SECT. iv.] AND PROPORTIONS OF ENGINES. 155 



305. For the present let the proportion of waste be 1 w, and let the friction 

 of the piston be denoted by F, then the power of the steam of a cubic foot of water, 

 of the temperature t, = 



5418 (459 + - M18(Q + OCr + g)- = 5418 (459 + (l - S+I) .. 



/ 



306. We may next ascertain the effect of EXPANSION ; which is easily com- 

 puted, for when the temperature does not sensibly alter during the action, the 

 force of the steam is inversely as the space it occupies ; therefore if b be its bulk, 

 its force, x any increase of bulk, and 8 x its variation ; then b + x : b : : p : -- . 



U "T* X 



and |-ilf = the power developed in expanding through the space 8 x ; and the 

 integral of this quantity is p b hyp. log. (b + x) + C ; but when x = o, p b hyp. 

 lo-. (b + .r) + C = o ; hence the power is 



P b hyp. log. - 



But, to return to our previous notation, suppose the volume of expansion to be n 

 times the volume of steam, and make x = n 1 b ; then since 



b = 76-58 (459+Q and p = 7 Q. 75/; 



we have s = p b hyp. log. b -~ = 5418 (459 + hyp. log. , 



for the additional power gained by the steam expanding. 



307. When the expansive principle is employed, that is, if the steam be to 

 expand during its action on the piston, an increased length of cylinder becomes 

 necessary ; and the reduction of effect which must follow from this cause has been 

 totally overlooked. If n be the bulk of the steam in its expanded state, when its 

 bulk corresponding to the force /and temperature t is unity, then 



5418 (459 +0 (l - " ( f + F >) w + s 



is the mechanical power of a cubic foot of water, when is the additional power 

 gained by employing the expansion of the steam. 



On the value of s being inserted in the expression it becomes 



5418 (459 + (l + hyp. log. - " (f ' + F) ) w =, 



for the mechanical power of a cubic foot of water, when the expansive force of the 

 steam is employed. 



