THE GEOMETRY OF MOTION 235 



velocity of P relative to O' is equal to the mean velocity 

 of P relative to O added geometrically to the mean 

 velocity of O relative to O'. If we take the interval of 

 time, and consequently the displacements, smaller and 

 smaller, mean velocities become in the limit the actual 

 velocities. These actual velocities have always the direc- 

 tion of the displacements P^P',, P^P.,, and 00.„ which 

 ultimately from chords become tangents to the corre- 

 sponding paths ; further, since the interval of time is 

 the same for all the displacements, the magnitudes or 

 speeds of these velocities are always proportional to the 

 sides Y^\, P^P,, and P,P', (or OO^) of the triangle 

 PjP'.,P.,. Hence the mean velocities and ultimately the 

 actual velocities always form the three sides of a triangle 

 which has its sides parallel and proportional to the sides 

 of the triangle P^P'^Pg, and this however small the latter 

 triangle becomes. The actual velocity of P relative to 

 O' thus forms one side of a triangle of which the actual 

 velocities of P relative to O and of O relative to O' form 

 the other two sides. In other words, the actual velocity 

 of P relative to O' is obtained from the actual velocities 

 of P relative to O and of O relative to O' by adding 

 them geometrically, or by the parallelogram law. Just 

 as the position of P relative to O' was found by applying 

 the parallelogram law to the steps O'O and OP (p. 211), 

 so we obtain the velocity of P relative to O' by applying 

 the same law to the velocities of P relative to O and of 

 O relative to O'. A very similar proof shows us that 

 the acceleration of P relative to O' may be obtained in 

 the same way from the accelerations of P relative to O 

 and O relative to O'. We thus obtain an easy rule — 

 that of the parallelogram law — for passing from the motion 

 of P relative to O to that of P relative to O'. 



The whole of this discussion may be looked at from 

 a somewhat different standpoint. We may suppose the 

 plane of the paper in which the motion of P about O 

 takes place to be always moved as a whole so that the 

 point O' remains stationary. In order to do this we must 

 always be shifting the paper so that O'., falls back on O'^, 



