128 
MR. MACQUORN RANKINE ON THERMO-DYNAMICS. 
the quantity of heat absorbed or emitted by the substance, in passing from any given iso- 
thermal curve to any other, shall be the same, whether the pressures and volumes be 
regulated according to the original curve EF, or according to the curve passing 
through the point B, 
(Solution.) The process by which the latter curve is to be deduced from the former 
is precisely the same with that by which one curve of no transmission is deduced from 
another, in the last problem. 
Fig. 8. 
(Demonstration.) Let GBH be the required curve. This curve, and the curve EF, 
in their relation to each other, may be called Curves of Equal Transmission. Through 
B draw the isothermal curve QAJi, intersecting the curve EF in A. Draw also any 
other isothermal curve Q 2 Q 2 , intersecting EF in D and GH in C. Through A,B,C,D, 
respectively, draw the four indefinitely-prolonged curves of no transmission, AK, in- 
tersecting Q 2 Q 2 in d, BL, intersecting Q 2 Q 2 in c, CM, and DN. Conceive the whole 
space between the isothermal curves QjQ^ Q 2 Q 2 , to be divided by other isothermal 
curves, into a series of indefinitely narrow stripes, corresponding to equal indefinitely- 
small variations of actual heat. Then, by the construction of the solution, the qua- 
drilaterals cut from those stripes by the pair of curves EF, GH are all equal ; and so 
also are the quadrilaterals cut from the stripes by the pair of curves of no transmis- 
sion, AK, BL. Therefore the area ABCD is equal to the area AB cd. The indefinitely- 
prolonged areas, MCDN, LcrfK, are evidently equal ; therefore, adding this pair 
of equal areas to the preceding, the pair of indefinitely-prolonged areas LBAK, 
MCBADN are equal. Subtracting from each of these areas the part common to 
both, ABR, and adding to each the indefinitely-prolonged area KRCM, we find, 
finally, that the indefinitely-prolonged areas KADN, LBCM are equal. 
But the former of those areas (by Prop. I.) represents the mechanical equivalent of 
the heat absorbed by the substance in passing from the actual heat Q 2 to the actual 
heat Qj through a series of pressures and volumes represented by the co-ordinates of 
the curve EF ; and the latter, the corresponding quantity for the curve GH ; there- 
fore those curves are, with respect to each other, Curves of Equal Transmission, which 
was required 
