276 Dynamics. 



But, since 



D : tf : : v' 



we have, by substitution, 



To find the maximum we put the differential of the first member 

 equal to zero, v' being regarded as variable ; we have thus 



Gal. 45. P dv/ - = > 



or 



V = 2 v' and -o' = | v. 



Substituting -D for v' in the above equation, we obtain 



r u = %pv> 



On the hypothesis, therefore, which we have assumed, the man 

 will do most work when he moves with half his greatest velocity, 

 and in this case the greatest effect will be p v. 



394. It appears, however, by direct experiments, that the 

 force of a man diminishes as the square of his velocity increases,! 

 in other words, that the effective forces are as the squares of the 

 diminutions of velocity from the point where the effective force 

 is nothing. Calling />', therefore, the force answering to the velo- 

 city i/, we shall have, according to this hypothesis, 



whence 



p'=p (j^rO" = p (T) 3 w ' putting v ~~ *' = w (n) ; 



and hence, 



since 



(7y\ 3 /w\* , 



J -D' or P(~) CD w) = r w, 



Taking the differential, as before, and putting it equal to zero, 

 and suppressing the constant factor we have 



t See note on the measure of forces* 



