F. A. P. Barnard on the Pendulum. 195 
2de dk. dD 
eee oe 
Z oO eee 
And, 
de dD 
*. =e 
Putting the mean density of the air= 1, and substituting a 
finite difference for dD, we shall find that the corresponding 
of the mean, the are of vibration will change one sixtieth part 
of the whole; that is 10 say, if the value of « is 2°, the arc 
fall off, or increase to the amount of 2°. : 
_ To compute the effect of such a change upon the quantity 47, 
We may regard the series in equation (1) as being sensibly con- 
stant, and then, representing the whole expression, except the 
denominator of the coéfficient fraction, by Q, and omitting the 
significant term m from the denominator, we shall have, 
4T=~ g and da je bebe 
a a 
Substituting ~«4T for Q and reducing, we have, 
ATdea 
d Pe Ff Sees ee en eet ® 
a 
Which, in the extreme case supposed above, gives a diminution 
of the daily acceleration equal to 058 sec. ‘This change is, un- 
fortunately, in the same direction as that of the circular error: 
but it is proportional to the quantity 4 7 itself, which is directly 
as the maintaining power; which, again, as 2 ape from equa- 
tion.(), is as the square of the arc. Hence, therefore, the im- 
of reducing the are of vibration, and the near approach 
to sensibility of the errors arising from its variations, when it 
Ssmall, Were the are only 1° on each side of the vertical, the 
7 ct would be between one and two hundredths of asecond per 
) 
Rot be an entire second in error in nine months 
ing the question how great a reduction of are is practicable ; but 
the Principle of the oceaainn exacts no larger motion than 
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