EFFICIENCY OF MEMBRANE 203 



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



fibre ; hence , , . . 



Z=2rsiirV 



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 



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

 written / I 



Now, if we subtract the (each side) from /, 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 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 l// 3 



and from this the preceding equation becomes 



1 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-s I 



- *= cos , 

 r 2r 



so that s = r - r cos 



2r 



=r (1 -cos 

 V 2rv 



/ l//\ 2 



Since cos =1 ^1 J approx., 



[( 1 / / \ 2 }~] 

 i - 1 1 -(-ML 

 2 \2r' J -I 



that is s= 112 , (2) 



8 T 



