238 
CALCULUS 
and along C, dx and dy are the same as along I\* 
Let ^ (x, y)=P [«, y, a> (x, y)~\ + R [x, y, o> (x, y)] (a?, y), 
y)= Q[x, V, «(«, 2/)] + y, w («, y)] u>2(x y y). 
Then (3) becomes 
(4) / ? dx + O dy. 
v 
This integral can be written in the form (§ 4, C): 
(6) XAt-fb'- 
s' 
The integrand of the last integral is seen to have the value: 
(6) |p _!!= Q, _ p 2 + ( q 3 - , + (b, _ p 3 )^, 
Oy 
where the subscripts against the letters P, Q, R indicate derivatives 
taken on the hypothesis that (as, y, z) are the independent variables. 
Let the positive sense of the normal to S be defined as that for 
which the direction angle y is acute. Then 
cos a = — aq/A, cos/? = — w 2 /A, cosy = l/A, A = Vl + <oi -f <o|- 
Hence the integral (5) can, by the aid of (6), be written in the form 
(7) 
f f KQi— -Pj) + (-Ra — Q3) Acosa+(P 3 — Ri) Acos/SfdS'. 
* A fuller explanation of this point is as follows. Let C be given in the para 
metric form : 
C: x =/(\), 2/ = 0(X) z = f(X), O^X^l. 
Then the integral (1) becomes 
1 
J(Px' + Qy'+ Rz')d\. 
But from (1) 
Z 1 = UiX 1 + u 2 y'. 
Hence this integral has the value 
1 
^{(P + Rw{)x> +(Q + Rw 2 )y 1 } d\. • 
0 
On the other hand, the curve r is represented by the equations 
r: * =/(X), V = 0(X), 
Hence the last integral is the same as the integral (3). 
0 < X < 1.