378 Sir W. Thomson on Periodic Motion 



in the other. Still with any finite value for E there is only 

 a finite number of modes for the motion subject to the con- 

 dition that the third transit of A through its initial position 

 completes the first period. Wilder and wilder vagaries we 

 have to think of if the first period is completed at the fourth 

 transit of A ; and so on. 



9. This terrible Frankenstein of a problem is all involved 

 in a very simple mathematical statement not including any 

 declaration that it is the first, or the second, or the third, or 

 other specified, transit of A that completes the first period. 

 It will probably be convenient to arrange so as to find a 

 transcendental equation which will have an infinite number 

 of finite groups of roots equal to the periods of the modes of 

 the periodic motions. 



10. The case of no gravity presents a vastly simpler problem,, 

 of which the main solution has no doubt been many times 

 found in terms of elliptic functions in the Cambridge Senate- 

 house and Smith's Prize examinations. The character of the 

 solution of this, as of all u adynamic" problems, is indepen- 

 dent of the absolute value of the given energy, and of cf> . It 

 depends only on the value of the ratio ^ /^oj which of course 

 may be either positive or negative. In the general solution 

 / v/r- i -^) is clearly a periodic function of the time ; and our 

 question of periodicity relatively to a fixed plane through I 

 resolves itself into this : — During the period of the variation 

 of yjr — </>, is the change of (/> either zero or a numeric com- 

 mensurable with 27r? A corresponding question occurs for 

 every case in which our " system " is free in space, without 

 any fixed guides, and with no disturbing force from other 

 bodies ; as, for example, in the question of rigorous periodicity 

 of the motion of three bodies such as the earth, moon, and 

 sun, or of any finite number of mutually attracting bodies, 

 such as the solar system, to be considered presently. 



11. An (idealized) ordinary clock with weight, and pen- 

 dulum, and dead-beat escapement, affords an interesting 

 illustration. For simplicity let the cord be perfectly flexible 

 and inextensible ; let the cord- drum be rigidly fixed on the 

 shaft of the escapement-wheel ; let the escapement be rigidly 

 fixed to the pendulum ; and let the pendulum be a rigid body on 

 perfect knife-edge bearings. Thus we have virtually two bodies, 

 each with one freedom : A the escapement-wheel, cord, and 

 weight ; B the escapement and pendulum. Each impact of 

 tooth on escapement is, in every clock and watch, followed by 

 a mutual recoil. This recoil probably in almost all practical 

 cases goes so far as to produce complete separation, followed 

 by several more impacts and recoils before the tooth escapes, 



