124 THEORY OF WORK AND ENERGY. [CHAP. VIII. 



133. Work done by a force in a finite displacement. 



Let a particle move from a point P to a point P by any path. 

 Choose on the path n 1 intermediate points P lt P t , ...lV-i 

 Let F^ be the magnitude of the resultant force acting on the 

 particle at any point between P and P lt 6 l the angle between 

 the line of action of F t and the displacement P Pi ; let F 2 be the 

 magnitude of the resultant force acting on the particle at any 

 point between P l and P 2 , 2 the angle between the line of action 

 of jP 2 and the displacement PiP 2 J proceeding in this way we can 

 assign the meanings of all the quantities that enter into the sum 



. COS 0, + F 2 . P X P 2 . COS 6, + . . . + F n . P n _ : P . COS 6 n . 



Now let the number n be indefinitely increased, and let all 

 the chords P r P r+l be indefinitely diminished. The sum just 

 written tends to a limit represented by 



i 



F . cos 6 . ds, 



where ds is any element of the path, F the resultant force on the 

 particle as it passes over the element, and the angle between 

 the line of action of F and the tangent to the path at the 

 element, the integral being a line-integral taken along the path. 



The integral just written down is clearly the sum of the 

 elements of work done by the force on the particle in an infinite 

 series of infinitesimal displacements equivalent to a displacement 

 from the point P to the point P. It is defined to be the work 

 done by the force in the actual displacement of the particle. 



If X, Y, Z are the resolved parts, parallel to the axes, of the 

 resultant force on the particle at any point of its path, dx, dy, dz 

 the resolved parts of an infinitesimal displacement from that point 

 to a neighbouring point, the expression for the work done may 

 be written 



fedx+Ydy 



where the integral is a line-integral taken along the path. 



134. Work done by a system of forces. Suppose a system 

 of forces to be applied to a system of particles. If we form a 

 sum of terms, each term being the work done by the forces on one 

 of the particles in the displacement of that particle along its path, 



