102 CAEPMAEL ON THE DETEEMINATION IN TEEMS OF A 



the series to be coutimied ouly as long as the quantity raised to power m-^n is positive, 

 w being a positive integer." 



The method which Caiichy eniploj^s to obtain a definite integral to represent tlie 

 value of what we here call <p (x, m, n) fails when m is an integer or zero, as it involves the 

 repeated integration with respect to a variable of an expression, in which the variable is 

 raised to power m-[-n — A — 1 where A is the greatest integer in ia-\-n. He assumes, how- 

 ever, without comment, that his results will hold for integral A^alues, but when vi is zero 

 or a positive integer, we shall find that this is not the case. 



In all that follows we shall suppose vi to be a positive integer, zero, or a negative 

 integer numerically less than n. 



If in the expression for q> (x, m, n) we put « equal to zero, it reduces to zero when x 



is negative, and to — .ï when x is positive. We will rexîreseut the expression in this 

 ° m 



case by (p {x, m). 



If we differentiate qi {x, m) with respect to x, we obtain f (x, m — 1) (supposing m be 

 not zero) ; 



Conversely, if we integrate <f> (x, m) with respect to x between the limits and x we 

 obtain — 



/ (p (x, m) dx:= cp (x, m + 1) — (^ (0, vi -\- 1) 



= (p (.r, W! + 1) (>) 



If in equation (i) m is equal to zero, there will be discontinuity in the value of <p {x, m) 

 at rt = . "We will proceed to consider this case further, and at the same time we will 

 somewhat enlarge the meaning of q) (x, 0). 



Let, then, (p {x, 0) represent a function which is zero when x is negative and unity 

 when x is positive, and which has any finite A'alue whatever when a; = 0. 



"We have, then, — 



/"<p (.1-, 0) dx = fl<p (.X-, 0) dx + Jl<p (.r, 0) dx 



Let A be the greatest and — B the least value of q> (.c, 0). Then J^cp (.i-, 0) dx must lie 

 between A c and — B c, and remembering that (p (.f, 1) represents zero when x is negative 

 and X when x is positive, we see that if we take c of the same sign with x, / ' q> (.'•, 0) dx 



must lie between <p (x, 1) — <p (c, 1) + Ac and ^p (x, 1) — (p (c, 1) — Be. If now we 

 diminish c without Jimit, cp (c, 1), Ac and Be all ultimately vanish, and we still have — 



Jg'q' (■'»■; 0) dx = <p (x, 1) (ii) 



If instead of integrating <p {x, m) between the limits and x, we integrate between 

 the limits x — J and x + |, we find — 



y.,_j <P (-c, m) dx = J„ cp (x, m) dx 

 = ip (x + i, m + 1) 

 or =^ cp (x, m, 1) 



