50 



vey it to B iu a certain unit of time, the body will move 

 over the line AB with a uniform velocity, and arrive at B 

 at the end of the given time. If it receives an impulse 

 in the direction of AC, of such an intensity as would 

 convey it to C in the same unit of time, at the end of the 

 time the body will be at C, having moved uniformly over 

 the line AC. If both forces, at the same instant, act on 

 the body at A, it will move uniformly over the line AD, 

 the diagonal of the parallelogram of which AB, AC, are 

 adjacent sides, and arrive at the point D at the end of the 

 given time. 



Instead of considering A as a moveable body on a 

 plane, let it be a fixed point on the surface of a sphere, 

 the sphere being free to turn in any direction whatever. 



The force applied at A (Fig. 2) in the direction of a 

 tangent to the arc AC, will caiise the point A to move 

 uniformly through the arc AC. The force applied at A, 

 in the direction of a tangent to AB, will cause the point 

 A to move uniformly through the arc AB. If both forces 

 be applied at the same instant, the point A will move 

 neither over AC nor AB, but over some arc AD 

 intermediate between the two ; and if the time be 

 indefinitely small, the arc AD will be the diagonal of an 

 indefinitely small parallelogram constructed on the same 

 principle as that which applies to compound rectilinear 

 motions. 



In order to pass from the consideration of the motions 

 of a point on the surface of a sphere, to the consideration of 

 the axes on which the sphere Avould turn in obedience to 

 the impressed forces, let AB (Fig. 3) be the axis on which 

 a sphere revolves with a velocity represented by the 

 magnitude of the line AB, and let a force be applied to 

 turn the sphere about the axis AC with a velocity repre- 

 sented by the magnitude of AC ; then it may be demon- 

 strated that the sphere will revolve about the axis AD, 



