Mr WARBURTON, ON SELF-REPEATING SERIES. 477 



fraction, whose numerator and denominator are recurrent; and the sign of repetition of the 

 series is 



a+/3 + l 



(- \)y = ( - i) (22) 



Therefore when a and /3 are either both odd, or both even, the series will have a negative 

 sign of repetition ; but when one of the two exponents is odd, and the other even, the series 

 will have a positive sign of repetition. 



10. From the first of the last three propositions several obvious and important corollaries 

 immediately follow. 



Let the general term of a series be a rational integer function of the index, and also let the 

 series generated be the right arm of a self-repeating series. Then, since the denominator of 

 the generating fraction is a positive integer power of (l - t), the denominator is a recurrent 

 function of t; and therefore the numerator also is a recurrent function of t. Thus, for 

 instance, when the terms of the series are the natural numbers, or their positive integer 

 powers ; the odd numbers, or their positive integer powers ; the figurate numbers of whatever 

 order, or their positive integer powers, or when the general term is any other rational integer 

 function of the general term of any of those numbers, such as will render the series a self- 

 repeating one, and when the series generated begins with the first term of the right arm, 

 then the numerator of the generating fraction will be a recurrent numerator. 



Thus, if we take the triangular of every triangular number ; that is to say, if instead of 



3.4 6x7 10 x 11 15.16 

 the series 1, 3, 6, 10, 15, 21, &c. we take the series 1, , , , , &c, we 



obtain the self-repeating series 1, 6, 21, 55, 120, &c, whose general term is obviously of four 

 dimensions, and consequently the denominator is (l - *)' ; and the corresponding numerator is 



readily found to be (l + t + f) ; therefore the generating fraction is — — - . And in like 



(1 — t) 



manner we may take the triangulars of the triangulars of the triangulars ; which are the series 

 1, 21, 231, 1540, 7260, 26796, 82621, &c, in the general term of which the index is of 8 dimen- 

 sions ; and the denominator therefore is (1- if; and the corresponding recurrent numerator is 



1 + m + 18f + 133« 3 + 78^ + 12* 5 + f. 



We may proceed in like manner with any other figurate series of an even order ; for in 

 this case the right arm and the left arm of the series have the same sign. But in the figurate 

 series of the odd orders, the right arm of the series consisting of positive terms, but the left 

 arm of negative ones, when we come to take the odd figurate of an odd figurate, we fall, in the 

 left arm, upon numbers differing in numerical value from those that we obtained in the right 

 arm ; and consequently the series obtained by a similar process from an odd figurate is not a 

 self-repeating series. 



11. Let the general term of the self-repeating series be the product of a rational integer 

 function of the index by the exponential (- 1)*; in other words, let the numerical values of 

 the rational integer function, corresponding to the consecutive integer values of the index, 0, 1, 



