522 BELL SYSTEM TECHNICAL JOURNAL 



Let us assume that the pendulum is sustained in oscillation by a succession 

 of short impulses, one for each swing applied at some phase angle (pi. If 

 the impulse is really short, the velocity will be increased to the value that 

 the pendulum had when it occupied the same position during the last swing. 

 This change of condition is represented by the short horizontal path on the 

 velocity-phase diagram and, as indicated, is accompanied by an advance in 

 phase A<pi. This can be interpreted as meaning that the period of a pendu- 

 lum sustained in oscillation in this way is reduced from its natural period in 



the ratio of — ^. It is obvious from the diagram that A</)i becomes 



ZTT 



smaller and that this ratio approaches unity as the phase of the applied 

 impulse approaches that of the maximum velocity — that is, when the 

 pendulum is in the center of its swing; and this is Airy's condition. It is 

 clear also that if the impulse is applied after (instead of before) the instant 

 of maximum velocity, the period will be correspondingly increased. From 

 the geometry of the figure, it can be seen that, in the neighborhood of the 

 optimum condition^ the deviation from natural period is very closely pro- 

 portional to the amount of the phase departure. 



The closeness of spacing of the turns of the spiral depends directly on the 

 Q of the resonant element. For a Q of 200, the turns will be packed ten 

 times closer than shown, and the corresponding A^ will be only one tenth 

 as great, other conditions being comparable. For a Q of a million or more, 

 A(p becomes very small indeed, especially when cp is properly chosen — and 

 the variation in A^, which is a measure of the variation in rate due to the 

 driving means, may be made vanishingly small. 



The importance of the above properties to timekeeping depends upon 

 how well conditions can be set up to realize them. At first wholly mechani- 

 cal means were employed and, with the advent of the dead-beat and detached 

 escapements and by careful design and operation, quite remarkable perform- 

 ance was obtained. 



A new approach in timekeeping methods was introduced by Alexander 

 Bain^ in 1840 when he first used electrical means for sustaining a pendulum 

 in oscillation. The importance of Bain's invention of the electric clock is 

 indicated by a long controversy over the priority of the invention with 

 Charles Wheatstone, who was working along similar lines at the same time 

 as a by-product of his extensive researches on the electrical telegraph. 

 A brief story of this controversy entitled ''The First Electric Clock" was 

 written for the one-hundredth anniversary of Bain's invention'^. The first 

 electric pendulum clocks could not compare in accuracy with the best 

 mechanically driven pendulums of the period but, in spite of a great deal 

 of initial skepticism on the part of those brought up in the mechanical 



