MR. MACQUORN RANKINE ON THERMO-DYNAMICS. 
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in d. Lastly, draw the ordinate CV c in such a position as to cut off from the space 
between the isothermal curves of Q 2 and Q 2 — a quadrilateral DC cd, of area equal 
to the quadrilateral AT$ba. 
Fig. 7. 
Then will C, where the last ordinate intersects the isothermal curve of Q 2 , approx- 
imate indefinitely to the position of a point in the curve of no transmission passing 
through the given point B, when the variation of actual heat IQ is diminished without 
limit. And thus may be determined, to as close an approximation as we please, any 
number of points in the curve of no transmission NBR which passes through any 
given point B, when any one curve of no transmission LM is known. 
(Demonstration.) For when the variation c>Q diminishes indefinitely, the curves 
q x q x , q 2 q 2 , approach indefinitely towards the curves Q 2 Q 2 respectively; and the 
small quadrilaterals bounded endways by the ordinates approximate indefinitely to 
the small quadrilaterals bounded endways by the curves of no transmission ; which 
latter pair of quadrilaterals are equal, by the first corollary of Proposition II. 
The symbolical expression of this proposition is as follows : — 
Let V A ,V B , V c , V D be the volumes corresponding to the four points of intersection 
of a pair of isothermal curves with a pair of curves of no transmission ; A and B being 
on the isothermal curve of Q,, C and D on that of Q 2 , A and D on one of the curves 
of no transmission, B and C on the other : then 
/*V B jp r Vc ^P 
^V(forQ=Q 1 )= tq-^V (for Q=Q 2 ) ) 
j Va m J V D ■ . . . (15.) 
or F b — F a =F c — F d I 
The second form of this equation is in the present case identical, because 
F b = F a ; F c =F b . 
(12.) Proposition IV. — Problem (see fig. 8). The forms of all isothermal curves 
for a given substance being given , let EF be a curve of any form , representing an arbi- 
trarily assumed succession of pressures and volumes. It is required to find , by the de- 
termination of points, a corresponding curve passing through a given point B, such, that 
