Maximum Effect of Machines. 285 



398. In those cases where n varies along with r, it will in 

 general vary in the same proportion, and we may therefore re- 

 present n by a? r, some multiple of r. For the sake of simplicity, 

 the friction f may be considered as absorbing a certain portion 

 of the impelling power, which will then be represented by p /; 

 and we may also regard the inertia of the machine, or i, as ap- 

 plied at the impelled instead of the working point ; that is, the 

 moment of inertia may be considered as proportional to i D 3 . 

 Now, if we make p /= 1, and D = 1, in the formula 



rpvd r 2 d 2 rfd 2 

 m D 2 -f- n d 2 + id* ' 



we shall obtain 



r d r 2 6 2 



m -\- i -j- xrd 2 ' 

 and making m + i s, we have 



r d r 2 d 2 

 s + xrd 2 



for the work performed. 



This is a maximum when the differential, r being considered 

 as variable, is equal to zero, which gives 



(d 3rd 2 ) (s -f- x rd 2 ) x d 2 (r$ r a d 2 ) = 0; 

 or, by reducing, 



sd %s rd 2 x r 2 d* =0; 

 that is, 



r2 j, * 



and by resolving this after the manner of an equation of the 

 second degree, we obtain 



_____ s I s s 2 ^ _ 



" xd 2 " h \s<5 3 x 2 d* 



r s 2 s 



xd 2 

 When x = 1, we have 



_ \/s d -f s 2 s 



