Lord Rayleigh on a Statical Theorem. , 455 



wards at Q, the reaction bears the same ratio to the force as the 

 displacement at Q would bear to the displacement at P when the 

 unsupported rod is bent by a force applied at Q. 



In this proposition the interchange of P and Q gives a differ- 

 ent, though of course an equally true statement. The first two 

 propositions are themselves reciprocal in form. 



The second and third propositions, as well as the first, admit 

 of the extension to the vibrations of systems subject to inertia 

 and dissipation ; but I do not here pursue this part of the subject. 



Our fundamental equations (C) may be arrived at with less 

 analysis and perhaps equal rigour by a somewhat modified pro- 

 cess. The conditions that the forces *F 3 , ^ 4 , &c. vanish, impose 

 linear relations on the coordinates, and virtually reduce the de- 

 grees of freedom enjoyed by the system to two. But for only 

 two independent coordinates we have at once 



%=Kf,+h^J ■■■■■() 



where the coefficients b 1<2 , £ 21 are equal. The equality of the 

 coefficients 6 ]2 , 6 2] is a consequence of the existence of an energy- 

 function, or may be proved de novo by taking the system round 

 the cycle of configurations represented by the square whose an- 

 gular points are 



*i=o\ *,=oi ^,=ii f,=n 

 *,=o/ +»=iJ f 2 =iJ t 2=0 ^". 



Prom (D) we may deduce the three propositions directly, or 

 mediately with the aid of (C), which is merely the algebraic so- 

 lution of (D). 



Finally, I would remark that essentially the same method, 

 though with a somewhat different interpretation, is applicable to 

 systems other than those contemplated in the preceding demon- 

 strations. In thermodynamics the condition of a body is regarded 

 as depending on two independent coordinates such as the tem- 

 perature and volume ; and by the principles of that subject it is 

 known that a function of that condition exists, representing the 

 work that can be got out of the system in reducing it to a 

 standard condition of volume and temperature, any communica- 

 tion or abstraction of heat being made at the standard tempera- 

 ture. The simplest course that can be taken is along an adia- 

 batic up to the standard temperature, and then along the iso- 

 thermal until the standard volume is attained. If the actual 

 condition of the body be defined by v + dv, t + dt, while the 

 standard condition corresponds to v, t, we have for the available 

 energy, or entropy (de), 



2de~dpdv + d(j>dt, 



