EFFICIENCY OF MEMBRANE 203 



the length of the arc may be regarded as the length of a radial 



fibre ; hence , / X 



/=2r siirV 



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 



2r 



the centre belonging to the arc /. This equation may also be 



written / Z 



X = 2r si 



Now, if we subtract the (each side) from /, we have 



I . (I 



sm( - 



2r \2r 



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 I and the divisions become rapidly very small as the sine 

 in the formula is developed by the involution of its arc. Hence 



I I 1 



sm 



2r 2r 6 



and from this the preceding equation becomes 



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 - s I 



r 2r 



so that s = r - r cos 



2r 



=r (1 -cos ). 



2r/ 



C 2r2 



r ( i / 1 



we have s =r\l- 11- - ( 



2 \2 



1 / 2 

 that is s=-~ (2) 



8 r 



