REPORT ON CERTAIN BRANCHES OF ANALYSIS. 875 



'•^P'^'^"* a^-2ab + b^ ^""^ b^-2ab + a^ ^ ^^^^.^ ^'^ ^^g^" 

 braically, as well as arithmetically, equivalent to each other. It 

 might be contended, therefore, that in this instance the sign co , 

 which replaces one of the two series, is no indication of a change 

 in the constitution of the generating function which is conse- 

 quent upon a change of the sign oi a — b ov b — a. But 

 though a^ — 2ab + b^\s equal to (a — bf, and b^ — 2ab + a^ 

 to {b — of ; and though a* — 2 a 6 + A^ is identical in value 

 and signification with b^ — 2 ab + a^ when they are considered 

 without reference to their origin, yet we should not, on that 

 account, be justified in considering (a — hf and (6 — of as 

 algebraically identical with each other. The first is equal to 

 (+ If {a — bf, and the second to (— 1)^ (a — bf; or the first 

 to (- If {b - of, and the second to (+ l)^ (jb - af. But the 

 signs (4- 1)^ and (— 1)^ are not algebraically identical with each 

 other, though identical when considered in their common result, 

 in as much as their square and other roots and logarithms are 

 different from each other *. It follows, therefore, that there is 



a symbolical change in the quantity denoted by -, j^ in its 



passage through infinity, which is indicated by the infinite value 

 of the equivalent series, in as much as it is not competent to ex- 

 press, in its developed form, the algebraical change which its 

 generating function has undergone. The same remarks will 

 apply to the series for {a — bf and {b — af, in all cases in 

 which n is a negative even number. When ra is a negative odd 

 number, the change of constitution of the genei'ating function 

 is manifest, and requires no explanation. 

 The two series 



and 



1 /i b b^ b^ b* ^ -\ 

 « L a a^ a^ a^ j 



_ 1 r a a^ a^ a'' \ 



a + b 

 1 



correspond to the same generating function, though one of 

 them is divergent, and the other convergent. But the divergent 

 series, whose terms are alternately positive and negative, cannot 

 be replaced by the symbol oo , in as much as it does not indicate 



• Thus, if a denote a line, (+ a)^ and (— a)« can only be considered as 

 identical in their common result a«. When (+ a)2 and (— a)» are considered 

 with reference to each other, they are not identical quantities, though equal to 



T 2 



