190 Mr DE morgan, ON INTRODUCING DISCONTINUOUS CONSTANTS 

 If therefore, the expression for a"x be 



«iC;,6 + x,Q^, + ... 



that for the sum of the series is discontinuous, and represented by 



(^a- - —px) C,, + (nx - ^vx)Q + ... 



vXi ' vXi ' 



I shall take the two instances given in my former paper, which 

 will of course be the most satisfactory, as the difficulty was prior to 

 the explanation. The first was the series 



X X of X x^ X* ^ 



1 + af I + of l+a;*l+a;^ 1+a;* 1+a^ 

 in which (ix = yx = , ax = a^ 



The equation of the series is 



a particular solution of which is ^x = x = nx, 



X . 1 



a particular solution of <()X - „ (ar*) being ^x =■ x — - = vx\ 



whence the expression for the series in question is 



x-^ I \\ 



X \X , 



which, if X lie between — oc and — 1) • 1 

 or + oc and + 1 / x' 



if X lie between - 1 and + 1, is x. 



To explain the cases where ax = x, return to the expression for the 

 series, which then becomes 



Ma;{l -(rixY\, 



giving in this case for a; = 0, the value 0, 

 and f or « = 1 the value 1, 



