42 



ANIMAL MECHANICS. 



3 0 . The line 2W + a = o, is also an asymptote to the 

 curve. 



4°. The time t has a maximum corresponding to the value 

 of w given by the equation 6w + a = o. 



5°. The value of t becomes equal to zero, and the curve 

 is a tangent to the axis of w 9 when jw + a - o. 



The first three of these conditions are fulfilled by observa- 

 tion, for — 



i°-2°. The time of holding up the arm must be always 

 positive, and must become zero, when the weight is infinite. 



3°. When zw + a = o, the arm is supported by a force 

 equal to half its weight applied at the hand, and being there- 

 fore in a condition of statical equilibrium, can be held up for 

 an infinite time. 



The last two conditions, however, are inconsistent with 

 observation, for — 



4°. The value of t, for all positive values of w 9 must 

 increase continuously as ic diminishes to zero ; and for negative 

 values of w can have no maximum but infinity, corresponding 

 to the condition for statical equilibrium, no + a = o. 



5°. The value of t can never be zero, except when w is 

 infinitely great. 



From the preceding investigation, I became satisfied that 

 my first idea, viz., that the arm dropped and was raised again 

 during repeated short intervals, was erroneous, and that a 

 simpler form of curve would represent better the relation 

 between t and w. This curve I afterwards found to be a 

 cubical hyperbola, which represents the observations equally 

 well, and is free from all the preceding objections. 



So far as the observations themselves are concerned, they 

 can be represented by any curve of an hyperbolic form, and 



