51 



which hes in the plane BAG, with a velocity represented 

 by the magnitvide of AD, AD being the diagonal of the 

 parallelogram ABDC, of which AB and AC are adjacent 

 sides. 



For this purpose assume any point on the siu-face of the 

 sphere, let the projection of this point on the plane BAG 

 be a, situated between AB and AG ; then ah, j^erpendicular 

 to AB, is the projection of the arc described by the 

 assumed point, in a given time, about the axis AB ; and 

 similarly ac is the projection of the arc described by the 

 same point, in the same time, about the axis AC, and if 

 the time be indefinitely small, the diagonal ad of the 

 parallelogram ahdc will represent the direction and mag- 

 nitude of the resulting motion, when both the component 

 forces are impressed at the same instant of time. 



But since ah is perpendicular to AB, and ac to AG, the 

 angle hac is equal to BAG, and the sides about these 

 equal angles are proportional, for they are proportional to 

 the velocities, therefore the parallelogram ABDC is 

 similar to the parallelogram ahdc. It may be shewn that 

 ad is perpendicular to AD, therfore AD is the new axis. 

 Instead, therefore, of taking the parallelogram ahdc, which 

 refers to the motion of a point on the surface of a sphere, 

 we may take the parallelogram ABDC, which refers to 

 the axes on which the sphere turns. Thus it appears 

 that if the adjacent sides of a parallelogram represent 

 respectively the axes on which a sphere turns in obedience 

 to two given forces, and also represent the velocities 

 which the forces impart, the diagonal represents the 

 direction of the resultant axis, and its magnitude re- 

 presents the resultant velocity. 



From the trigonometrical relations of the sides and 

 angles of the figure, it is easy to deduce the most im- 

 portant propositions of the doctrine of compound rotatory 

 motion. Thus:— 



