147 
THEORY OF TIDES. 
able point must be situated before or behind it, in order to re- 
present the actual magnitude of the periodical force by the 
relative situation according to the law or the primary force 
concerned, and to find an expression for this distance in terms 
of the sines of arcs increasing equably, in order to obtain the 
situation and velocity of the body at any time, provided that 
we suppose it to have attained a permanent state of vibration. 
Scholiums. If the oscillating body be initially in any other Different case, 
condition, its subsequent motion may be determined, by consi- 
dering it as performing a secondary vibration with respect to a 
point vibrating in the manner here supposed, which will con- 
sequently represent its mean place ; but if there be no resist- 
ance, the body will have no tendency to assume the form of a 
regular simple vibration, rather than any other. 
Theorem B. 
If the resistance be simply proportional to the velocity, a Forced vibra- 
pendulum with a vibrating point of suspension may perform ^sthe 
regular vibrations, isochronous with those of the point of sus- velocity, 
pension, provided that, at the middle of a vibration, the point 
of suspension (A) be so situated as to cause a propelling force 
equal -to the actual resistance, the extent of the vibrations being 
reduced in the ratio of the whole excursion of the point of 
suspension (BC) to its distance from the middle at the begin- 
ning of the motion of the pendulous body (DC) : and it will 
ultimately acquire this mode of vibration, whatever may have 
been its initial condition. 
Let FG, fig. 3 and 4, be the supposed length of the thread carrying 
the point of suspension, and draw FE passing through D ; then 
if HC=EG be the extent of the vibration, it will be maintained 
according to the law of the cycloidal pendulum. Draw the 
concentric circles BI, DK, HL : novt the initial force may be 
represented by HD, which determines the inclination of the 
thread j and at any subsequent part of the vibration, if the 
centre be advanced from D to M, the time elapsed will be ex- 
pressed by the arc IN 5 DI and MN being perpendicular to 
AB >, and taking HL similar to IN, the perpendicular LP will 
show the place of the pendulous body, and PM the force, 
which may be divided or resolved into two parts, PQ. and QM. 
But PQ is to LK, or HD, as PC to LC, or HC; consequently 
L 2 i this 
