^76 THIRD REPORT 1833. t 



any change in the constitution of the generating function. They 

 may both of them, therefore, be considered as representing the 

 vahie of this function, though in one case only can we approxi- 

 mate to its arithmetical value by the aggregation of any number 

 of its terms *. 



Similar observations would apply to the series 



/ , i\n « r I n b « (w — 1 ) 6^ . o ~1 

 (a + 5)" = «» |l -f - + -hV-^ + ^^-j 



when n is not a positive whole number. In all such cases, the 

 developement will sooner or later become a series, whose terms 

 are alternately negative and positive, and which will be di- 

 vergent or convergent, according to the relation of a and b to 

 each other. More generally we might assume it as a general 

 proposition, " that divergent series which correspond to no 

 change in the constitution of the generating function, will have 

 their terms or groups of terms alternately positive and nega- 

 tive :" and conversely, " that divergent series which correspond 

 to a change in the constitution of the generating function, will 

 have all their terms or groups of terms affected with the same 

 sign, whether -f- or — , and the whole series may be replaced 

 by the symbol oo ." 



In both these propositions the change of which we speak is 

 that which corresponds to those values of the symbols which 

 convert the equivalent series from convergency to divergency, 

 and conversely. 



I am not aware of any proof of the truth of these important 

 propositions which is more general than that which is derived 

 from an induction founded upon an examination of particular 

 cases. But such or similar conclusions might be naturally ex- 

 pected to follow from the fundamental principles and assump- 

 tions of symbolical algebra. If the rules of algebra be perfectly 

 general, all symbolical conclusions which follow from them must 

 be equally true : and those rules have been so assumed, that 

 when the symbols of algebra represent ai'ithmetical quantitiesj 

 the operations with the same names represent arithinetical 

 operations, and become symbolical only when the correspond- 

 ing arithmetical operations are no longer possible. It will be 

 essential, therefore, to the perfection of algebraical language 

 that it should be competent to express fully its own limitations. 



* The equations s = and « = -r 7- will equally give us 



s ;= in one case, and s = r in the other, whatever be the relation 



, a + b a + 



of and b. 



