336 Prof. Sylvester on a Theorem concerning Discriminants. [June 15, 



where >//(#) must be supposed less than unity, in order that the following 

 transformation may hold : 



( 



J. f' J9Q- 



-*) (sin 



1-2 cos mO, 

 where 



The remainder of the process will be evident from the two former 

 examples. 



V. " On a Theorem concerning Discriminants." By J. J. SYLVES- 

 TER, F.R.S. Received May 27, 1865. 



Let F(, b, c } d)=a*d*+4a 3 c+4<f 63 a 2 6 2 -6 abed, and let a, b, c, d 

 be four quantities all greater than zero, which make this function vanish. 



(1) The cubic equation in x, F (a, x, c,d) will have two positive roots 

 (6, bj) ; so F (a, b lt x, d) will have two such roots (c, cj, F (a, x, c lt d) 

 two such (6 1S 6 2 ), F (a, 6 2 , x, d) two such (c lt c 2 ), and so on ad infinitum ; 

 we may thus generate the infinite series b^ c l 6 2 c a 



Similarly, beginning with the equation F (a, 6, x, d), and proceeding as 

 above, we shall obtain a similar series, c', b', c", b" . . . ; and combining the 

 two together, and with the initial quantities b, c, we obtain a series pro- 

 ceeding to infinity in both directions b" c" b' c' b c b l c l 6 a e 2 



(2) The four quantities 



Ya U* 37' Id 1 



where F represents F (a, b, c, d), will present one or the other of the three 

 following successions of sign, 



0000 

 (3) When the last is the case, i.e. when the differential derivatives all 



