ANIMAL MECHANICS. 



277 



positive, and these equations will represent, for all values of y l 1 

 positive or negative, an ellipse and hyperbola, having the same 

 axes. 



Fig 74' 



Let C be the common centre of the ellipse and hyperbola, 

 Fig. (74), found by the equation (54), by measuring off 

 OC = S. Take CA = m and CB = n, and construct the el- 

 lipse and hyperbola corresponding to these lines as semi-axes, 

 the line OX being the real axis of the hyperbola. Then, as 

 before, the work done by the quadrilateral muscle in rotating 

 the moveable bone round any axis ab, cd, or ef t will be pro- 

 portional to the square of these chords ; and if the chord of 

 rotation intersect the hyperbola, the muscular contraction will 

 restore the skew quadrilateral to the plane ; but if the chord of 

 rotation intersect the ellipse, the contraction of the muscle 

 will convert the plane quadrilateral into a skew muscle ; and 

 finally, the contraction of the muscle will produce no effect 

 at all, if the axis of rotation pass through the extremity of 

 the axis of the ellipse and hyperbola. 



2 Q . If the expression (55) 



- (b - b') 2 X sin j3 sin j5' - 4bb' Y cos /3 cos 3' 

 be equal to zero ; the semi-axes m and n will both become 



