86 
Proceedings of the Royal Society of Edinburgh. [Sess. 
I shall take next the dynamical equation of motion. The most material 
additional element to the ordinary theory consists in the addition of 
frictional terms. Before referring to them further, however, I shall clear 
out of the way a secondary point, viz. the influence of the pendulum 
spring. It is necessary to put its effect upon the pendulum in a clear 
light, because it is an important feature in all clocks, but especially 
in Kiefler, since there it is the instrument for conveying energy to the 
pendulum and of locking the train. In what follows motion in two 
dimensions only is contemplated, the genesis and effect upon time-keeping 
of a twisting motion about the axis of the rod of the pendulum being 
reserved for subsequent consideration. 
Among other features, bending the spring produces dissipation of energy. 
This, however, may be amalgamated with air-resistance and other sources 
of loss, which are considered below. I shall also disregard the kinetic 
reactions of the spring itself, — both its mass and movement being very 
small as compared with the pendulum — and shall generally retain only 
terms of the first order of the angular displacement of the pendulum. 
The diagram (fig. 5) shows the forces called out, first in the spring, then 
upon the pendulum, and, finally, the kinetic reactions of the pendulum which 
result from the forces applied. 
6 is the inclination of the pendulum to the vertical ; 
x, y, vertical and horizontal co-ordinates of the top of the pendulum 
with respect to the point of fixture of the spring ; 
l, length of spring, so that x = l\ 
d, distance from top of pendulum to its centre of mass ; 
T, S, L, the tension, shear, and bending couple on the spring at its point of 
attachment to the pendulum ; 
X, Y, L 0 , the same things at its upper point of fixture ; 
£, rj, current co-ordinates at any point of the spring ; 
P, maintaining force, applied directly to the pendulum ; 
then the couple at the point is Ed^/dg 2 , where E is a constant ; and we 
have from consideration of the spring the equations 
X - T + S0 = 0, 
Y-T0— S = 0, 
E d*r,/d& = L + Y(x - i) - X(y - v ). 
We may remark that S is of the same order as P the maintenance, and 
hence S/T is of order 10~ 4 . Hence S$ may be omitted in the first of these 
equations. 
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