EFFICIENCY OF MEMBRANE 203 



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



fibre ; hence . n / A 



l=2r snr 1 !-- 



2r 



where / = length of fibre, r = radius of the circle of curvature, and 



^ 



X chord of the arc /, because - is the sine of half the angle at 



2r 



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



written / I \ 



X =2r sm(--V 

 \2r) 



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



, 

 2r V2r/J 



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 / and the divisions become rapidly very small as the sine 

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



/ i i/zy 



sin - - 



2r 2r 6 \2r / 



i 



and from this the preceding equation becomes 



1 / 3 

 Z-A = -.., ....................................... (1) 



24 r 2 



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 



= cos , 

 r 2r 



so that s = r - r cos - 



2r 



I 

 =r 1 1 -cos 



2r. 



i i/zy 



Since COS 2~r = ~' l ~~2\2r) a PP rox " 



we have 8 = r [l - (l - i (^ 



1 I 2 

 that is s= -~ ............... , .......... . ......... (2) 



8 r 



