MR. MACQUORN RANKINE ON THERMO-DYNAMICS. 
135 
curve CC shall bear the same proportion to the whole ordinate which the difference 
£Q bears to the whole actual heat corresponding to the ordinate ; for example, let 
AA : AJA : : Q, : SQ 
AA : AJT 2 : : Q 2 : §Q, &c. 
Then draw an ordinate V b B 6, parallel to OY, cutting off from the space between 
the isothermal curves A 2 Q 2 , a 2 q 2 , a quadrilateral area A 2 B ba 2 equal to A 1 D,D 2 A 2 , the 
area of the curve DD between the ordinates at A 4 and A 2 . 
Then if the difference &Q be indefinitely diminished, the point B will approximate 
indefinitely to the intersection required of the isothermal curve A 2 Q 2 with the curve 
of no transmission passing through A 4 ; and thus may any number of points in this 
curve of no transmission be found. 
(Demonstration.) Let A,Mj be the curve of no transmission required. Let a 3 c 3 , 
a 4 c 4 be any two indefinitely-close ordinates of the curve CC, corresponding to the 
mean actual heat Q 3 4 . Let a 3 m 3 , a 4 m 4 be curves of no transmission, cutting the 
curves a 2 q 2 , A 2 Q 2 , so as to enclose a small quadrilateral area e. Then by the con- 
struction, and Proposition I., 
The area a 3 c 3 c 4 a 4 = the indefinitely-prolonged area m 3 a 3 a 4 m 4 ; 
and by the first corollary of the second proposition and the construction, 
the area e SQ, area a 3 d 3 d 4 a 4 
m 3 a 3 a A m 4 Q 3 4 area « 3 c 3 c 4 « 4 
Therefore the area e=the area a 3 d 3 d 4 a 4 ; but the area A,D 1 D 2 A 2 is entirely made up of 
such areas as a 3 d 3 d 4 a 4 , to each of which there corresponds an equal area such as e ; 
and when the difference &Q is indefinitely diminished the area A 2 BZ>« 2 approximates 
indefinitely to the sum of all the areas such as e, that is, to equality with the area 
A^iDaAa. Q.E.D. 
The symbolical expression for this proposition is found as follows : — 
The area A,D,D 2 A 2 ultimately=^Q.J^ 1 (for V=V A ) 
/»V B f [ p 
the area A 2 B ba 2 ultimately =SQ.\ (forQ=Q 2 ) ; 
divide both sums by SQ and equate the results ; then 
f|iT(forQ=ft)=f^Q(forV=V I ), .... (21.) 
which denotes the equality of two expressions for the difference, F, — F 2 , between the 
thermo-dynamic functions for the curve of no transmission A,M, and for that passing 
through the point A 2 . 
