542 GENERAL WALKER ON PENDULUM OPERATIONS FOR DETERMINING 
was at first deduced from observations of the disappearance and reappearance of both 
edges of the disc ; but after a short time, this was considered unnecessary, and one edge 
only was observed, the same throughout each set of swings. 
Colonel Herschei. substituted for the single large disc a system of multiple discs 
consisting of several pairs of small circles, arranged symmetrically on opposite sides 
of a central vertical line, and painted white on a piece of black cardboard which was 
attached to the bob of the clock pendulum. He designed a large variety of systems, 
one of which is here shown. He observed the times of disappearance and re-appear¬ 
ance of several pairs of discs, eventually retaining five pairs only, of which the general 
mean was taken as the moment of coincidence. The United States’ officers adopted 
one of Colonel Herschel’s discs, but observed on only one side of the tail-piece and 
not on both sides as he had done. In the revisionary operations at Kew and Green¬ 
wich, a single large disc, of which the image was made of nearly the same diameter as 
the tail-piece, was employed, and observations of disappearance and re-appearance 
were made on one side only, as in India. 
In the Indian and the revisionarj operations the times of the three first and the 
three last coincidences in each set of swings were observed, and the means were 
employed to indicate the moments of commencement and conclusion of the set; the 
observed intervals between successive coincidences gave the divisor to the duration of 
the set to find the total number of intervals which is wanted in calculating the 
vibration-number. In Colonel Herschel’s operations one or two discs were observed 
* Very great precision in the determination of the moment of coincidence is unnecessary. If V be 
the vibration-number of a pendulum, R the clock vibrations in a mean solar day, and N the clock 
vibrations during a set of swings in which there are n intervals between visible coincidences, then 
V = R h - -y j and dY = 2R dN. 
Let R= 86,6.30, let the duration of the set of swings be 6 hours and the interval between coincidences 
6 minutes, giving n = 60, then 
dY - -022 dN. 
Thus an error of 4 seconds in N. which is improbably large, would not affect the vibration-number 
by as mnch as 01, which is but a fraction of the probable error from other causes. 
