1872.] 



Elementary Principles in Animal Mechanics. 



331 



fibre of the original muscle, will produce the same mechanical effect as 

 the whole skew muscle. 



6. "When the equivalent fibre 0' is drawn, it is always found to be so 

 placed in relation to the axis of rotation as to produce the maximum work 

 possible. 



7. Two generators can be assigned on the locus hyperboloid, dividing 

 the surface into two regions ; in one of these regions each generator is an 

 axis of stable equilibrium, and in the other region each generator is an 

 axis of unstable equilibrium ; the generators separating the two regions are 

 axes of neutral equilibrium. The axis of maximum work is a unique 

 axis lying in the middle of the region of unstable equilibrium. 



8. Every skew muscle is a supplemental contrivance fQr the purpose of 

 procuring a line of force joining points O andO' already occupied by other 

 structures, so that it would be impossible to apply the force directly. 



Let 1 1' be the shortest line joining A B and A' B', and take this line for 

 axis of z s and the lines bisecting the angles between A B and A' B' for axes 

 of x and y, the origin being placed at the point of bisection of 1 1'. 



The equation of the locus hyperboloid is 



y_ ffl V)+(l-r)(. 2 - c >0, (1) 



where m is the tangent of half the angle between the bones, c is half the 

 line I T. 



The equations of the bones are 



yy — mx=O s ^y-\-mx = 0, 



AB \*-.c =0, A ' E V + c =0. 

 The equations of the rectangular planes passing through the bones are 

 y-mx + X(z-c) = 0A , 

 y + mx+X'(z+c) = Q s f K 

 where X and X' are parameters connected by the relation 

 XX =m 2 — 1. 



If cc, oc be the coordinates of the extremities of any fibre whose length is 

 p, and if we write 



(?)~-<--»*G>j 



then the biquadratic which determines the generator of maximum work is 

 -K(m 2 -H).X 4 "] 

 4- 2c { mX(m l + 1 ) - mX'(m l - 1 ) } . X 3 \ 

 + 2c(vi--\){mX(m i -l)-mX'(m x +})}.X f * ' ' 4 ' 

 + (m--l)(m l -l)K = 0. J 



The sums (3) were calculated from measurements made on freshlv dis- 



