472 Mr WARBURTON, ON SELF-REPEATING SERIES. 



Sect. II. Self-repeating Series defined. 



2. I call a series self-repeating, if when extended without limit in opposite directions, it 

 admits of separation into two similar arms, each beginning with a finite term ; either no zero- 

 term, or one or more zero-terms, lying between these two equal finite terms. The right arm 

 extends from left to right, and increases as the positive index of the series increases ; the left 

 arm extends from right to left, and increases as the negative index increases. One arm repeats, 

 and contains, arranged in reverse consecutive order, the terms of the other arm ; either all, or 

 none of, the terms so repeated having their signs changed. 



3. Let C be the finite term with which the right arm begins ; and let C be the term, in 



O X 



that arm, corresponding to the positive index x ; and let « denote the number of the zero- 

 terms that lie between the two arms. Then C is the finite term with which the left arm 



begins. 



If y is some integer number or other, constantly odd, or constantly even, for the same 

 series, then between the term C in the right arm, and the term C in the left arm, there 



x - (z + 1 + x) 



will subsist the following equation : 



C=(-1)VC , . . . . (i) 



x -(z + l + x) 



which admits also of the form 



C =(-i)C (l*) 



z-(z + l) -* 



We may call each of these equivalent equations the Equation of Repetition, or the Repeating 

 Equation of the series ; and (— Vjff its sign of repetition. 



4. In the sequel, I shall principally have occasion to treat of series, which, besides being 

 self-repeating, are also recurring series. 



Sect. III. Certain elementary properties of Recurring Series, in general. 



5. Let the terms of the numerator, and of the denominator, of a proper* fraction, and 

 also those of its developement into series, proceed according to the positive integer powers of t; 

 and let this developement, for brevity, be called the right arm of the recurring series ; and let 

 the coefficient of the first and of the last term, as well in the numerator as in the denominator, 

 be finite; and, in the denominator, let the coefficient of the first term be 1. Then the 

 coefficient of the first term in the right arm of the series is obviously finite. 



Now by means either of the scale of relation, or of any knowledge we may possess what 

 function the general term is of the serial index, let the terms which correspond to negative 

 integer indices be constructed. Then the first term of finite magnitude, taken in that direction, 

 will be the term whose index is the difference between the dimensions of the numerator and of 

 the denominator, taking that difference with a negative sign ; and all the terms intermediate 



* By proper, I mean that the dimension of t, in the numerator, is lower than it is in the denominator. 



