362 Mr. T. Tate on certain Modifications of the Form of 



the work expended in friction^ / being the friction on the piston 

 supposed to be one inch in the section ; Wj = the work performed 

 by the air in the pump upon expanding from the pressure E„_i 

 to E„; Uq= the work accumulated in the air in its transfer ; then 



„ fE 2fb It,, E„_i 



Here «= j^J "' = 1^' Uc^= j^aEn-iloge-^ 



= To « (-7-1)"' E log,^*, for E„ = ( -^) "e when the re- 

 12 \a + b/ ^^ a ' " \a + b/ 



h wv 

 sistances of the valves are neglected; "3=^=0^ '~n~> where w is 



17*0 /ig 



the weight of a cubic foot of atmospheric air, and v the velocity 



of discharge, the vis viva of the air in passing through the lower 



valve being neglected ; 



.jL[„,,E../,-«-^)E(i-C-fJ},o,;-±*+^.^']. (1) 



And when n is very great, we find the work expended in exhaust- 

 ing the receiver 



^ f j,/v , oj'\ «(« + S)E, « + 6 , nb wv^X ,„. 



= -^nb{E + 2f)--^-^\os,— +—^~^. . (2) 



2. To find the work expended in exhausting the receiver of 

 the pump No. 1. 



In this case we have for the work of the nth single stroke, 



U„ = M + Mj + Mg + W3. 



Here « = t-k ^'^j where a' is the volume of the air in the barrel 



when its pressure becomes E, but a'= — ^—; 



b / a \«-' 1 ^, 



•■•"=l2to; ^' "' = 12-^^' 



1 fT?i E 1 , Tw / « \"~'t.i a + b 



(- 



)n-l 



3~ 1728 2^~1728 2^Va + Ay 



^ i rZ-p, 1 wv^\^ a y-' 



I 



