198 



ANIMAL MECHANICS. 



But t and a become equal to each other, and to half the arc 

 contained between the tangents when 0 indefinitely approaches 

 the circle ; hence the foregoing equation becomes 



2r (x - y) + {x 2 + y 2 ) = o ; 

 and, as x and y are indefinitely small, the term x 2 + y 2 disap- 

 pears, being of the second order, and the equation becomes, 

 finally, 



2r (x - y) = o 



Q. E. D. 

 Let xyZ, Fig. 42, 

 represent a circular 

 fibre of a sphincter 

 muscle, and let any 

 point 0, indefinitely 

 near the fibre, be taken, 

 and from this point 

 imagine two tangents 

 drawn to the circle. 

 When the whole fibre 

 contracts there will be 

 produced a tangential 

 strain at each point of 

 the circle ; and it is 

 necessary for equilibrium that forces perpendicular to the 

 circle shall act at each point. We are required to find the 

 relation between the system of tangential and perpendicular 

 forces. 



It must be remembered that the sphincter fibres can pro- 

 duce tangential strains only, for a muscle can only contract ; 

 and that the perpendicular forces, which equilibrate the tan- 

 gential strains must be produced by a distinct set of radiating 

 fibres. 



