152 Royal Society. 
possible by the methods which had been before resorted to: for the 
determination of the precise centre of oscillation of a body vibrating 
as a pendulum, depending as it does on the regular figure and uni- 
form density of that body, involves difficulties which might be re- 
garded as insurmountable. Capt. Kater fortunately discovered the 
means of solving this problem, by the application of a mathematical 
property already known to belong to the centre of oscillation, but 
which had never hitherto been practically employed with this view ; 
namely, that this centre and the centre of suspension are reciprocal to 
one another : that is to say, that if a body, vibrating as a pendulum, be 
inverted, and suspended by its former centre of oscillation, its former 
point of suspension will become its centre of oscillation in its new 
position; and the vibrations in both positions will be performed in 
equal times. This property, therefore, furnishes an easy method of 
determining the exact distance between these two points, in a body 
of any form, or however irregular may be the densities of its different 
parts; for it will be only necessary, for that purpose, to provide a 
second axis of suspension, placed by estimation very near to the 
centre of oscillation, while the body is vibrating on its first axis, 
and also capable of adjustment as to distance, and as to its being kept 
in the line passing through the first axis, and the centre of gravity : 
thus by repeated trials of the number of vibrations performed, in a 
given time, by that body, when suspended on either of these two axes, 
and by altering the place of the moveable axis until this number be- 
comes the same in both positions, we obtain a final adjustment which 
gives the exact distance between the centres of suspension and os- 
cillation in that body; a distance equivalent to the length of a sim- 
ple pendulum performing the observed number of vibrations in a cer- 
tain time. 
The mode of suspension adopted by Capt. Kater was the knife- 
edge, of which he points out the various advantages and disadvan- 
tages, and the methods he took for overcoming the difficulties of the 
inquiry. By employing the method of coincidences he found that 
the number of vibrations made by the pendulum in twenty-four 
hours might be obtained to within half a second of the truth in the 
space of eight minutes: and he then applied the usual correction 
for the extent of the arc of vibration, and also for the height of the 
place of observation above the level of the sea. 
5. This paper was followed by another, ‘“‘ On the length of the 
French Metre estimated in parts of the English Standard :’ in de- 
termining which he employed the same micrometer microscopes as 
were used in the pendulum experiments, bringing them alternately 
over the metre and over the standard scale, placed in the same plane 
parallel to and in contact with one another; care being taken that 
their temperatures were the same. 
6. In the following year (1819) Capt. Kater gives an ‘* Account of 
experiments for determining the Variation in the Length of the Pen- 
dulum vibrating Seconds, at the principal stations of the Trigonome- 
trical Survey of Great Britain :” a paper which is full of laborious 
calculations, founded on the observations therein detailed. The in- 
vestigation of the diminution of terrestrial gravity from the equator 
