332 



ANIMAL MECHANICS. 



test whether or not the actual axis of rotation in the limb of 

 the animal coincides with the mix. maximorum axis. 



Let PP f = p, be any fibre joining two points, (x, y, z) and 

 (x', y% z r ) on the bones AB and A'B' '. It has been already 

 ' proved that the moment of this fibre, to turn A B' round the 

 intersection of the planes passing through the two bones, is 



ff 



moment = — ; 



V 



where /,/', denote the lengths of the perpendiculars let fall 

 from each point upon the plane passing through the other 

 point. If we express /, /', in terms of the coordinates, we 

 obtain 



ff 4 (mx + cX') {mx' + cX) 



p ~~ P V^X 2 + m 2 + i V\"> + m*"+ 



ff 4 mc (xX + x'\') + k 



p p ' v/X 2 + m 2 + i y/\ 2 + + i 



(85) 



(86) 



where 



k = m 2 xx' + c 2 (m 2 - 1). (87) 



Differentiating (86) with respect to X and X', and remember- 

 ing the condition 



XdX' + X'dX = o. 



we obtain the following equation of condition to determine 

 the position of the generator which gives us the maximum or 

 minimum amount of work, 



cm (X 2 + m 2 + 1) (X' 2 + m 2 + 1) (xX - xX') (88) 

 - (m 2 + 1) [mc (xX + x'X') + k] (X 2 - X /2 ) = o 



This is the condition required and expressed for a single 

 fibre. The equation of condition for the entire muscle, com- 



