Prof. Barnard on the Elastic Force of Heated Air. 159 
gether the most favorable to the production of power, nor abso- 
lutely the best as it respects economy. Moreover they involve 
an absolute pressure greater than is required for the attainment of 
a higher mean effective pressure, and therefore a larger amount of 
useful labor. 
By differentiating the formula already referred to, on page 248 
of the last volume of this Journal,* in reference to m and 7 suc- 
cessively, we shall find that neither the ratio of capacities of the 
two cylinders, nor the degree of compression of the air as as- 
sumed by Mr. Joule, is that which is most favorable to power un- 
der the given working temperature. 
If we make m variable, the maximum Boe occurs when 
al ( (t+ "jy —1T"1 a 
al (2y—1)r ye 
or a wide range of working temperatures and for a great va- 
riety of values of 2, m falls but little below unity. It seems to 
increase with 7’, and to diminish as 7 diminishes: but it is only 
in extreme cases ’ that it becomes less than -90, while it is usually 
above ‘95, and is sometimes higher than 1-00. In the present 
case, with 7” = 1220°-7 and /=-25, it is ‘907, and the dependent 
Mati will be as follows: 
= 967°-66 (=508°-66 F.) 
eats 447 as to ihe aquare inch. 
p=128 
To compare the working under these conditions with the for- 
mer, in point of economy, we have to observe that the expendi- 
ture of heat is directly as m(t/’— 1’), and the useful effect, direct- 
ns 
ly as P. The economy will therefore be as —_——. ale) The val- 
ues of this expression for the two cases are as the numbers 430 
and 456, nearly ; or as 1: 1-057, the condition of maximum pow- 
er relatively to m, being nearly 6 per cent., the most favorable to 
By ee with respect to /, we shall find that a max- 
imum occurs W 
* This formula i > “mien and geet to a more orm — —— page — of 
the last number of the Journal. may be rendered still 
by substituting for the symbol & ‘ the last form given) its ae =. It me 
becomes 
4 ” MW mm 
P= ef a RereeH 
in which IT is the pressure against which the engine works, 
s. the 
working piston, m is the ratio of the supply pets to the pie Bck cylin 
section, fis the fractional part of th of the stroke before cut-off, and x and 1! 
values assigned them in the text. 
<a 
er in cross 
sega tes 
