Chap. 7] 



GRAVITATIONAL METHODS 



117 



red. chron. 



— ^obs. + 



S 



86,400 



T, 



obs.j 



(7-22a) 



where S is the rate of the chronometer in seconds in sidereal time per 

 sidereal 24 hour day (positive if losing, negative if gaining), so that the 

 rate correction itself is 0.00001157 ST. The correction for chronometer 

 rate on the coincidence interval n is 



S 



^red. chron. — ^ 



86,400 



n(2n - 1) 



^ + 86,400 ^"^ 



(7-226) 



provided the pendulum swings more slowly than the chronometer does. 

 For rates less than 2| seconds per day, equation (7-226) may be simpliiSed 

 to 



"red. chron. 



n — 



S 



86,400 



n(2n - 1) 



(7-22c) 



If *S < 2| sec, the effect is less than 0.1 milligal. No correction for 

 chronometer rates and no time comparisons are necessary if measurements 

 are made simultaneously on two field stations and if their flash boxes are 

 connected to the same chronometer, or if radio time signals sent out by a 

 central astronomical clock or pendulum are recorded simultaneously. Of 

 course, rate corrections are likewise unnecessary when a gravity pendulum 

 is used for radio transmission from a base station. 



2. The amplitude correction follows from formula (7-l6h), so that the 

 reduced period Tred. a=o = ^"(1 — a/16 ••••), in which a is the average am- 

 plitude during an observation. It may be considered as the arithmetic or 

 geometric mean of the extreme amplitudes or may be obtained from 

 "Borda's relation." If ao is the initial amplitude and a/ the final amplitude, 



/ X 2 fm + a/Y 



or 



(6) a = ocdOif 



, . 2 _ sin (ao + a/) sin (oo — a/) 

 2(loge sin ao — loge sin a/) 



which, for small angles, is 



2 2 



ao — a/ 



2(log« ao - log« a/) 



(7-23a) 



