34 Mr . Davies Gilbert's observations on the steam engine. 
Let AB represent the portion of the 
cylinder through which the piston is 
urged by the entire force of the steam, 
and let its length be expressed by unity. 
Then let the supply of steam from the 
boiler be cut off by the valve at ( v ) and 
conceive the piston to have descended 
to B through the space (jc) urged by 
the expansive elasticity of the steam, 
the density of which will then be 
and assuming that densities and elasti- 
cities are proportionate, — J- will be 
i “r x 
2 2 *o 99 the fluxion of the efficiency, and the 
3 2.386 efficiency itself equal the nat. log. of 
5 2.792 J 
6 „ 2 94 g stroke from A to B will be 1 -f- nat. 
7 3-°79 log. 1 cc, giving this very curious 
8 3,197 result, that when x is infinite, the effi- 
,o 3.398 ciency also exceeds any assignable limit. 
But it is obvious that several important circumstances have 
been omitted : 
The reduction of temperature, and consequently of elasti- 
city, caused by rarefaction, as the steam expands : 
The increased relative effect occasioned by the reaction of 
the imperfect vacuum : 
The increased relative burden of the air pump. 
The relative increase of friction. 
So that although this mode of working is highly advan- 
tageous, yet it is probable that, in practice, not more than 
from one half to two thirds of the cylinder can beneficially be 
occupied by expansive steam. 
