200 Mr. DE morgan, ON DIVERGENT SERIES. 



Before showing some consequences of this and similar results which will be interesting as 

 extensions of known theorems, I proceed to verify my assertion that this series, being double, 

 and not = 0, will show signs of discontinuity. Let us consider the series 



^ " T+1^ "^ 1 + (6 + c)= "^ r+ (6 + 2cy '^ •••' 



which is convergent when c a? is or negative. This series is a solution of 



' y e 





hx . 



f' cos ijce' dw r' sin JCe ' dx 

 whence y = sin a / — ; ^ cos x / — j;^ . 



There is nothing in this result, as long as the final value of x is negative, to hinder the 

 computer from finding the value of y by the method of quadratures, and comparing it with the 

 result of the convergent series. And even when x = 0, the part of the first integral which comes 

 from between a = - a and ,r = 0, a being infinitely small ; is rendered evanescent by the factor 

 sin ,1', as special examination will show. If then we make a? = 0, and if we venture to change the 

 sign of c, and put the two results together, we have, remembering that the term (1 + 6') occurs 

 twice, 



1 1 /-o / 1 1 A • , . 



S ; H = - / h Sin xe dx 



1 + (6 + pcf 1+6^ -'- „ VI - e" 1 - e-'^l 



/•o . , 1 ^ 1 



= - / Sin if £ 'da; = — ; or 6 j-. r, = 0\ 



J_^ 1+6- 1 +(6 +xcY 



a false result, but agreeing with the theorem already discussed, and which I think may now 

 be described as follows. The double series S(px is, if its two sides be perfectly continuous, = : 

 and any method which proceeds by neglecting discontinuity will end in .S*^*' = 0, true or false. 



But perhaps it may not be evident at once why I say we have neglected discontinuity in the 

 preceding process : if so, the following explanation will be necessary. 



A continuous equation is one in which the two sides are algebraical equivalents, that is, 

 in which the right to use the sign of equality is independent of the value of any letter or letters. 

 If this right be destroyed by the passage of any one letter over a given limit, there is obviously 

 discontinuity. Now iiy^x = cf)X + <p{x+ 1) + ... bea continuous equation, or if \//a' = (hx + \|/(.i?+ I) 

 be universally true, we may convert it into 



y^/X = - (p(x - 1) + \//(.K - 1), or \j/x = - <p{x - 1) - <p(i- - '-') - ... : 



if this be granted, then S<px = o. Conversely, if S<px be not =0, then ^^x = <px + ... being 

 true, y\/X = — (p(x — 1) — ... is false. Also, if the assumption of the permanence of any equation 

 make S<px = 0, then, whenever this last is not true, it follows that such assumption of per- 

 manence is erroneous. In the preceding result, we have assumed the permanence of the equation 



e*' dw I 1 



/ 



+ . 



- e" 1 + b^ 1 + (b + cY 



for all values of c. The error of our result is manifest : this permanence then has no existence. 

 And the warning is that when c is made negative, we integrate over a diverging series : in 

 fact, our process assumes the ordinary development of (1 - e'')~' when c and .r are both negative, 

 or £">], and integrates that development. 



There have been two discontinuities occurring in the preceding ; the first dependent upon 

 the introduction of m, the second that just considered. The first may be treated as merely 



