188 ME. MACQIJOEN EANKINE ON THE THEEMO-DTNA^nC THEOET 
They may therefore be applied with confidence to engines working withm those limits, 
and probably somewhat above them; but for pressures much exceeding 120 lbs. on the 
square inch, and ratios of expansion exceeding 16, it is advisable for the present to use 
the exact formula. . 
The foundation of the approximate formulae for work, and for mean efi^ective pressure. 
is the fact, that for pressures not exceeding 120 lbs. on the square inch, or 1(280 lbs. 
on the square foot, the equation of the curve CDG is very nearly 
P OC V~^ ' 
This equation is very convenient in calculation, because the sixteenth root can be 
extracted, with great rapidity, to a degree of accuracy sufficient for practical pui-poses. 
by the aid of a table of squares alone; and by a little additional labour, without anv 
tables whatsoever. 
Let r, as before, be the ratio of expansion ; then evidently, 
1 (^b I 
Final pressure I ’ 
Gross work per lb. steam-= Siren. OBCDTO—W+psV^—^^ 'vdp-^p^Vz,^^.^ _ ^ 26 . 
- 16r-^) ^p^llr'^ - 16r-^) J 
Effective work per lb. steam — =^^jvdp-\-{p^~P3)v^\. ^ _ ^ 27 .) 
=i;,{_p,(17r-'-16r-i^)™i?3} J 
Mean gross pressure "—-\-pz—Pi{^'^'>f' ’ 
Mean effective pressure, ox yvoi)s. GxMic iooi, w—— jPi(17/ 16> io) p^- ■ (-0 
It is evident, that if the pressure of exhaustion p^ be given, and any two out oi the 
following three quantities,— the initial pressure p,, the mean effective pressure ii\ the 
ratio of expansion r,— the fourth quantity can be calculated dhectly, if it is one or otaev 
of the pressures, p,, w, or by approximation, if it is the ratio ol expansion /’. 
The approximate formula for the expenditure of heat, in foot-pounds per pomid ot 
steam, which has been found by trial to agree very closely with the exact formula within 
the limits already specified, and when the feed-water is supplied at a temperature ot 
from 100° to 120° Fahe., is as follows: 
so that the heat expended per cubic foot of space traversed hy the piston, or, the pressure 
per square foot of piston to which the expenditure of heat is equivalent, is 
(81.) 
This gives for the efficiency. 
W IV 17 — 16r-TV rp^ 
151 “l5ljo,’ 
(S2.) 
