ANIMAL MECHANICS. 



199 



At the points of contact x, y, the tangential strains act in 

 the directions. Ox and Oy, and they must be equilibrated by 

 a perpendicular force acting in the direction CO, passing 

 through the centre. Let T represent the tangential strain at 

 each point x and y ; the resultant of these two forces is 



R = 2Tcos TOO, 

 Since the arc xy is small, we have the intercepts between 0 

 and the circle, and between the circle and chord xy, equal to 

 each other; let the tangent, arc, or semichord be called a, and 

 the intercept c ; then we have 



cos TOO = - = £ 

 a p 



and therefore 



9 



If P represent the perpendicular force acting on each unit 

 length of the circle, the resultant R just found must equili- 

 brate the force P applied to the whole arc intercepted between 

 x and y ; and hence we have 



R = zPa. 



Equating these two values of R, we obtain, 



and, finally, 



2Pa = 2T- 

 9 



T 



r~ (3.0 



Hence, if the tangential strain be given, the perpendicular 

 force will vary inversely as the radius of the circle, or directly 

 as its curvature ; and if the perpendicular force be given, the 

 tangential strain will vary directly as the radius of the circle. 



