282 



ANIMAL MECHANICS. 



Integrating these expressions between + 0 and - 0, we obtain, 

 after some reductions, the following values, which may be 

 readily computed, for given values of A and B:— 



-rr i ( • • n 21 tan ± (a + Q) 



JL = * < 2sm a sm H cos l a log e . f-f ~ 



R \ 66 tan 1 (a - 0J 



17- 1 ( • • /j • 9 i tan i (a + 0) ,~ % 



Y = ~r> - 2sm a sm 0 + sm 2 a lo^. f-r 77V (60 



R ( & tan f (a - 0)J v 7 



rr 1 { . 1 tan 4 (a + #)) 



Z = — { 2cos a sin 0 - sin a cos a log e . 5-7 77-} 



22 fa tan J (« - 0)( 



The best examples, in Nature, of the rotation, by means 

 of a quadrilateral muscle, of limbs round an axis passing 

 through a fixed socket, and perpendicular to the bisector of 

 the angle made by the extreme fibres, are to be found in the 

 wings of birds, which are depressed by the contraction of the 

 great pectoral muscle, in the manner here considered. I shall 

 give a few examples, to illustrate the general principles just 

 given. 



(a). Wing of the Albatross. — In Fig. 76, the bone AB 

 represents the origin of the pectoral muscle, from the fur- 

 culum, A, and sternum, B, and A' B f represents the insertion 

 of the muscle into the humerus ; both bones being placed in 

 the same plane, when the wing is extended, previous to the 

 contraction of the muscle. The centre of the shoulder-joint 

 is shown at S, and the bone A'B' is rotated, by the action of 

 the pectoral muscle, round the axis ST, perpendicular to OX, 

 the bisector of the angle A OB. 



We are required to calculate, from the preceding theory 

 the position of the axis of rotation, corresponding with the 

 maximum work done, when the system is in a position of 

 unstable equilibrium ; or, in other words, we are required to 

 find the centre of the conic (59). 



