1865.] The Proposed Pendulum Operations for India, 253 
it is always possible to calculate the length of the simple: pendulum 
which would vibrate in the same time as a given compound pendulum, 
the latter may be used for precisely the same purpose as the former. 
Besides this, there are several other conditions supposed to hold 
good, which in practice are never attained, viz. the are of vibration 
has been assumed to be indefinitely small, the length of the pendulum 
to be constant, 2. e. unaffected by temperature, and the oscillations 
made in vacuo and at the level of the sea. Corrections have therefore 
to be computed and applied to the observations, for each of these 
assumptions. 
The time of vibration* in a circular are is expressed in terms of the 
length of the pendulum, the force of gravity, and a series of ascending 
powers of the are of vibration. The are is always small, but still not 
so small that the terms depending on it can be wholly neglected; the 
first term, however, of the series is all that is ever appreciable in 
practice. Again, the observations are generally continued for a con- 
siderable time, and the change in the are of vibration has to be taken 
into account. It has been shewn mathematically, on a certain sup- 
position regarding the resistance of the air, and found to be the case 
practically, that the arc decreases in a geometric ratio, whilst the times 
increase in an arithmetic ratio, and on this principle the correction} 
to the observed time of oscillation is computed. 
Secondly, a correction must be applied for the temperature of the 
pendulum : a change of temperature will, of course, by altering the 
length of the pendulum, affect the time of its vibration. This cor- 
star JEfia G) e+ (2) (wa) + 
GHEY 
in which t = time of one oscillation. 
a = semi-circumference of a circle whose radius is unity. 
‘ t = length of the Pendulum, 
g = force of gravity. 
a = arc of semi-vibration. 
+ The formula for this correction is iy 
M Sin (A+a) Sin sin (A —a) 
* 32 Log Sin A—Loe Sina Sin a 
n= numberof oscillations made in a day ; M = logarithmic modulus =0.43842945 ; 
A the initiul, and u the final semi-ares of vibration, Correction always additive. 
in which 
