310 Astronomical and Nautical Collections. 
assigned to it, on the very erroneous supposition of a homogeneotis 
sphere of water. 
The third section investigates the ‘ effects of resistance in vi- 
bratory motions, whether simple or compound,” and reduces into 
a somewhat more technical or fashionable form the propcsitions 
which the author had before deduced from a geometrical mode of 
representation, but with considerable extensions and improve- 
ments: and as a corollary tending to illustrate the accuracy of bis 
formulz, he has applied them to the problem of a pendulum mov- 
ing with a resistance proportional to the velocity, which had been 
left incomplete by Euler. He has shown that the resistance, in 
Captain Kater’s experiments, could only have caused an error of a 
second in about fifty years: a quantity certainly altogether insig- 
nificant, but which could not with propriety be wholly neglected, 
while it was known that its magnitude was determinable; and 
while its insignificance remained undemonstrated. 
He then proceeds to compute the effect of periodical forces with 
or without resistance, and shows that the effects of such forces on a 
pendulous or vibratory body are always most considerable when 
the period of the force approaches very near to that of the vibra- 
tion: a proposition which is illustrated by the sympathetic vibra- 
tions of the pendulums of clocks, and in the motion of the inverted 
pendulum, invented by Mr. Hardy, as a test of the steadiness of a 
support, which shows, when it is well adjusted to the rate of a 
clock, that no pillar can be so steady as not to communicate to it 
a very perceptible motion by its regular, though extremely minute, 
and otherwise insensible change of place. 
The theorem most immediately applicable to the case of the tides is this, 
dds ds 
(K): “ the equation, a +A 7 + Bs + Msin, Gt = 0, may be satisfied 
by taking s = aavine gp Ms epicpay sin. (Gt = arcta — AG 2” 
/ ([GG—BF + AAGG B-GG 
which is also extended by a subsidiary approximation to the case of a resistance 
varying as the square of the velocity. 
Inthe fourth séction of the article we find the *« Astronomical deter- 
mination of the periodical forces which act on the sea or on a lake,” 
affording the equations which by means of Theorem K, could give 
