90 
Proceedings of the Royal Society of Edinburgh. [Sess. 
this hereafter. I shall merely mention here that over the range of arcs 
that are considered I find that the rate of decline of motion of a free 
pendulum, subject only to air-resistance and the dissipation of energy in 
the spring, is satisfactorily represented by the solution of the equation 
x" + KX + Xx' 2, + 7l 1 X = 0. 
The quantity k/u ranges from 05 x 10~ 4 to 08 x 10 -4 approximately, for 
different pendulums, and A is such that in the neighbourhood of the 
semi-arc a^lOO' 
^ ^ to k I a o 2?l2 1 
that is to say, the second term with respect to the third bears much the 
same order as the fourth term would bear to the term ?i 2 (sin x — x) 
= — \n 2 x z which is required to complete it. Both may be treated in the 
same way. Deferring this point for the moment, when the action of the 
maintenance, represented by the term R, is included, the motion over a 
portion of every oscillation may be considered to differ from that of the 
free pendulum to which the equation above applies. For example, there 
is the friction of unlocking, and in the case of Riefler there is the pivoting 
on the knife-edges for a portion of the swing. All such additions, of what- 
ever origin and character, supplementing the equation of motion of a free 
pendulum, are considered as a part of R. But such a feature as the 
friction of the crutch on its pivots, in Synchronome, and of the link that 
joins it to the pendulum, would be included in the left-hand member of 
the equation, because these are continually present. 
Consider, then, the fundamental equation of motion 
x + kx + n 2 x = R . . . . . (8) 
The chief peculiarity of this equation is that the term R, being set in 
action by the pendulum unlocking the maintenance, has not a given period 
of its own but is strictly of equal period with the motion taken up by the 
pendulum itself. This feature modifies the bearing of the work throughout. 
As mentioned above, in the best clocks k/u is less than 10~ 4 . Hence 
any approximation including ( k/u ) 2 may be considered complete, running 
as it does below the order of: 0 S 001 per day in the rate. But for mathe- 
matical purposes it will often be convenient to give exact solutions. 
Taking the suggestion from Airy,* let us write the solution 
x — a sin r = a 0 exp v . sin r 
x = n'a cos t = 7i'a 0 exp v . cos r .... (9) 
where 
r — n't + e. 
* Trans. Cambridge Phil. Soc ., iii (1827), p. 105. 
