376 Sir W. Thomson on Periodic Motion 



5. To prove this theorem, suppose the number of freedoms 

 to be ?'. Any configuration, Q, is fully specified by i given 

 values for the i coordinates respectively. Suppose now the 

 system to pass through some configuration, Q, at two times 

 separated by an interval T, and to have the same velocities and 

 directions of motion at those times. The path thus travelled 

 in this interval is an orbit, and it is periodically travelled over 

 in successive intervals each equal to T. To find how to pro- 

 cure fulfilment of our supposition, let the system be started 

 from any configuration, Q, with any i values for the i velocity- 

 components (or rates of variation per unit-time, of the i coordi- 

 nates). To cause it to return to Q after some unknown time 

 T, we have i—1 equations to be satisfied : and to cause i—1 

 of its velocity- components to have the same values at the 

 second as at the first passage through Q we have i—1 equa- 

 tions to satisfy ; and, in virtue of the equation of energy, the 

 remaining velocity-component also must have the same value 

 at the two times. That the total energy may have the prescribed 

 value, E, we have another equation. Thus we have in all 

 2 i—1 equations, among coordinates and velocity-components. 

 Eliminate among these the i velocity-components, and there 

 remain i — 1 equations among the i coordinates which are the 

 conditions necessary and sufficient to secure that Q is a con- 

 figuration of an orbit of total energy E. Being i — 1 equations 

 among i coordinates, they leave only one freedom, that is to 

 say they fully determine one path; of which, in the language 

 of generalized analytical geometry, they are the equations. 

 The or any path so determined is an orbit of total energy, E. 

 Thus is proved the Theorem of § 4. 



6. The solution of the determinate problem of finding an 

 orbit whose total energy has the prescribed value, E, is, in 

 general, infinitely multiple, with different periods for the 

 infinite number of different orbits determined by it. 



7. A simple illustration with only two freedoms, will help 

 to the full understanding of § 6 for every case, of any number 

 of variables. Consider a jointed double pendulum consisting 

 of two rigid bodies, A and B : one (A) supported on a fixed 

 horizontal axis, I ; the other (B) supported on a parallel axis, 

 J, fixed relatively to A : and for simplicity let G, the centre 

 of gravity of A, be in the plane of the two axes. Call H the 

 centre of gravity of B. Let <£ be the angle between the plane 

 I J and the vertical plane through I, which we shall call I V ; 

 and let yjr be the angle between the plane J H and the vertical. 

 The coordinates and velocities of the system in any condition 

 of motion are <f> , yjr, <£, ^. The potential energy of the sys- 

 tem, in kinetic units, will be yWs, where W denotes the sum 



