"54 



THE EAR. 



In other words, the radius of the arc and the chord of the arc will change, 

 while the length of the arc remains constant. But the length of the arc may 

 be regarded as the length of a radial fibre ; hence — 



I — 1r sin" 



A 



\2r/ 



where I = length of fibre, r = radius of the circle of curvature, and A = chord 



of the arc I, because — is the sine of half the angle at the centre belonging to 



2r 



the arc I. This equation may also be 



written — 



\ 



2rsin(^). 



Now, if we subtract the one equation 

 from the other, we have — 



l-X = 2r 



I 



Yr ~ sin 



(£) 



/ 



which gives the difference between 

 the chord of the arc and the curve. 

 But as the curve is very slight, r is large in comparison with 1, and the 



divisions becoming rapidly very small as the sine in the formula is developed 



by the involution of its arc. Hence — 



I I 



2r 2r \ 2r / 



and from this the preceding equation becomes - 



Z-Wt 



I s 



(!)• 



Again, let s be the distance of the centre of the arc from the centre of the 

 chord, then the degree of curvature is found by the equation — 



r 



So that 



s I 



= COS -— . 



r 2r 



By evolution of the cosine- 



s = i — 



r cos — 

 2r 



= r(l-cos(l-) 



(2)- 



Now eliminate r from equations 1 and 2, and we obtain 



l-k-J 



I 



This equation gives the amount of shortening of the chord which occurs 

 when the curve of the arc is increased ; that is to say, it gives the extent to 

 which the two ends of the fibre are drawn together. Now, if s, the displace- 

 ment of the middle of the fibre, be very small in comparison with 7, then I — A 

 obviously becomes very small in comparison with s. Conversely, the very 

 small increase in the magnitude Z - A must cause a relatively great increase of 



