THE GEOMETRY OF MOTION 



225 



13, p. 220). As in the first case the direction of actual 

 acceleration at V^ is that of V^T or the tangent at V^, it 

 is clear that as a rule acceleration will not be in the 

 direction of velocity,^ but will act partly in the direction 

 of velocity and partly at right-angles to it. This result is 

 so important that the reader will, I hope, pardon me for 

 considering it from a slightly different standpoint. Let 

 us imagine the acceleration to be such that throughout it 

 never stretches IV, and let us try to analyse this case a 

 little more closely. Obviously if IV be never stretched, if 

 the speed be never spurted, the point V can only describe 



Fig. 14. 



Fig. 15. 



a circle, for IV remains uniform in length. Uniform speed 

 can, however, be conceived associated with a point moving 

 in any curved path whatever. Let Fig. 14 represent this 

 path, and let Fig. 1 5 be the circular hodograph, corre- 

 sponding points of the two curves being denoted by the 

 same subscript numerals attached to the letters P and V. 

 Now, since all the acceleration in this case depends 

 upon the change in the direction of motion, or the change 

 in the direction of the tangent to the path, we must stay 

 for a moment to consider how this change in direction, or 

 the bending of the path, may be scientifically described 

 and measured. Now if we pass, for example, from the 



1 At V:i, for example, IV... appears to coincide with the direction of the 

 tangent at V:j. In this case the whole effect of acceleration is instantaneously 

 to spurt without shunting. 



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