ASTRONOMICAL AND GEODETIC OBSERVATIONS. 33 



FORMULA]] AND METHOD OF REDUCTION. 



To render the results obtained at different places comparable with each other, 

 the observed number of vibrations require the following corrections, that for rate 

 of clock having first been applied. 



Reduction to Infinitely Small Arc. 



The duration of a vibration in any small arc is always greater than in an infinitely 

 small arc, the correction to the observed number of vibrations is therefore additive. 

 Let A = the initial semi-arc of vibration 

 a = the terminal semi-arc of vibration 

 iV= number of vibrations in a given time; 



then the correction =N^^-^ '' ^^+") '''' jA-a) _ ^ M shv^ 1° _ 4!=^ 



32 (log. sin A — log. sin a) 32 log. A — log. a 



At Cambridge the number, iV, of vibrations in a mean solar day is about 86420, at 



Port Foulke about 86550, and since if, the logarithmic modulus := 0.4342945, the 



M siti? 1° 

 logarithm of the factor iV • — becomes [9.55295] and [9.55361] respectively 



for these localities. 



Correction for Temperature of Pendulum. 



For a higher temperature than the adopted standard temperature, the pendulum 

 becomes longer, and the number of vibrations are diminished ; the correction to N 

 is therefore positive, for a lower temperature than the standard temperature, the 

 correction is negative. , 



Let e = coefficient of expansion of the material of the pendulum bar 

 t =^ observed temperature 

 ^0= standard temperature 



then the correction = iV — [t — <„) 



The average temperature of the pendulum, when swung at Cambridge, was about 

 71°, and at Port Foullie about 23° Fah. I have therefore adopted 50° Fah. as a 

 convenient standard temperature. 



Reliable determinations of e for 1° Fah. seem to vary between 0.0000104 and 

 0.0000105, taking the mean and using iV as above we find for the coefficient of t — t^ 

 the value 0.4511 for Cambridge, and 0.4518 for Port Foulke, or the logarithmic 

 factors [9.65428] and [9.65494] respectively 



Correction for Buoyancy. 



As the pendulum was not swung in a rarified medium to ascertain the correction 

 for buoyancy and resistance experimentally, we use the coefficient determined by 

 Bailey (see Vol. VII, p. 27, Memoirs Eoyal Astronomical Society). 



