1865.] . The Proposed Pendulum Operations for India. 253 



it is always possible to calculate tlie 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 purposes as the former. 



Besides this, there are several other conditions supposed to hold 

 good, which in practice are never attained, viz. the arc of vibration 

 has been assumed to be indefinitely small, the length of pendulum 

 to be constant, i. 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 arc is expressed in terms of the 

 length of the pendulum, the force of gravity, and a series of ascending 

 powers of the arc of vibration. The arc 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 arc 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 correctionf 

 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. Tbis cor- 



* , =\/H 1+ 0y sina ^ + (ia) 8 ( s » a f) + - 



Qst^y^y] 



in which t = time of one oscillation. 



ir= semi-circumference of a circle whose radius is unity. 



7 = length of the Pendulum. 



g = force of gravity. 



a = arc of semi-vibration. 



f The formula for this correction is 



M. Sin (A 4- a) Sin (A-cc) . 



n.^r -^ 5^ — * — t a^~„ m which 



32 Log Sin A — Log Sin «• 



n = number of oscillations made in a day ; M = log i. e. modulus = 04342945 ; 

 A the initial and a the final semi-arcs of vibration. Correction always additive. 



