MR. MACQUORN RANK1NE ON THERMO-DYNAMICS. 
139 
mission, abm, a'b'm!, through any part of the diagram, cutting out of it a quadrilateral 
stripe, abb' a!. Let be the mean actual heat corresponding to the upper end aa! of 
this quadrilateral stripe ; Q 2 , that corresponding to the lower end, bb'. 
The area of this indefinitely-narrow stripe representing a portion of the heat con- 
verted into motive power, is found, according to the principles and notation of the 
third corollary to Proposition II. and of Proposition III., by multiplying the differ- 
ence between the actual heats by the difference between the thermo-dynamic functions 
for the curves of no transmission that bound the stripe, thus : — 
^E=(Q 1 — Q 2 )£F ; 
while the area of the indefinitely-prolonged stripe, maa'm', representing part of the 
total heat expended, is, according to the same principles, 
and that of the indefinitely-prolonged stripe mbb'm!, representing part of the heat 
given out, is 
m 2 = q 2 if. 
Integrating these expressions we find the following results 
whole heat expended, 
heat given out, 
motive power given out, 
efficiency, 
H 
PFb 
1 Jf Qi^F; 
H,= 
; f a 
’Fb 
CFb 
Jfa Q ^ F; 
E 
=H 1 — H 2 =|* Fb 
Jf a 
(Q,— Q 2 )dF ; 
>Fb 
Hj~ C F b 
r Fb 
I QAF 
JFa 
. . (28.) 
formulae agreeing with equation (28.) of a paper on the Centrifugal Theory of Elasti- 
city* ; it being observed that the symbol F in the last-mentioned paper denotes, not 
precisely the same quantity which is denoted by it in this paper, and called a thermo- 
dynamic function, but the product of the part of that function which depends on 
the volume, by the real specific heat of the substance. 
(24.) First Corollary. Maximum Efficiency between given limits of Actual Heat. 
When the highest and lowest limits of actual heat at which the engine can work 
are fixed, it is evident that the greatest possible efficiency of an engine without a re- 
generator will be attained when the whole reception of heat takes place at the 
highest limit, and the whole emission at the lowest ; so that the true indicator dia- 
gram is such a quadrilateral as is shown in fig. 6, and referred to in the second corol- 
lary of Proposition II. ; bounded above and below by the isothermal curves denoting 
* Trans. Roy. Soc. Edinb. vol. xx. 
T 2 
