LINE INTEGRALS AND GREEN’S THEOREM 217 
doing no work. Thus we arrive at our final definition: The work 
done by g in the particular case in hand shall be 
(1) W= FI cos <(,. 
A second form of the expression on the right is as follows. 
X and Y be the components of g along the axes 
that the path AB makes with the posi 
tive axis of x. Then the projection of 
g on AB is equal to the sum of the pro 
jections of X and Ton AB, or 
F cos ip ~ X cos r + Y sin r. 
On the other hand, 
x 2 — x i = l cos r, 2/2 — 2/i = ^ s i u T - 
Hence 
(2) W=X(x 2 -x l ) + T(y i -y l ). 
\V\ 
Let 
Let r be the angle 
Fig. 50 
The General Case. If C be any regular curve, divide it into n arcs 
by the points s 0 = 0, , • • •, s„_!, s n — l. Let g* be the value of % at 
an arbitrary point of the Zc-th arc, and let ip' k be the angle from the 
chord (s k - x , s k ) to the vector g*- Then the sum 
n 
F k cos ip k l k , 
k= 1 
where l k denotes the length of the chord, gives us approximately 
what we should wish to understand by the work, in view of our 
physical feeling for this quantity. The limit of this sum, when the 
longest l k approaches 0, shall be defined as the work, or 
(3) 
W = lim 2) K cos if,[ l k 
h 
Since 
it is clear that the above limit is the same as 
lim = 1, 
As* 
l 
lim y)F k cos ip k As k = I F cos «p ds. 
”=* "èi J 
For, the conditions of Duhamel’s Theorem are fulfilled if 
= F k cos ip k As k , /? fc = F' k cos <p[ l k , 
since 
Pk __ (F'k\fcos\p' k \i l k 
<*k \ F J\ cos ïpj\±sj’ 
and thus 
lim = 1.