158 On the maximum of mechanic powers, ane* 



a circular inch, «c* will express the weight of the at- 

 mosphere on the piston of the cylinder, which is 

 therefore the power of the engine, and answers to 

 P in the former case. And hy an easy eomputation, 

 if /'represent the depth of the pit in fathoms, it 

 will he found that 2y//' will nearly express the 

 weight of the water in pounds, which is to be raised 

 through a gi\-cn space, by the power of the cylin- 

 der, and which therefore answers to .r. Now in the 

 "usual theorems that have been deduced for ascer- 

 taining the different values of c, f\ and p, a& and 

 2/) j' have been made equal to each other, so that 

 the weight and power must have been supposed in 

 eqidllhrlOy which is never the case. But let us al- 

 low the weight of water in the pump to be overcome 

 by the superior v/eight of the atmosphere in the cy- 

 linder the moment the steam is condensed, and then 

 the case becomes precisely the same as when the 

 weight P is suspended at one end of the lever ; and 

 like that weight the atmosphere ^vill descend with 

 an accelerated motion, and raise the column of wa- 

 ter at the opposite end. 



Now since the value of P is here given in terms 

 of c the diameter of the cylinder, it will be necessary 

 to substitute another quantity for t in the general 



equations. Let then ^-^'-f— ^ — J""^^ ^^ equal dt 

 then V-\-d (a&-\-d):=.t\ and therefore the equations 

 ^J^V^t, and ^-^^^-ot.^^^^^^-:^^ ^ become v^2F+ 



3P^+^^_.p_^ and v-^°^-4-a8P.t,i^-3^-^p ^ respective- 

 ly ; and by putting 2/;^ for x, and a& for P', \ye 



shall then have ^ff—y/od'c' + 3adc''-i-d^' — ac" — d 

 for a general equation when the effect is greatest 



after a gheti time, and ojr,y-£!f^5±lf^E!z^-il'^ 



when the effect is greatest after passing through a 

 given space ; and from which equations may be de- 

 duced the following values of c, p, and J] viz. 



M'hen 



