Mr WARBURTON, ON SELF-REPEATING SERIES. 473 



between this finite term and the first term in the right arm of the series, will be zero-terms ; 

 and, consequently, the number of such zero-terms will be a unit less than the difference between 

 the dimensions of the numerator and of the denominator, taking that difference with a positive 

 sign. 



For let the generating fraction be, 



A + A 1 t + + A a t a 



l+B 1 t + + B b t b% (2) 



and let the right arm of the recurring series be, 



C+Ct+ +Ct*+&c (3) 



1 x 



and let the coefficients of the terms corresponding to negative integer indices be 



C, C, C, &c. 



-1 -2 -3 



Then, supposing w to be less than b, there will subsist between any (b + 1) consecutive 

 terms of the series extended as above directed, in accordance with the scale of relation, the 

 following equation : 



= 1xC + jBxC+ +J?xC+fixC+5x(?+ + B x C. . (4) 



x 1 x-1 x x + 1 -1 x+2 -2 b -(b-x) 



ButA = lxC + BxC+ +5xC; . . . . . . . . (5) 



x x 1 x-1 x 



••• A= -YBxC + BxC + +B*C "i (6) 



x Lx + 1 _l x + 2 -2 b -(b-x)J 



In equation (6), substitute for x, in succession, the values, 



(6-1), (6-2) ,(« + l), a. 



Then, first, A = - B x C ; and since B is finite, therefore, if A = 0, C = 0. 

 6-1 6-1 6 b-\ -1 



Secondly, A •» - VB X C + B ■* C "|l and since B is finite, therefore, if A and A are 



6-2 Lj_l _l b -2-J b 6-1 6-2 



each equal to 0, C and C will be each equal to 0. 

 -1 -2 



And in like manner it may be shewn, that if A , A, , A, are each equal to 0, 



6-1 6-2 a+l 



then C, C, , C , will be each equal to 0. 



-1 -2 _ [6 _ (a+1) j 



Lastly, A= - B x C ; but A and B are finite, .*. C is finite. 

 a b -(6-0) a b -(b-a) 



Hence, if % denotes the number of the zero-terms, intermediate between C and C , we 



nave the equation, 



x = b - (a + 1). . . . . . (7) 



6. In like manner may the converse proposition be proved ; namely, that if a recurring 

 series contains x consecutive terms, each equal to zero, and these zero-terms are immediately 



preceded by a finite term, C , and are immediately followed by another finite term, C, 



-(*+i) 



61—2 



