220 On the Measure of J Fork in the Theory of Energy. 



1 /2T\ 



be-F 2 ( — ) 2 : that at the end of the third interval will be 



2 \n J 



1 /3T\ 2 



- F 2 ( — ) , &c. : and therefore the work done in the first interval 



2 \n J 



TIT 2 . 3 T 2 



— will be -x F 2 -K-: that during the second interval will be - F 2 -^ ; 

 n 2 n 2 ?r 



5 T 2 



that during the third interval will be - F 2 -^, &c. It has alreadv 



6 n 



been pointed out, however, that the foregoing definition of work 

 implies that the amount of work done in equal intervals of time 

 will be the same. It follows, therefore, that the definition and 

 measure of work above propounded contradict each other in the 

 case we have been considering. 



2. Suppose a body whose mass is M to be moving in a certain 

 direction with a velocity Y v and that the force F is applied to the 

 body in the direction of its motion. Professor Maxwell proves 

 that, if during the small time T the body moves through the 

 space s, and has acquired at the end of T the velocity V, we shall 

 have 



F 5 =i(MV 2 -MV^ 



an equation, be it remembered, which holds independently of 

 the magnitude of T, provided F be uniform. 

 If we put V=V 1 -t-i>, we shall have 



work=Fs=iM(fl 2 +2flV 1 ) (1) 



Now v is the pure product of the force F acting on the body M 

 during the time T ; whence it appears that, adopting the mea- 

 sure of work above proposed, the work done by the force F on 

 the body M in the time T involves the variable quantity V p 

 which is entirely independent alike of F, of M, and of T. 



3. The right side of the expression (1) will always be positive 

 so long as Vj and v have the same sign, i. e. so long as the direc- 

 tions of the force and the initial velocity conspire. But if the 

 force and initial velocity have opposite directions, and T and V 1 

 are both finite, the right-hand side of (1) will first be negative ; 

 as the motion proceeds it will become zero ; and it will finally 

 become and continue positive. It results, therefore, from the 

 above measure of work, that the work done in a finite time by a 

 finite force acting upon a body of finite magnitude which is free 

 to move, may be zero. 



4. The proper work of force is to generate or destroy mo- 

 mentum*; and the work done by the force in a given time will 



* No doubt force has another effect — that, namely, of causing a body to 

 describe, or of preventing its describing, space : but of these two effects, 

 viz. the description of space and the generation of momentum in any in- 



