THIRD REPORT — 1833. 



1 



and b 7 1, we have r = (a — 1) (6 — 1) = -_^. 



_ ^ ^ {(« -1)2(6 -l)2}-4- 



and the value of s (^4) is determined by the combination of the 

 two last developements. In a similar manner, if « Z 1 and b Z.\, 

 z (^i) will be formed by the combination of the two first. If 



aZlamU7l,thenr = (l-«)(6-l)=^-^^— ^^±-^_^: 



and the value of s {s:^ is formed by the combination of the first 

 and third developement. And if a 7 1 and i Z 1, then the value 

 of z (sfg) will be formed by the combination of the second and 

 third developements : in other words, the selection of the de- 

 velopements is not arbitrary, in as much as {(1 — a)^}"* and 

 {(a — 1)*}~* ought not to be considered, as we have already 

 shown, as identical quantities. 



These combinations of the convergent and divergent series 

 form all the four values of z, of which it appears that one value 

 alone is correct for any assigned relation of a and 6 to 1, being 

 that which arises from the combination of the convergent series 

 for K and K' only. The considerations, however, which deter- 

 mine the selection of the correct developement of z are as de- 

 finite and certain when the general series are employed as when 

 that value is determined directly from the definite integral 

 which expresses the value of z. It would appear to me, there- 

 fore, that not only was the employment of divergent series 

 necessary for the determination of all the values oi z, but that 

 when the theory of their origin is perfectly understood they 

 are jDcrfectly competent to express all the limitations which are 

 essential to their usage. The attempt to exclude the use of 

 divergent series in symbolical operations would necessarily im- 

 pose a limit upon the universality of algebraical formulae and 

 operations which is altogether contrary to the spirit of the 

 science, considered as a science of symbols and their combina- 

 tions. It would necessarily lead to a great and embarrassing 

 multiplication of cases ; it would deprive almost all algebraical 

 operations of much of their certainty and simplicity ; and it 

 would altogether change the order of the investigation of results 

 when obtained, and of their interpretation, to which I have so fre- 

 quently referred in former parts of this Report, and upon which 

 so many important conclusions have been made to depend. 



Elementary Works on Algebra. — There are few tasks the 

 execution of which is so difficult as the composition of an ele- 

 mentary work ; and very few in which, considering the immense 

 number of such works, complete success is so rare. They re- 

 quire, indeed, a union of qualities which the class of writers 

 who usually undertake such works are not often competent to 



