202 
MR. LUBBOCK ON THE PENDULUM. 
have given the errors which would arise in the length of the simple pendulum 
corresponding to given deviations of the knife edges : it is difficult to make 
the results intelligible without the use of symbols ; but I may add, that the 
effect of a small deviation of one of the knife edges in azimuth is quite insen- 
sible : this is not the case with a deviation in altitude : a deviation of a degree 
in altitude increases by 3 the vibrations in twenty-four hours : a deviation from 
horizontally in the agate planes has a more sensible influence than either of 
the former deviations : a deviation in horizontally in the agate planes of 10' 
increases by about 6 the vibrations in twenty-four hours : both these deviations 
have the effect of rendering the distance between the knife edges greater than 
the true length of the simple pendulum. I have also considered the case in 
which the agate planes are fixed on the pendulum and vibrate on a fixed knife 
edge ; and I find, as might be expected, that the length of the simple pendu- 
lum is equal to the distance between the planes. 
Let 0.r, O y, Oz be rectangular coordinate axes meeting in the point O in the 
plane (x y), upon which the pendulum rests ; let the plane (x z) be vertical and 
the plane (x y) nearly horizontal, and let the axis of rotation coincide with the 
line Ox. Let g be the force of gravity, s the angle which a vertical line 
makes with the axis O z, a the distance of the centre of gravity from the line 
Ox, and M (a 2 -f k 2 ) the moment of inertia of the pendulum about the axis 
Oj: ; then, according to the analysis of M. Poisson, (Traite de Mecanique, 
p. 1 1 G.) the length of the simple pendulum which oscillates in the same time 
a 2 + k* 
is . 
a cos 
Let G x t , G y t , G z t be the three principal axes which intersect each other 
in the point G, and let the equations to the axis Ox referred to the coordinate 
axes G x p G y t , G z { , which are fixed in the pendulum and move with it, be 
y t = x t tan 5 -f- /3 
tan 8' 
z. — x. j — f- y. 
S and S' are small angles which may be considered as the deviations of the 
knife edge in azimuth and altitude. 
If a y = b x |3 
