H54 



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 



= Zr sin 



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

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



the arc Z. This equation may also be 

 Avritten 



A = 2i 



Now, if we subtract the one equation 

 from the other, we have 



/ 



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 Z, and the 

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

 by the involution of its arc. Hence 



and from this the preceding equation becomes 



I A-' ? 



L - A - tt r 2 



(I)' 



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 



So that 



r s I 



= cos . 



r 2r 



= r -r cos 

 2r 



By evolution of the cosine 



(2). 



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



2 



Z-X = | s - 



z 



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 Z, then Z X 

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

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



