LINE INTEGRALS AND GREEN’S THEOREM 221 
where (x' k , y[) is any point of the k-th. arc. Then this sum ap 
proaches a limit when n becomes infinite, the longest arc approach 
ing 0. For, let x = w(s), where s k increases with k. Then 
Now 
(4) 
&X h = u> (8 k ) - co (**_! ) = A S k COS r*. 
« X 
lim V p(x k , y k ) cos T k As k = / Pcos r ds 
«=* fc=i J 
is the ordinary integral of Pcost, a continuous function of s. 
Hence the limit (3) exists and is equal to the integral (4). We 
(a'.jC) 
J Pdx. 
write : 
(5) lim V P(x[, y[) Ax* = I Pdx 
(«, ») 
If C is divided into n arcs and the extremities numbered in the 
inverse order, the new variable (3) approaches as its limit the 
negative of the former limit. Thus reversing the sense of the inte 
gration reverses the sign of the line integral, or 
(a, l) 
J’rdx = -j 
Pdx, 
(6) 
(a, b) (<f\b') 
the curve C being the same in both cases. 
The limit of (3) is precisely of the type (1), and thus may be written: 
(7) 
/ 
P cos t ds. 
When, however, the sense of the integration is reversed, t is re 
placed by t ± 77-, and so the sign is changed. 
The line integral 
<«', 5') 
(«'. &') 
is defined in a similar way. Finally, 
(9) lim 5) tit) Ax k + V'k) ^Vk] = I Pdx + Qdy 
k = l <«%) 
or I Pdx -j- Qdy, 
for this limit is evidently equal to the sum of the line integrals (5) 
and (8).