On the Oscillation of the Pendulum. 239 
In the reasoning which follows the radius Cm representing the 
semi-oscillation of the pendulum is supposed, as indeed it always 
is, exceedingly minute compared with the earth’s radius; it is 
also very minute compared with CC’ which measures the rotary 
movement of C during a semi-oscillation. 
being the centre of motion, the bisecting point of each are of 
oscillation, its rotary velocity about the earth’s axis will evidently 
be that of the pendulum itself. 
Even if we suppose the pendulum as commencing its series of 
oscillations at either of the points g or m and thence having their 
rotary velocities, the very first vibration will check or accelerate 
it to that of C. 
While the pendulum is moving from, C to m, suppose the centre 
of motion C to move to C’ by the earth’s rotation; the point m of 
the dial; will only have the velocity due to the radius of the 
earth’s small circle Pm. Its motion will be mm’ while that of C 
is CC’ due to the radius PC. 
But the pendulum having the rotary velocity of the point C 
will advance, in rotation a distance, from its original plane mq, 
equal to CC’. Instead therefore of striking the dial circle at zm’ it 
will strike it in advance of it a distance m/m” equal to CC’—mm’. 
That is to say, the distance the pendulum advances each vibra- 
tion is equal to the difference of the arcs of the circle described 
about the earth’s axis by the points C and m, during the vibration. 
he foregoing proposition has been derived from the considera- 
_ tion of the pendulum vibrating in the plane of the meridian. 
But it can be established in a more general way by considering 
a semi-vibration in any other plane Cm’. (Fig 2. 
In this case the pendulum will gain in the direction of rotation 
upon the point m/ (where it would strike the dial circle, if the 
earth was motionless) a distance om” in the direction of rotation 
due to the difference of rotary velocities of the points C and m/ 
orz. Such an increment in the ordinate from x will cause an 
increment mm’ of arc the same as just found for the vibration in 
the plane of the meridian. — 
_ For, call V the rotary velocity of C, the time of a semi-oscilla- 
ion, ¢ : the radius PC=R’ and that Pm=R”. (Fig. 1.) 
“ut 
The rotary velocity of the point m will be ey and the dif- 
R/—R” 2 
Jerence of velocities of C and m, V ftom - The distance 
m'm’’ gained by the pendulum will therefore be (in case of vibra- 
: : z ) R/—R” 
tion in t e tneridian) = Ve (=a): 
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