29, 30] DEFINITE INTEGRALS. 51 



^-coordinates are # 13 # 2 ^-i> multiply the length of each chord 



PS-I PS, v - ^ 



by the value of the point function f(p) at some point TT S in the 

 arc between p s -i)p s , and take the sum for all the arcs into which 

 the curve has been subdivided, then if this sum approaches a 

 finite limit as the number of subdivisions increases indefinitely, 

 this limit is called the line-integral of the point-function f(p) 

 along the curve AB, and is denoted by 



/(p)fe-lii 



A n=oo i 



If y(jp) = 1., the integral represents the length of the curve 

 AB 



'B 



ds=s AB . 



If in forming the line-integral we had multiplied the values of 

 /(TT S ) by the ^-projection of the chord, instead of by the chord 

 itself, we should have arrived at the integral already defined, 



(p) dx = lim S/(TT S ) (x s ^_ a ), 



n=<x, i 



except that/(_p) is a function not of x alone, but of the point on 

 the given curve corresponding to x. It will in general happen 

 that as we go continuously along the curve from A, x will not 

 increase continuously but will increase to a certain value G, and 

 then decrease. As x decreases, 

 however, reassuming previous 

 values, we are still continuing 

 along the curve and reaching 

 new points and corresponding 

 values of / (p) which are to be 

 used in the integral. The func- 

 tion f, which would otherwise Fm * 8 * 

 not be uniform in x, becomes uniform when defined in this manner, so 



that if we interpret in the ordinary manner the integral lf(p) dx, 



we must separate it into several integrals, in each of which x 

 varies in one direction throughout, taking in each the values of 

 / belonging to the corresponding part of the curve. If however 

 we write 



/v r _ // 



(V _ v \ 8 s ~ l 1 

 (X S X 8 .,)- fc 



42 



