of a Finite Conservative System. 377 



of the masses, and z the height of their centre of gravity in 

 any configuration of the system, above its lowest. Suppose 

 now A to be placed in any particular position <£ ; and let it 

 be required to find what must be the position, yfr , of B, and with 

 what velocities, $ Q , yfr we must start A and B in motion, so that 

 the first time <f> has again the same value, <£ , that is to say when 

 A has made one complete turn in either direction, the system 

 shall be wholly in the same position ($0^0) an( i moving with 

 the same velocity (</> 0; 'M ( m the same direction understood) 

 as at the beginning. This implies only two equations, -v|r = -v/r ; 



and either ^ = -^ , or (f> — (j>o (because either of these' implies 

 the other in virtue of the equation of energy). And we have 



just two disposables, -*|r , and either ^ or (£ (the given total 



energy E determining either <p Q or yfr when the other is 

 known). The solution of this determinate problem is clearly 

 possible, unless E is too small : but it is not generally unique. 

 We may have solutions with the velocities of A and B started 

 each in the positive direction, or each negative, or one nega- 

 tive and the other positive. If A is a flywheel of very great 

 moment of inertia, and B a comparatively small pendulum 

 hung on a crank-pin attached to it, and if for simplicity we 

 suppose the crank to be counterpoised, so that the centre of 

 gravit} r of A is in its axis, it is clear that, according to the 

 greater or less value given for E, B may turn round and round 

 many times before A comes again to its primitive position </> . 

 But it is clear that, though not generally unique, our problem 

 of finding periodic motion with just one complete turn of A 

 in its period has no real solution unless E is large enough ; 

 has many solutions for large enough values of E ; but has 

 not an infinite number of solutions for any finite value of E. 



8. Again, let the condition be that, not the first time, but the 

 second time A passes through its initial position, both coordi- 

 nates and both velocities have their primitive values. When 

 the given value of the total energy is not too great, the 

 periodic motion which we now have will be purely vibratory ; 

 and the solution clearly duplex. But if E be great enough, A 

 may still merely vibrate, while B may go round and round, first 

 in one direction and then in the other, within the period of A's 

 vibration. If the condition be that not at the first and not at 

 the second, but at the third transit of A across its initial 

 position, both coordinates and both velocities have their pri- 

 mitive values, we may, with sufficiently great total energy, 

 have still wilder acrobatic performances, both bodies going 

 round and round sometimes in one direction and sometimes 



