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 = H sin A, 



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 A. 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 



