Lord Rayleigh on a Statical Theorem. 453 



Let the system be referred to the independent coordinates 

 Tjr p yjr^, &c., reckoned in each case from the configuration of 

 equilibrium. Since only small displacements are contemplated, 

 yjr l &c. are small quantities whose squares are to be neglected. 

 Then, if M^, ^ q , &c. are the impressed forces of the correspond- 

 ing types, the equations of equilibrium are of the form 



"»tl + « 22 ^2 + «23^3 +..--= %, l 

 ^1^1 + «32^ 2 + ^33^3 + * ' ' = ^3> 



(A) 



being the same in number as the degrees of freedom. It may 

 be observed that forces of a constant character need not be in- 

 cluded in M^j &c. ; for the effect of such is only to alter the con- 

 figuration of equilibrium, and may be supposed to be already 

 accounted for in the estimation of that configuration. 



If the system be conservative, as is here supposed, the coeffi- 

 cients in the equations (A) are not all independent ; for in order 

 that an energy-function may exist, any coefficient such as a r3 

 must be equal to the corresponding a sr . 



The solution of equations (A) is 



V* 1 =A 11 ¥ 1 + A 12 <p 2 +... 



(B) 



where y denotes the determinant 



A - dv 

 k »~da n 



A 12 =|j,&e., 



in which, therefore, by a property of determinants, 



A rs =A sr . 



In the application that we are about to make it will be sup- 

 posed that all the forces but two vanish, for example that ty 3 , 

 ^ 4 , &c. vanish. Under these circumstances we obtain from (B) 





(C) 



