Me WARBURTON, ON SELF-REPEATING SERIES. 485 



Which coefficients, 256 being positive, are to be with signs alternately positive and negative ; 

 so that when t m 1, the fraction becomes, 



— [256 A'O 9 - 93 A 2 9 + 29AV - 7A 4 9 + lA'O 9 ], .... (29) 



instead of the elegant, but more complex expression, 



— — [Z56A l 9 - 128A 2 9 + + 2 4 A 5 9 - 2 3 A 6 9 + 2 2 A 7 9 - 2A 8 9 + 1A 9 9 1. 



1024 L J 



In the former expression, the labour of computation is less, by a half, than it is in the 



latter; and the proportion of the labour saved becomes greater and greater, as the numbers 



are raised to a higher power. 



12. I should enter upon a vast field of enquiry, were I to enlarge upon the application of 

 this theorem to the different cases where the denominator of the generating fraction is repre- 

 sented by different recurrent functions of t. It will be sufficient to point out as a fertile soil 

 for new theorems, the cases where the recurrent denominator is a power, or rational integer 

 function of {l ± cos (0) x t + f}. 



13. The third, or converse proposition suggests several important corollaries. It is too 

 obvious to require proof, that if two or more rational integer functions of t are severally 

 recurrent, their product will be a recurrent function of t ; and that in the product of any two 

 such functions, the sign of recurrence will be positive or negative, according as the two func- 

 tions agree or differ in their respective signs of recurrence. 



Hence it follows that if two or more proper, or improper, fractions have for their respective 

 numerators and denominators recurrent functions of t, then, if the highest dimension of t, in the 

 product of all the numerators, is lower than the highest dimension of t in the product of all 

 the denominators, the series arising from the developement of this fractional product, will be the 

 right arm of a self-repeating recurring series. For the product will then be a proper fraction, 

 having a recurrent numerator and a recurrent denominator. This obviously includes the case 

 of one of the factors being a recurrent integer function of t, since it may be considered as a 

 fraction, whose denominator is 1, the limit of all recurrent integer functions. 



Let a and b be the respective dimensions, as to t, of the recurrent numerator, and of the 

 recurrent denominator, of a proper fraction ; and let n be any integer not greater than 

 b - (a + 1). Then the series generated by this fraction will be the right arm of a self-repeating 

 series; and if a new series is constructed by adding together every (ra + 1) consecutive terms 

 of the original series, the new series will be self-repeating. For the process directed to be 

 performed is equivalent to multiplying the numerator of the fraction by the recurrent factor, 



(l + t + t? + + f); and, by supposition, the dimension of the product, as to t, is less 



than 6, the dimension, as to t, of the denominator. And, in like manner, with the like limita- 

 tion, may the recurrent factor be (l — t + t* — f =•= f). 



Thus if we add together every n 1 consecutive terms of a figurate series of the n th order, n 1 

 being any integer not greater than n, we shall obtain a self-repeating series, between the arms of 

 which (n-n l ) zero terms will intervene. For instance, if we add together every two consecutive 



