218 
CALCULUS 
We have, then, as the expression for the work, 
(4) 
l 
--/• 
W= F cos t// ds. 
A second formula for the work is obtained from (2), namely, 
(5) W= lim V (X' k Ax k + Y[ Ay k ). 
n=0 ° ¿=1 
This limit can also be expressed as an integral. Since 
lim- ^ X - k - — — 1 and lim——= 1, 
As k cos r k As k sin r k 
we see that, on setting 
«* = ( x k cos T k + Y k sin T k ) As k , p k =X' k bx k + Y' k Ay k , 
the conditions of Duhamel’s Theorem are fulfilled, and hence the 
above limit has the value : 
i i 
(6) J(X cos r -f Ysin r) ds or f(x ^ 4 Y d £^ ds. 
Thus the limit (5) is seen to exist, and to have for its value the 
integral (6). The limit (o) is an example of a line integral, and is 
expressed by the following notation, § 3: 
(a',)/) 
(7) W—Jxdx + Ydy or Cxdx + Ydy. 
(a, b) C 
The extension to three dimensions is immediate. Formula (4) 
requires no modification whatever. Formula (7) is replaced by the 
following: 
(a7/, c') 
(8) W = IXdx + Ydy 4 Zdz or Cxdx+Ydy + Zdz. 
(a, b, c) C 
Example 1. To find the work done by gravity on a particle of 
mass m which moves from an initial point (x 0 , y 0 , z 0 ) to a final 
point (x lf y u z x ) along an arbitrary twisted curve, C. 
Let the axis of z be vertical and positive downwards. Then 
W 
x=0, Y= 0, Z=mg; 
= Cxdx 4 Ydy 4 Zdz = j mgdz = mg(z l — z 0 ).