ANIMAL MECHANICS. 



37 



My first attempt to calculate a formula to represent the 

 Law of Fatigue, in the case of the arms extended horizontally, 

 was based upon the periodicity of muscular action, as proved 

 by the muscular susurrus, explained in the preceding section, 

 and may be thus stated : — 



Each portion of the muscle contracts, relaxes, and goes 

 through all its changes thirty-two times in each second, and 

 does so in succession, so as to keep the ends of the fingers 

 steady ; but the amount of work given out by the whole 

 muscle is evidently the same as if all the particles went through 

 their changes simultaneously, and not in succession. 



If the whole muscle acted simultaneously, the arm would 

 fall like a compound pendulum during the ^nd part of a 

 second, and then be raised to its original level by the renewed 

 contraction of the muscle. It may be shown, as follows, that 

 in this case the arc through which the arm falls is that which 

 would correspond to the centre of oscillation falling freely 

 for ^nd of a second. 



For, if W denote the weight of the arm, x the distance of 

 its centre of gravity from the centre of the glenoid cavity, w 

 the angular velocity acquired after falling for any time, and / 

 the moment of inertia of the whole arm ; we have, by D'Alem- 

 bert's principle, the following equation : — 



Idw = Wxdt. (10) 



If 6 denote the angle made with the horizontal line at any 

 moment by the arm, and k denote its radius of gyration, and 

 I the length of the equivalent simple pendulum ; we have at 

 once from equation (io) — 



dt ie r 



multiplying both sides by 2^, and integrating, we obtain 



