and the Conservation of Energy. 37 



F should be a function of a form such that equation (3) may 

 hold. We have now to inquire what this form must be, or, 

 in other words, within what limits F may be allowed to vary 

 so that the equation (3) may still hold. 



Now we have not supposed the constitution of A and B, or 

 their relations to each other, to vary in any way except in 

 regard to space and in regard to time ; and we have every 

 reason to believe that these are the only variations which take 

 place in the ultimate molecules of matter. Hence we need 

 only consider variations with regard to space and time. 



Now if F be any function of time, then, since some time 

 must have elapsed between the exertion of the two amounts of 

 energy represented by the two sides of equation (3), it follows 

 that for every value of F in the right-hand expression the 

 time will be greater than for the corresponding value of F in 

 the left-hand expression ; and therefore the sums of the two 

 sets of values, or the two integrals, cannot be equal. Hence 

 F cannot be a function of time. 



We have therefore only to consider variations in space. 

 Now, if we confine our attention to one plane, we know that 

 any variation of B's place in that plane may be represented 

 by a change in the values of x and 6; where x is B's distance 

 from A, and 6 the angle which the axis of x makes with some 

 fixed line in the plane*. Then it is easy to show that F must 

 not vary with 6. For if it does, let us suppose that when B 

 has come to rest, and before it is allowed to return, it is made 

 to rotate about A through an angle dd, and again brought to 

 rest. Then the circumstances of A and B are unchanged ; 

 for the kinetic energy given to B during the rotation has been 

 taken out again in stopping it. But if B is now allowed to 

 return towards A, then, for every value of F in the right-hand 

 expression, the value of 6 will be greater or less by dd, the 

 amount of the change, than for the corresponding value in the 

 left-hand expression; and therefore, as before, the two inte- 

 grals cannot be equal. Similarly, if we take coordinates x, 0, </>, 

 in three dimensions, it will follow that F cannot be a function 

 of<£. 



Hence we are left with the conclusion that F can only be a 

 function of r ; in other words, the force with which A acts 

 upon B always tends towards A, and varies, if it varies at all, 

 according to the distance from A only. But this is the defi- 

 nition of a central force. 



[The proof just given, that F cannot vary with 6, appears 



* We here make no assumption except that the force varies as B's 

 position in the plane varies ; which is essential to every theory on the 

 subject. 



