52 



AC : CD = sin ADC : sin DAC 

 or AC : AB = sin BAD : sin DAC 

 or v' : V = sin BAD : sin DAC 

 because AB and AC represent the velocities v and v'. 



Thus the component velocities are to each other reci- 

 procally as the parts into which the whole angle, between 

 the component axes, is divided by the new axis of 

 rotation. 



Again AC : AD = sin ADC : sin ACD 

 = sin BAD : sin ABD 

 or v' : v" = sin BAD : sin BAC. 



Let ALBK (Figs. 4 and 5) be a sphere rotating about the 

 axis KL, in the direction HAFB, and let an impulse be 

 applied at A, in the direction AR, tending to make the 

 sphere rotate about the axis HF. 



If the point at A, at the instant of impact, could be 

 carried to B in a time infinitely short, the impulse at A 

 would then be equivalent to two equal forces acting on 

 the lever AB at equal distances from the fulcrum C, and 

 in directions AR, BR", parallel to each other, the line AB 

 therefore would remain at rest. But since the sphere 

 cannot rotate with a velocity infinitely great, AB (Fig. 4) 

 tends to assume the position a b immediately after impact, 

 and the point A begins to move in the diagonal of an in- 

 definitely small parallelogram (Fig. 5), in which 



Ad : Aa = velocity about KL : velocity about HF. 

 The axis AB, therefore (Fig. 4), has not remained at rest, 

 but has moved through an indefinitely small angle, ACa. 

 The maximum efi"ect of disturbance on the line AB is 

 produced when the particle struck is at A, acting in 

 the direction AR. When the particle which receives 

 the impulse at A arrives at F by the rotation of the 

 sphere, it tends to move in the cUrection FR' parallel to 

 AR ; here it has no effect on the position of the axis AB, 

 but it tends to turn the whole sphere about that axis in 



