36 Mr. W. R. Browne on Central Forces 



equation 



^ (»»-«»)= r^FAfc 



Let us suppose b to be such that v = 0, so that 



Fdx (1) 



m 2 C a+ < 

 2 V H 



Then, when B arrives at distance a + Z>, its velocity, and there- 

 fore its kinetic energy or vis viva, will be reduced to zero. 

 There is therefore a loss of energy, so far as B is concerned. 

 But now let us suppose that B is left free to return towards A, 

 and that it passes back again over the space b. Then, if F 

 continues to act, A will exert during the motion an amount of 

 energy on B, or will do an amount of work upon B, which 

 will be represented by 



f" Fdx; 



Ja+b 



and when B has reached the distance a, it will have gained a 

 velocity V, given by the equation 



^V 2 =f a Fdx (2) 



1 Ja+b 



Now if Y = — v ly then V 2 = v\ : hence we shall have the two 

 particles in the same position as at first, and the kinetic energy 

 of B will be the same as at first. Therefore there will 

 have been no loss or gain of energy on the whole ; and the 

 energy is then said to have been conserved during the motion. 

 At the time when B's velocity is zero, the energy of the system 

 is represented by the potential energy of A — that is, the 



power A has of subsequently doing the work 1 F dx upon B. 



Ja+b 



At other times during the motion, the energy of the system is 

 partly potential energy of A, partly kinetic energy of B. 



We thus see that it is essential to the Conservation of 

 Energy that V 2 should = v±. But by equations (1) and (2) 

 this is equivalent to the equation 



f*a+b na 



I F^=j Fdx; (3) 



Ja Ja+b 



these two expressions representing the two amounts of energy 

 exerted, as described above. 



It is therefore essential for the conservation of energy that 



