Mr. DE morgan, ON DIVERGENT SERIES. ]99 



mathematician has lately amused himself with preserving the first part of the air of ' God save the 

 King' for posterity by means of a case of Fourier's integral ; and any one who lias studied the pro- 

 perties of the series A cos .p + B cos 2 x + ... knows that a sturdy computer, who is not afraid of the 

 method of quadratures, might hand down the means of recovering the profile of his own face from 

 its equation : and that in a form which no analyst could tell at sight from the equation of a circle, 

 a parabola, or some other continuous curve. Nor is such discontinuity a mere possibility : it is 

 constantly occurring in the higher branches of mathematics, and its detection and treatment forms 

 the most distinctive feature of the most recent school. Surely then it is time to pay attention at the 

 outset of every plan of investigation to the possibility of the occurrence of discontinuity in inverse 

 operations. 



I do not see how absolute error is to be avoided without such a precaution. Defining 

 E<p,v as (p(a!+ 1), nothing is clearer than the right to use the symbol E, and those derived from it, 

 algebraically : all the fundamental symbolic definitions are satisfied by it. If we are to assume, as 

 of necessity, that E~''(pw can be nothing but (p{a! — n), the symbol S(pai must represent 0, as 

 shown : and experience points out that it actually does so in every case in which there is no 

 discontinuity. But in certain cases, as I shall show, S(px does not represent 0, but another 

 solution of \l/ (x + l) = \{/ a; : there is then some flaw in the demonstration, which I take to be the 

 assumption without reserve ot E ~ " (px = (p (^x — n) . 



I might give other ways of expressing S(f>x, all ending in the same result, that, unless some 

 special mode of introducing and allowing for discontinuity be adopted, it represents 0. But this 

 paper is already too long, and I therefore pass on to some cases in which it does not represent 0. 



Let us consider the series. 



which is both ways convergent. We have the two following results, 



f"^-ii-m''^invdv=- ' , ^, , r£-*-''"'sin«rf« = -; ^ 



Jf, 1 + (6 ± kef Jf, 1 + (6 - key 



that one being taken in which e" is raised to a negative power. Let 6 lie between mc and 

 (m + I) c : then we have 



— -^ r7 + 7r-^—T2 + -- = r (6-*'""''° + £-"-"'* + ...) sin «d« 



1 + (6 - mc)- l+(6-m-lc/ ^ ' 



•'ft 



1 1 



1 + (6 - m + 2c) 1 + (6 - TO + 1 c)^ ^0 

 in which integration is performed on convergent series only. Hence, 



S — -, = / sin«du= / — i-^ sin «d«, 



where to' = TO + ^. Now to'c - 6 is numerically less thanlc; and Legendre has shown (see my 

 ' Differential Calculus,' page GGQ) that if g be not greater than h. 



i: 



-i:; -T7.sinvdv 



2h J -J TTfl- 



e'' + e * + 2 cos — 

 h 



whence r „ or - 



+ e ''+licos(2m + l \ it e' + e ''-2 cos 



is the value of the series, which is, as it should be, a solution of -.^ (6 + c) = x|/t 



