LINE INTEGRALS AND GREEN’S THEOREM 223 
along the shore in the sense we are defining as positive always has 
the water on his left. 
If, however, the axes are oriented as shown in Fig. 52, then the 
positive sense is defined in just the opposite way. 
Both cases may he brought under a single defi- — 
nition as follows. Let the inner normal, n, be 
drawn at an arbitrary point P of C, and let the 
tangent at P be taken as positive when it is v 
oriented to the above normal as the positive axis 
of x is oriented to the positive axis of y. Then C 
is described in the positive sense when the point P moves in the 
sense of the positive tangent. 
Returning to the above integrals, we see that 
X 
Fig. 52 
the line integrals being taken in the positive sense. Hence we 
have the result: 
the line integral being taken in the positive sense. 
If the region S is cut by a parallel to the axis of y in more than 
two points, it may be divided up into regions, each of the former 
type,* and the above result applied to each 
of these. On adding the equations thus ob 
tained, the final result is found to be Equa 
tion {A), written for the original region. 
In a closely similar manner it is proved 
that, if Q, together with dQ/dx, be contin 
uous within and on the boundary of S, then 
Fig. 53 
(®) 
/f d ^ dS= J Qdy - 
(Q 
From equations (A) and (B) we infer that 
cP_dQ' 
dy dx. 
//' 
dS = — / Pdx + Qdy. 
* More precisely, we restrict ourselves to such regions <S as have this property, 
since these are sufficiently general for the needs of practice.