ANIMAL MECHANICS. 



The following geometrical construction for the work done 

 for any position of the axis S Y 9 Fig. 7 1, is easily remembered, 

 and applies to every case. Describe the circle XTO, whose 

 diameter is OX = b, the bisector of the angle made by the 

 muscular fibres. From Y, the point where the axis of rotation 

 intersects the bisector YOX, draw YTa tangent to the circle 

 XTO ; I say, that the work done by the muscle, during a 

 small rotation of the bone AB round the axis SY, is propor- 

 tional to the square of the tangent YT. For the work done 

 is proportional to 



a cos (f> (a cos $ + b), 



and, 



YO = a cos <j), 

 YX = a cos 0 + 6, 



also, 



( YTf =Y0* YX. Euc. iii. 36. 



Therefore, 



Work done, varies as (Yiy. Q. E. D. 



If the axis of rotation be supposed to move parallel to 

 itself, the work done will be greater, the farther off the point 

 Fis from the circle, either beyond 0, or beyond X. When 

 it passes through either 0 or X, the work done becomes zero ; 

 and for positions like S'Y' lying between 0 and X, the work 

 done becomes negative, and is proportional to the square of 

 the ordinate Y'T' ; and the negative work done will be a 

 maximum, when the axis of rotation passes through the centre 

 of the circle XTO; or through the point of bisection of the 

 bisector of the vertical angle A OB. 



The mechanical interpretation of the preceding facts, is — 

 i°. That if the axis of rotation, round which the bone AB 

 is compelled to turn, be placed beyond O or X in either 

 direction, no amount of contraction of the muscle can alter 

 its position, for any change of position would be equivalent 

 to a total lengthening of the fibres, which is impossible. 



