SECT. IV.] FUNCTIONS COINCIDING BETWEEN LIMITS. 431 



the values of f(x] are nothing when x is not included between a 

 and 6; the limits of integration with respect to a, in the preceding 

 equation (B\ become then a = a, a = 6; since the result is the same 

 as for the limits a = oc , a = oo , every value of &amp;lt;/&amp;gt; (a) being nothing 

 by hypothesis, when a is not included between a and b. We have 

 then the equation 



The second member of this equation (B ) is a function of the 

 variable x\ for the two integrations make the variables a. andp dis 

 appear, and x only remains with the constants a and b. Now the 

 function equivalent to the second member is such, that on substitut 

 ing for x any value included between a and b, we find the same 

 result as on substituting this value of x in &amp;lt;f&amp;gt; (x) ; and we find a nul 

 result if, in the second member, we substitute for x any value not 

 included between a and b. If then, keeping all the other quantities 

 which form the second member, we replaced the limits a and b 

 by nearer limits a and & , each of which is included between a and 

 6, we should change the function of x which is equal to the second 

 member, and the effect of the change would be such that the 

 second member would become nothing whenever we gave to # a 

 value not included between d and 6 ; and, if the value of x were 

 included between a and 6 , we should have the same result as 

 on substituting this value of x in &amp;lt;j&amp;gt;(x). 



We can therefore vary at will the limits of the integral in the 

 second member of equation (B&quot;). This equation exists always for 

 values of x included between any limits a and b, which we may 

 have chosen; and, if we assign any other value to x, the second 

 member becomes nothing. Let us represent &amp;lt;t&amp;gt;(x) by the variable 

 ordinate of a curve of which x is the abscissa ; the second member, 

 whose value is /(a?), will represent the variable ordinate of a second 

 curve whose form will depend on the limits a and b. If these 

 limits are oc and + 20 , the two curves, one of which has &amp;lt;j&amp;gt;(x) for 

 ordinate, and the other f(x], coincide exactly through the whole 

 extent of their course. But, if we give other values a and b to these 

 limits, the two curves coincide exactly through every part of their 

 course which corresponds to the interval from x = a to x = b. To 

 right and left of this interval, the second curve coincides precisely 



