REPORT ON CERTAIN BRANCHES OF ANALYSIS. 267 



Convergency and Divergency of Series. — The subject of di- 

 vergent series, their origin, their interpretation and their use 

 in analysis, is one of great importance and great difficulty, and 

 has been and continues to be the occasion of much controversy 

 and doubt. I shall feel it necessary, for such reasons, to notice 

 it somewhat in detail. 



If the operations of algebra be considered as general, and 

 the symbols vi'hich are subject to them as unlimited in value, 

 it will be impossible to avoid the formation of divergent as 

 well as of convergent series : and if such series be consi- 

 dered as the results of operations which are definable, apart 

 from the series themselves, then it will not be very important 

 to enter into such an examination of the relation of the arith- 

 metical values of the successive terms as may be necessary to 

 ascertain their convergency or divergency ; for, under such 

 circumstances, they must be considered as equivalent foi'ms 

 representing their generating function, and as possessing, for 

 the purposes of such operations, equivalent properties. Thus, 

 if they result from the division of the numerator of an alge- 

 braical fraction by its denominator, then they will ^jrorfwce the 

 numerator when multiplied into the denominator or divisor : if 

 they result from the extraction of the square or cube root of 

 an algebraical expression, then their square or cube will pro- 

 duce that expression ; and similarly in other cases, no regard 



, 2r v 



which is not 2 tt v — 1, but ^z /^^., which, thouarh it includes the 



former, is not included by it. It appears to me, however, that there exists a 

 fundamental error in the attempt which has been made by Mr. Graves to 

 generalize the ordinary logarithmic formulie upon the same principles which 

 have been applied by Poinsot to the generalization of the trigonometrical series 

 which have been noticed in the text. He assumes / {&) ^= cos 6 + -v^l sin 3 

 := e and makes the series for/(^) and/"' (&), combined with the equa- 



tion/ (x &) — a. value of/ {&f, and therefore/"' / ^ = 2 r 5r + ^, the foun- 

 dation of his logarithmic developements : in other words, he makes e '^~' a 

 periodic quantity the base of his system of logarithms, an assumption which 

 is essential to the truth of the formula/"'/^ = 2 ?• ar -f and to the gene- 

 ralization of the series for/"' 6 by means of it; an hypothesis which is al- 

 together at variance with our notions of logarithms as ascertained by the ordi- 

 nary definition. The logarithms of + 1 and of (+ 1)"' alone, for very obvious 

 reasons, can be considered as possessing such a character. 



Though I have felt myself called upon to state my objections to the fun- 

 damental principle assumed in this memoir of Mr. Graves, and consequently 

 to many of the conclusions which are founded upon it, yet I think it right at 

 the same time to observe that it displays great skill and ingenuity in the con- 

 duct of the investigations, and is accompanied by many valuable and ori- 

 ginal obscrvationb upon the general principles of analysis. 



