146 ' 
THEORY OF TIDES. 
suspension (xAD), as the length of the thread (AE), supposed 
fo -carry this point, is to the difference of the lengths of the two 
threads. 
In representing the vibrations, we may disregard the curva- 
ture of th6 paths, considering them as of evanescent extent, 
the forces being still supposed to depend on the inclination of 
the threads. Let F be the intersection of AB with the vertical 
line EF, then, upon the conditions of the theorem, BF will 
be equal to AE, since BC : AD = AE : AE - AB, and, by the 
properties of similar triangles BC : AD = BF : BF - AB. Con- 
sequently the inclination of the thread AB will always be the 
same as if F were its fixed point of suspension ; and the body 
B will begin and continue its vibrations like a simple pendulum 
attached to that point, the true' point of suspension accompa- 
nying it with a proportional velocity, so as to be always in the 
right line passing through it and through F. It is obvious, that 
when the thread suspending the moveable point of suspension 
is the longer of the two, the vibrations will be in the same 
direction } when the shorter, in the contrary direction. 
Illustration. Scholium 1. The truth of this proposition may easily be 
illustrated, by holding any pendulous body in the hand, and 
causing it to vibrate more or less rapidly, by moving the hand 
regularly backwards and forwards. 
Application to Scholium 2. The same mode of reasoning is applicable to 
other forces, oscillations of any other kinds, which are governed by forces 
proportional to the distances of the bodies concerned, from a 
point of which the situation, either in a quiescent space, or 
with respect to another moveable point, varies according to the 
law of the cycloidal pendulum, or may be expressed by the 
sines of arcs varying with the time : such forces always pro- 
ducing periodical variations, of which the extent is to that of 
the excursions of the supposed point of suspension in the ra- ; 
tio of n to ?i — 1, n being to l as the squareof the time of the i 
forced, to that of the time of the spontaneous vibration, and 
when n — 1 is negative, the displacement being in a direction 
opposite to that of the supposed point of suspension. Conse- 
quently, when a body is performing oscillations by the operation f 
of any force, and is subjected to the action of any other peri- 1 
adical forces, we have only to inquire at what distance a move- : 
able 
