 former integral has been shown to be positive, the latter is necessarily 

 positive. 



x 2 



To demonstrate that continually increases as x increases from to q, 



take the differential coefficient with respect to x ; the sign of this differ- 

 ential coefficient is the same as that of 



2xW —5x 2 (2qx + q 2 r 2 x + 2x z ) } 



that is, of 



2T> 2 -5x 2 (2q + q 2 T 2 + 2x 2 ), 



that is, of 



2q 2 -(6q + 3q 2 T 2 )x 2 -8x* ; 



this is positive when x = ; when x — q it becomes q 2 (2 — 6q — 3q 2 T 2 — 8q 2 ), 

 which we shall now show to be positive. We have by supposition 



and 



3 less than?!, 

 % .2 10 10 22' 



2-6^-3^r 2 -8^=2-^-^+3 2 V + 8^ ; 



3 27 21 



and as q does not exceed — , this is greater than 2— — — — and so is 

 lo Jo Z 2. j 



positive. Hence we see that when V=0, the value of is necessarily- 

 positive, and cannot be zero. 



clY 



22. We have shown in the preceding Article, that — cannot vanish 



simultaneously with V ; the demonstration is rather complex, and perhaps 

 for this reason Ivory attempted to establish the proposition by unsound 

 general reasoning. Liouville does not give the proposition, although it is 

 naturally required to make the discussion of the admissible values of p 

 and t complete. 



23. Thus we may state the results of Arts. 14, 16, and 21 in the fol- 



