230 THE GRAMMAR OF SCIENCE 



the radius IV,. Takin<j P^ closer and closer to P,, and 

 therefore V, to V,, mean values become the actual values 



.> 4' 



at P^ and V^ ; we therefore conclude that the actual spin 

 of the tangent at P^ is the ratio of the actual speed of V 

 at V^ to IV^, or, in other words, to the speed of P. Thus 

 the spin of the tangent is the ratio of the speed of V to 

 the speed of P. But the speed of V is the magnitude of 

 the acceleration, which in this case is all shunt. Hence 

 we conclude that the rate of shunting at P is properly- 

 measured by the product of the spin of the tangent and 

 the speed of P (or in symbols, shunt acceleration = S X U, 

 U being the speed of P). But we have seen above that 

 the curvature is the ratio of the spin of the tangent to the 

 speed of P (or in symbols curvature = S/U). Combining, 

 accordingly, these two results we see that the shunt 

 acceleration in this case is properly measured by the 

 product of curvature and the square of the speed. ^ This 

 acceleration takes place in the direction V^T^, or is per- 

 pendicular to the direction of motion at P. 



A little consideration will show the reader that the 

 expression we have deduced for the acceleration per- 

 pendicular to the motion would not be altered were the 

 speed to vary between P^ and P.. For, returning to Fig. 

 13, we note that IV^ is to be changed to IV,. This can 

 be conceived as accomplished in the following two stages 

 (p. 223): (i.) rotate IV^ round I without changing its 

 length into the position IV. ; (ii.) stretch IV^ in its new 

 position into IV,. The first stage corresponds to the type 

 of motion we have just dealt with, or shunt acceleration 

 without spurt ; the second stage to the case of spurt 

 acceleration without shunt. In the limit when IV. is 

 indefinitely close to IV^, the first stage gives us the element 

 of acceleration perpendicular to the direction of motion, 

 and the second stage the element of acceleration in the 

 direction of motion. By the above reasoning the former 



1 If r be the radius of curvature (see the footnote, p. 228), then \jr will 

 be the curvature, and if we term this element of acceleration normal accelera- 

 tion, we have, by the above results, the three equivalent values : normal 



U2 



acceleration := — =S X U =rS'^. 



r 



