274 THIRD REPORT — 1833. 



admit of no arithmetical interpretation. And it will be after- 

 wards made to appear that such series do not include in their 

 expression, at least in many cases, all the algebraical conditions 

 of their generating functions. Before we proceed, however, to 

 draw any inferences from this fact, it may be expedient in the 

 first instance to give a short analysis of some of the circum- 

 stances in which such series originate. 

 The series 



] 1 6 62 



+ -2 + 73 + &C- 



a — b a a^ a^ 



is convergent or divergent according as a is greater or less 

 than b. As this series is incapable, from its form, of receiving 

 a change of sign corresponding to a change in the relation of 

 a and b to each other, it would evidently be erroneous in the 

 latter case if it admitted of any arithmetical value, in as much 

 as it would then be equivalent to a quantity which is no longer 

 arithmetical. In this case, therefore, the series may be replaced 

 by the symbol go , which is the proper sign of transition, (see 

 page 237,) which indicates a change in the constitution of the 

 generating function, of such a kind as to be incapable of being 

 expressed by the series which is otherwise equivalent to it. 

 The same observations apply to the equation 



r, b n(n-\) b^ 



(a-bf = a'' < 1 - rt — + — ; ^ . -2 



^ ^ l_ a \ . 2 or 



~ 1.2.3 • ^ + ^""'J' 



as we have already stated in our remarks upon signs of transition, 

 in page 237. It will be extremely important, however, to examine, 

 both in this and in other cases, the circumstances which attend 

 the transition from generating functions to their equivalent 

 series, in as much as they will serve to explain some difficulties 

 which have caused considerable embarrassment. 

 The two series 



If, 2b , 3b^ 4 63 -| 



a^ [^ a a'' a'' J 



I If, 2b . 3b'' 4*3 



(« - bf 



and 



J _ If 2 a 3a^ 4 a^ \ 



- af~ b^X '^ b + ~b^ "^ 63 + *'^-/ 



(b 



will be divergent in one case, and convergent in the other, 

 whatever be the relation of a and b, though they both equally 



