304 On an Extension of the Principle of Least Action. 



Secondly, suppose the elasticity of the chains and rods to be 

 taken into account, then the work done in stretching them must 

 be estimated in terms of w, and added to the expression for U, 

 thereby introducing terms of the form k w 2 , where A: is a quantity 

 depending on the rise, span, and elasticity of the chains ; U must 

 then be made a minimum as before. 



Having briefly indicated the mode of applying the principle 

 of least action to various problems, I return to its demonstra- 

 tion. 



From the description given of the process by which equili- 

 brium is attained, it is apparent that if the principle of Least 

 Resistance be given, the principle of Least Action follows, and 

 vice versa; and since the principle of Least Resistance is well 

 known, I have assumed it. But inasmuch as that principle 

 has perhaps never been satisfactorily proved, at least in its 

 general form, it will be well to give a direct demonstration of 

 the principle of Least Action in the case where the body is per- 

 fectly elastic. 



Let X, Y, Z be the components of one of the forces acting on a 

 free perfectly elastic body, u, v, w the displacements of its point 

 of application parallel to three rectangular axes, U the work 

 done in the body, then 



2U = S(X« + Yi; + Zm;); 

 but U may be expressed as a homogeneous quadratic function 

 of the forces ; therefore 



. 



»-* *S+tS+»S « 



comparing which expressions for TJ, we see that 



dV _ dV dV __ 



dX~ U ' dY ~ V > 1Z- W ' 

 Now conceive the body, instead of being free, to be immove- 

 ably attached at certain points to some fixed object, then we 

 shall have for these points 



dV dV d_V 



dX ' dY ' dZ ' 

 that is, the variation in U, due to a change in the resisting 

 force at the fixed boundaries of the system, is zero. And the 

 same is true for a change in the resisting force anywhere within 

 the mass; for conceive the body to be divided into two parts by 

 a surface of any form passing through the point, and let U 1 , U 2 



be the works done in the two portions, then —~ and —~ are 



dJL dJL 



evidently equal and of opposite sign, that is, -?=? = 0, as before. 



