198 Mr. DE morgan, ON DIVERGENT SERIES. 



look for its value at w = 0, from the limit of A^- JyV -{■ A.^w^- ... as well as from that of 

 A +^,.r + ... Accordingly, when there is continuity, all the presumptions of superior safety 

 which the alternating series presents may be applied to this intermediate case. 



SECTION V. 



On Double Injinite Series, in which the Terms are infinitely continued in both 



Directions. 



One look at the series 



... + <^ (.r - 3) + ^ (.r - 2) + (.r -l)+(pa) + (p{x + \) +<p{ai + '2) + (i>{,v + 3) -\- ... 



will show that, whenever it can represent a definite function of on, which preserves its properties for 

 different values of x, it must be a solution of the equation \// (.» + l) = \// r . Various modes of 

 proof applicable however only to functions and processes of complete continuity, show that, in 

 all cases to which those proofs apply, the repre.scntation of the above is simply 0, or rather 



— either 0, or, in particular cases, -. And certainly, in all cases, it can be reduced to the 

 ■v.v ^ 



limiting form + + + ..., so that, if not always =0, the warning given in another part of this 

 paper is confirmed. Throughout this section, let Scp.v stand for a double series of the above form. 

 For <hx write (px.a" and divide by a'' wiiich gives ... + <p {ic - 1) a'' + cp.v + (p {.v + 1) a + ... 

 Now 



1 a , a + a'' (f)"x a + 4 o^ + o" <p"'ai 



,^,r + ,^(.r + l).a + ... = p3^<^^+^73^,<^.''+(Y3^-^+ (1 - a)' ^T^ + - 



in which it need only be noted here that the numerators of the functions of a all read backwards 

 and forwards the same in their coefficients. Now by the same rule 



(pi-.v)+(f>(-x-l).a+ ... = j— ^0 (- ,r) - ^^y, 0' ( - -r) + ... 



change x into —x in the last, and a into a'', add the result to the preceding equation, and 

 substract (p .v , which gives S {(j) x . a^) = a" {0 + + + ...). Again, taking the calculus of 

 operations, let E (px = (p (x + I), then, of all perfectly continuous answers, E~'(px must mean 

 <p{x-\). The whole operation performed upon cpx in S<px is ... +£-' + £"+£' + ... or 



L , or 0. But it must not be forgotten that, in cases in which discontinuity is 



]_£-' i-£;' 



possible, it does not follow that E~"(px always signifies (.r - w). For if we were to assume, 

 for instance 



/ TT /•= sin (a - «) u , N 

 ■<px = (p(x - S) + I — + / — dvj \px 



we should be justified by the result E'E-^(px = <px, whenever a - (.r + 3) is negative, tliough 

 when a - x is positive, the preceding would not be the same as <p {w - 3) . 



This is an important point, not only in reference to the calculus of operations, but to every 

 case in which inverse operations are employed. There is, I am well aware, among mathematicians, 

 something like a disinclination to provide beforehand for discontinuity, which first showed itself in 

 the struggle against admitting discontinuous functions into the solution of partial differential 

 equations. But it should be remembered that, in our time, trigonometrical series of the most 

 continuous form have been shown to represent functions of the most capricious discontinuity. A 



-s , 



J 



