140 
the parallel, and consequently that the vertical passing through 
the point of suspension and centre of the earth may be con- 
ceived to move through a succession of small angles in a series 
of planes perpendicular to the axes of the great circles above 
referred to. These, which may be called directive axes, lie 
on the surface of a cone whose axis coincides with that of 
the earth, and whose angle equals twice the latitude. In the 
model, a graduated circle fixed on the vertical shows the de- 
viation of the meridian from the plane of oscillation, after a 
period of time indicated by an hour circle attached in the 
usual way to the axis of rotation. If the successive positions 
of the directive axis were taken indefinitely near to one ano- 
ther, the expression for the azimuthal motion would be 
A= Hsin, 
where A is the angle made by the plane of oscillation with the 
meridian, after the earth has described the angle H round its 
axis, and ) is the latitude of the place of observation. The mo- 
del is so constructed as to enable the directive axis to be placed 
in the positions it occupies at the termination of periods of 
half hours, and the error in the value of A produced by this 
approximation is so small as to be almost insensible on a 
model of the size of the present one, and much less than that 
necessarily arising from defects of construction.” 
The Rev. Charles Graves, D. D., communicated a formula 
containing a symbol which denotes rotation through a given 
angle, and round a given axis, by means of rectangular co- 
ordinates and differential coefficients. 
‘«‘ Sir William Hamilton, by his calculus of quaternions, 
has arrived at a simple mode of denoting rotation round an 
axis. 
“* Using Q to denote the quaternion whose amplitude is 0, 
and whose axis has given directive cosines, he finds that 
