THEORY OF TIDES. 
148 
this part of the force will always be employed in generating the 
regular velocity ; and QM is equal to KR, which is the sine 
of the angle KNR or RCL to the radius KN=DI=AC, and 
which therefore varies as the velocity, and will always be equal 
to the friction, provided that it be once equal to it, the ratio of 
the forces concerned in any two succeeding instants being 
always such as to maintain a regular vibration. 
If the pendulum be initially in any other situation than that 
which is here supposed, its subsequent motion may be deter- 
mined by comparison with a point vibrating regularly : and if 
we wish for a general view of the case in an early stage, the 
effect of resistance in these secondary vibrations with re- 
spect to such a point, may be neglected : but since they are 
not supported by any sustaining force, they will evidently be 
rendered by degrees smaller and smaller, so that the pendulum 
will ultimately approach iufinitely near to the regular state of 
vibration here described, which may therefore be considered as 
affording a stable equilibrium of motion. 
Magnitude of Scholium 1. Supposing the relation of the resistance to the 
displacement. ve ] oc j t y to \y Q altered, the relation of the sine AC to the cosine 
CD must be similarly altered, the force equivalent to the re- 
sistance varying as the sine, and the extent of the vibrations, 
and consequently the velocity, as the cosine of the displacement 
BI : but the relation of the sine to the cosine is that of the 
tangent to the radius : so that the tangent of the displacement 
will be as the mean resistance. And the sine of the displace- 
ment AC is to the radius BC as the greatest resistance is to the 
greatest force which would operate on the pendulous body if 
it remained at rest at G. 
Different de- Scholium 2. This proposition may also be deduced from the 
wionstration. former, by representing the resistance as a force tending to a 
moveable centre of attraction, analogous to the point -of sus- 
pension of a pendulum, so as to create a new vibration liable 
to an equal resistance ,* or still more simply in the present in- 
itance, by attributing the whole actual resistance to the prin- 
cipal vibration, and considering the subordinate vibration as 
exempt from it. The resistance at G may evidently be repre- 
£resented by the force acting on a pendulum of the length AG 
at the distance AC from the vertical line, and the correspond- 
ing excursion of the pendulous body must be represented, ac- 
cording 
