ANIMAL MECHANICS. 



321 



through the socket, and the origin, and insertion, respectively, 

 must be at right angles to each other. 



With the socket of the joint as centre, describe a sphere, 

 and let AB and A'B' (Fig. 85) be the spherical arcs, which 

 are the projections of the bones upon the surface of the 

 sphere. Let Ibe the 

 intersection of the 

 planes containing 

 the two bones, and 

 let A A' be the pro- 

 jection of any mus- 

 cular fibre; let I A = j 

 a, IA! = a', and let 

 the angle between 

 the planes be /. If 

 the plane IA'B' be 

 made to revolve Fi S 8 5 



round the intersection of the two planes through a small 

 angle w, so that A and B' assume the positions A" and B" \ 

 then let 



A' LA" = w, 

 AA'=x, 

 AA" = y. 



\ip be the original length of any fibre, and A, A\ the 

 distances of its extremities from the socket; and also, ifp'be 

 the length of the contracted fibre corresponding with the 

 rotation through the angle w, then we have 



but 



p z = A} + A' 2 - 2AA' cos x ; 

 p 2 = A % + A /U - 2 AA' cos y ; 



cos x = cos / sin a sin a + cos a cos a 1 ; 



cos y = cos (/- u)) sin a sin a 4 cos a cos a. 



Y 



