MR. MACQUORN RANKINE ON THERMO-DYNAMICS. 
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The algebraical expression of this result is that the equation (15.) holds for any pair 
of curves of equal transmission, as well as for a pair of curves of no transmission ; or, 
in other terms, let F A , F B , F c , F D be the thermo-dynamic functions for the curves of 
no transmission passing through the four points where a pair of isothermal curves 
cut a pair of curves of equal transmission : A, B being on the upper isothermal curve ; 
C, D on the lower ; A, D on one curve of equal transmission, B, C on the other : then 
F b -F a =F c -F d (16.) 
(13.) Proposition V. — Theorem. The difference between the quantities of heat 
absorbed by a substance, in passing from one given amount of actual heat to another, at 
two different constant volumes, is equal to the difference between the two latent heats 
of expansion in passing from one of those volumes to the other, at the two different 
amounts of actual heat respectively, diminished by the corresponding difference between 
the quantities of expansive power given out. 
(Demonstration) (see fig. 9). Let be the isothermal curve of the higher 
amount of actual heat ; Q 2 Q 2 that of the lower. Let V A , V B be the two given 
volumes. Draw the two ordinates V A u A, V b 5B, and the four indefinitely-prolonged 
curves of no transmission AM, am, BN, bn. The quantities of heat absorbed, in 
passing from the actual heat Q 2 to the actual heat Q 15 at the volumes V A and V B , are 
represented respectively by the indefinitely-prolonged areas MA am, NB6w. Then 
adding to each of those areas the indefinitely-prolonged area rc&BAM (observing that 
the space below the intersection R is to be treated as negative), we find for their 
difference 
NB&rc- MA«m=NBAM— rc6BAam=(NBAM— nbam)- (V b BAV a - V B baV A ); 
but NBAM and nbam represent the latent heats of expansion from V A to V B , at the 
actual heats Qj and Q 2 respectively ; and V b BAV a and V B baV A represent the power 
given out by expansion from V A to V B at the actual heats Q! and Q? respectively ; 
therefore the proposition is proved. Q.E.D. 
This proposition, expressed symbolically, is as follows. AQ being the difference 
MDCCCLIV. S 
