298 Mr. J. J. Sylvester on a Fundamental Rule in 



in magnitude, but alternating in sign, so long as the differences 

 of the indices /, g, h, k are constant ; and as an easy deduction 

 from this sub-corollary, if 



be the characteristic equation of a recurrent series, andif/+A=^+ h, 

 —^ — * +h s ' — - will be constant ; and as a particular case of this 



deduction from the sub-corollary to the second corollary of the 

 fundamental theorem, we have 



•*■ n+l — *-n-\ » J-n-fl 



t. e. 



an a constant, 



p^T w+ «y. = 



which is Eider's theorem. See Terquem's Nouvelles Annates, 

 vol. x. p. 357, and November 1852. 



I was led up to a knowledge of the fundamental theorem (be 

 it new or old) by some recent researches connected with my 

 new Rule of Limits, considered with reference to the conditions 

 which must be satisfied when one of the limits found by the rule 

 comes into actual contact with a root ; a contact which I can 

 demonstrate is always possible, as well for the superior as for 

 the inferior limits, and with so much the fewer equations (as 

 distinguished from inequations) of condition between the coeffi- 

 cients of the assumed auxiliary function which the application of 

 the rule of limits requires, as there are fewer pairs of imaginary 

 roots in the function whose roots are to be limited. 



I may add that the fundamental theorem is an immediate 

 result of the representation of the terms of the convergents to a 

 continued fraction under the form of determinants. Thus, ex. gr. 

 the determinant 



a 1 



-16 1 



-1 c 1 



-1 d 1 



-1 e 1 



is obviously decomposable into 



a \ x d \ + a \ x el 



-16 1 -lei -lb -If 



-I c -1/ 



