272 



ANIMAI MECHANICS. 



perbola, whose axis major is OX, the bisector of the angle 

 made by the muscular fibres. 



Let this hyperbola 

 be a Ob, cXd, Fig. 72 ; 

 then the square of any 

 chord ab, cd, will be 

 proportional to the work 

 done, by the contraction 

 of the muscle, in rotating 

 the bone round the chord 

 considered as an axis. 



When y % is negative, 

 equation (51) denotes a 

 circle described on the 

 bisector OX as diameter; and the square of the chord ef\$ 

 proportional to the work done, by the contraction of the 

 muscle, in rotating the bone round the chord ef. 

 The denominator A, used in equation (48), is 



A = b sin j3 sin (j3' - 0) - b' sin j3' sin (/3 - 0) ; 



where 



A = A cos 0 + B sin 0, 

 A = (6 - b') sin j3 sin |3'. 

 B = b sin j3 cos /3' - V sin ]3' cos j3. 



If B = o, the equation (48) becomes much simplified, and is 



reduced to the following : — 



(\n in <*> 3 . r , . , ,s , 1t C cos 2 OdO 

 jbpdO = — . sin p sin p {a cos <j> + 0) (a cos 0 + b )\ - 



„ n n> fsin 2 6(10 , x 



+ — . a 2 cos 2 0 . cos p cos b -< ~ ( C2) 



2 J A cos 0 



The third term of equation (48) disappears altogether, for 

 it consists of pairs of elements, of the following form : — 



sin 0 cos OdO sin 0 cos OdO 



A cos 0 + B sin 0 A cos "0-2? sin 0 



