310 E. D. Preston — Deflection of the 



tion, we know that the whole number of oscillations is even. 

 Hence the uncertainty, of half a second throws no doubt on 

 the number made. However, if the variations in temperature 

 are excessive, it may be necessary to take this into account 

 when the intervals are long ; for example, one degree centi- 

 grade changes the time of oscillation by one hundred thou- 

 sandth part of a second ; therefore an interval of 15000 oscilla- 

 tions, in which there was a change of say 5°, would be differ- 

 ent from the normal period by about f of a second ; and this 

 with the uncertainties arising from the other varying condi- 

 tions, would make it extremely doubtful, how many oscillations 

 were actually observed. But if intermediate transits are taken 

 every two hours, none of the conditions can change enough to 

 endanger t*he count. 



The observations are reduced to similar conditions at the 

 different stations, as regards temperature of the pendulum, 

 pressure of the atmosphere, amplitude of oscillation, and rate 

 of time piece. The temperature and pressure coefficients are 

 those employed by Professor C. S. Peirce, and which he de- 

 termined experimentally for these particular pendulums. The 

 corrections for arc were calculated by several different formulae, 

 all of which however gave practically the same result. That 

 of Borda was given the preference on account of the rapidity 

 with which the numerical computations could be performed. 

 It supposes that the arcs decrease in a geometrical ratio, while 

 the time increases in an arithmetical one. Peirce's formula 

 assumes that the differential coefficient of the arc, with refer- 

 ence to the time as the independent variable, may be expressed 

 in terms of the ascending powers of the arc and constants. 

 These constants are to be determined from the curves of decre- 

 ment themselves, for each set of swings ; or in case no abnor- 

 mal decrement occurs, mean values for the constants may be 

 employed to correct all the swings. Weddle's rule, which finds 

 a value for the mean square from six equidistant values, gives 

 approximately the same result. 



The varying conditions of rate, temperature, pressure, am- 

 plitude, elevation, and latitude, all influence the period by 

 nearly the same amount, when the conditions change by cer- 

 tain simple units. One second per day, one degree centigrade, 

 one inch pressure, one hundredth of the radius of amplitude, 

 one hundred meters of elevation and ten minutes of latitude, 

 all changing the period of a seconds pendulum by about one 

 hundred thousandth part of itself. 



In the determination of differential gravity much labor can 

 be saved by beginning each swing at the same amplitude, and 

 making it consist of the same number of oscillations. This 

 makes the corrections for arc the same for all, and in the 



