the Density of the Earth. 519 
The motion of the ball answering to one division of the arm 
= » anc h if m B is the motion of the ball answering to 
d divisions on the arm, — ^ 8j8; =-^~; and there- 
fore, the time of vibration, and motion of the arm, must be cor- 
rected as follows : 
If the time of vibration is determined by an experiment in 
which the weights are in the near position, and the motion of 
the arm, by moving the weights from the near to the midway 
position, is d divisions, the observed time must be diminished 
in the subduplicate ratio of 1 — — to 1, that is, in the ratio 
°f 1 — 7^ to i ; but, when it is determined by an experiment 
in which the weights are in the midway position, no correc- 
tion must be applied. 
To correct the motion of the arm caused by moving the 
weights from a near to the midway position, or the reverse, 
observe how much the position of the arm differs from 20 di- 
visions, when the weights are in the near position : let this be 
n divisions, then, if the arm at that time is on the same side of 
the division of 20 as the weight, the observed motion must be 
diminished by the ~ part of the whole; but, otherwise, it must 
be as much increased. 
If the weights are moved from one near position to the other, 
and the motion of the arm is 2 d divisions, the observed mo- 
tion must be diminished by the part of the whole. 
If the weights are moved from one near position to the other, 
and the time of vibration is determined while the weights are 
in one of those positions, there is no need of correcting either 
the motion of the arm, or the time of vibration. 
