188 MR W. J. M. RANKINE ON THE 
To reduce the logarithm of the number of cubic metres occupied by one kilo- 
gramme to that of the number of cubic feet occupied by one pound avoirdupois, 
add 1:20463. 
The logarithms are given to five places of decimals only, as a greater degree 
of precision is not attainable in calculations of this kind. 
The second table is for the purpose of calculating the mechanical action of 
steam in expansive engines. 
The first column contains values of the fraction of the entire capacity of the 
cylinder which is filled with steam before the expansion commences (being the 
quantity — 1 of the formule), for every hundredth part, from 1:00, or the whole 
cylinder, aes to 0°10, or one-tenth. 
If 7 be the entire length of stroke, /’ the portion performed at full pressure, 
and ¢ the fraction of the entire capacity of the cylinder allowed for clearance, then 
1 
U Ste 
3 i U 
Tee ate ee a 

The entire capacity of the cylinder is to be understood to include clearance at 
one end only. 
The second column gives the reciprocals of the quantities in the first, or the 
values of the ratio of expansion s. 
The third and fourth columns, headed Z, give the values of the quantity 
eee 
l—¢ 1-¢ 
of the steam to its action at full pressure, without allowing for clearance. The 
third column is to be used for initial pressures of from one to four atmospheres ; 
and the fourth for initial pressures of from four to eight atmospheres. 
The deduction to be made from the quantity Z for clearance is cs, or the 
product of the fraction of the cylinder allowed for clearance by the ratio of expan- 
sion. Hence, to calculate from the tables the net mechanical action of unity of 
weight of steam, allowing for the counter-pressure of the waste steam P,, as well 
as for clearance, we have the formula 
P, V, (Z—es)—P, V, (l—e)s 
being equivalent to the formula (47.) of this paper. 
1 
sof Article 23, which represents the ratio of the entire gross action 

