ANIMAL MECHANICS. 



197 



fibres forms the entire muscle. Sphincter muscles are usually 

 placed round an orifice, which is closed by the contraction of 

 the fibres. These two systems of fibres counteract each other, 

 and by means of their alternate action the orifice may be 

 opened or closed to any required extent. 



We shall first consider the theory of a single circular fibre, 

 but before doing so, it is necessary to prove an elementary 

 theorem in Geometry, of which we shall make use, not only 

 in the theory of sphincter muscles, but also in the theory of 

 jnuscles forming curved surfaces. 



If from any 

 point 0, outside a 

 circle, Fig. 41, two 

 tangents be drawn 

 and the chord join- 

 ing their points of 

 contact, and the 

 diameter passing 

 through 0 be also 

 drawn : 



Then let 



r = radius, 

 t = tangent, 

 a = half the chord, 



x = intercept between circle and point 0, 

 y = intercept between circle and chord : — 

 I say that when the point 0 approaches indefinitely near 



the circle, the intercepts x and y will become equal to each 



other. 



For 



t 2 = (2r + x) x (Euc. iii. 36). 

 a 2 = (2;* - y)y (Euc. iii. 35). 

 Subtracting these equations from each other, we have 

 2V (x ~ y) + (x 2 + f) = t 2 - a\ 



