224 THE GRAMMAR OF SCIENCE 



P, ; the instantaneous change in OP, was in the direction 



4 ' '-'4 



of the tangent at P^, and was measured by the slope of 

 the time-chart at O^ (see Fig. lo). In precisely the same 

 manner the instantaneous change in IV^ will be along the 

 tangent at V^, and will be measured by the slope of the 

 time -chart for V's motion at the corresponding point. 

 Thus actual acceleration appears, as in our first discussion 

 of the matter, as the velocity of V along the hodograph. 

 Now, however close V^ is to V^, whether we give a stretch 

 and a spin or add the small step V^Vg, the final result of 

 the two processes will be the same. Hence we can either 

 look upon actual acceleration as the velocity of V along the 

 hodograph, or as the combined mode in which IV is being 

 actually stretched and spun.^ Either method of treating 

 acceleration leads to the same result, and both possess 

 special advantages for describing various phases of motion. 

 In the first case actual acceleration is represented by a 

 step ; the bearing of this step denotes the direction and 

 sense in which V is moving, or the velocity with which 



IV is changing ; the number of units of length in this 

 step denotes the number of units of speed with which 



V is moving, or the number of units of speed being 

 actually added per unit of time in the given direction to 

 the velocity IV of P. By " added in the given direction" 

 we are to understand that the increments of velocity are 

 to be added geometrically or by the parallelogram law 

 {e.g. IVg= ^^4 + ^4^5> ^"<^ ^J^is however small V^V^ may 

 be in conception). 



^ 14. — Curvatiire 



In the spurt and shunt method of regarding accelera- 

 tion, on the other hand, actual acceleration will be specified 

 by two factors : (i) the rate at which velocity is being 

 spurted or IV being stretched ; (2) the rate at which 

 velocity is being shunted or IV being spun about I (Fig. 



^ What we have here stated of acceleration applies just as much to change 

 of position. Turning to Fig. 9, we may look upon the change of position of 

 OP as measured by the velocity of P along its path, or by the manner in which 

 OP is being actually stretched and spun. 



