REDUCTION OF PENDULUM OBSERVATIONS. 
119 
swings, and as a matter of fact consecutive swings show just 
this difference. Therefore the discrepancies are explained 
by the variations of the time-piece, both at elevated stations, 
where they amount to 20, and at normal ones, where they 
are as low as 4. The conclusion would then seem to be that 
swinging through the entire 24 hours is necessary, unless 
time can be obtained from a fixed observatory, where the 
clocks are well protected from changes of temperature, and 
even when the swinging is done in observatories, with the 
good temperature conditions that usually exist, the discrep¬ 
ancies from the clock seem to be equal, if they do not over¬ 
shadow, those arising from the other variable conditions. 
The method of reducing the time of a vibration to what 
it would be were the oscillating body to move in a very 
small arc involves purely geometrical considerations, and 
is given in all works on mechanics. Some modifications in 
the application of the formula have been introduced by 
modern experimenters, but the influence of these modifica¬ 
tions on determinations of relative gravity is probably with¬ 
out material significance. The law of decrement of the arc 
is that of a geometrical progression, or at least such a pro¬ 
gression represents the descent with sufficient accuracy. Of 
the different transformations of the formula for the purpose 
of systemizing the computations and making the work 
more expeditious, it is unnecessary to speak. They all come 
within the requisite degree of accuracy, both for absolute 
and relative work. Prof. Peirce however assumes a differ¬ 
ent law of descent, and corrects each set of swings with 
quantities that are derived from observations of the decre¬ 
ments of arc from these same swings. His method is to ex¬ 
press the differential coefficient of the arc with respect to the 
time as the independent variable in terms of constants and 
ascending powers of the arcs. These constants, determined 
either graphically or algebraically, being substituted in the 
integral formula which represents the sum of all the arc 
corrections, gives the correction for the whole swing. This 
method, which certainly answers most thoroughly the 
