38 Mr. W . R. Browne on Central Forces 



quite general. But it is easy to show that any particular law 

 of force which can be imagined, other than that of a central 

 force, is inconsistent with the conservation of energy. Thus, 

 suppose the force to vary according to B's distance from some 

 other point in the plane than A; then that distance can always 

 be expressed in terms of the coordinates of its extremities,, and 

 therefore in an expression involving 6, which is inadmissible. 

 Again, suppose the force to vary according to the perpendi- 

 cular distance of B from some line in the plane. Then, if B 

 move parallel to that line the force is constant, while if it be 

 perpendicular it varies from zero; and it is easy to see that if 

 B moves perpendicular to that line, and if, before it is allowed 

 to return, it is rotated till the line AB is parallel to that line, 

 then the two integrals will not be equal. Again, suppose the 

 force to act upon a certain line only, so that when B is off 

 that line no force acts upon it ; then, if we suppose the return 

 journey made parallel to that line, the energy on that journey 

 is zero.] 



We have throughout taken F as the force between A and B, 

 resolved along the line joining them. We have still to con- 

 sider the possibility of there being another component always 

 at right angles to this line. This component, if it exists, will 

 produce a rotation of B round A, which will increase B's 

 kinetic energy ; and as there will be nothing to balance it, this 

 increase will go on for ever; so that the conservation of energy 

 would not be true in this case. 



I have thus proved, I believe, the proposition with which I 

 started — namely, that the doctrine of central forces and that 

 of the conservation of energy are mutually interdependent, so 

 that one is not true without the other. In general, as remarked 

 at the beginning, the existence of central forces is assumed, 

 and the conservation of energy deduced from it. But the 

 process may be reversed. The conservation of energy may 

 be considered to rest, as a general law of nature, on the broad 

 basis of observed facts, such as the conversion of heat, elec- 

 tricity, chemical actions, &c. into mechanical work, and the 

 reconversion of mechanical work into these other forms of 

 energy. There can be no doubt that the evidence of this 

 character is of very great weight ; and I am myself disposed 

 to accept it as conclusive. But it must be pointed out that, 

 unless the above investigation be false, it involves our accept- 

 ing a mechanical definition of matter substantially to the fol- 

 lowing effect * : — " Matter consists of a collection of centres 



* This definition has been already given in a pamphlet entitled ' The 

 Foundations of Mechanics' (Charles Griffin and Co., 1882). 



