126 
MR. MACQUORN RANKINE ON THERMO-DYNAMICS. 
of no transmission will differ from an area whose upper boundary is an isothermal 
curve, by an indefinitely small area of the second order. 
To express this symbolically, let 
be the ratio in question, for a given indefinitely-close pair of curves of no transmis- 
sion. Let the change from one of these curves to the other be made by means of 
any indefinitely-small changes of actual heat and of volume, <5Q, W. Then by the 
general equation 7> the following quantity 
»F=|={ 1± |^+£JPrfv}SQ+g.SV= I ^.SQ + g8V . . . (11.) 
is constant for a given pair of indefinitely-close curves of no transmission, and is, 
therefore, the complete variation of a function, having a peculiar constant value 
for each curve of no transmission, represented by the following equation : — 
„ (r/H fl+$'.Q jr , P/P 7Tr . 
F =nH — 5— rfQ + W rfV < I2 -> 
This function, which I shall call a Thermo-dynamic function, has the following 
properties : — 
H=jcWF (13.) 
is equivalent to the general equation (6.) ; 
t/F=0 ............... (14.) 
is the equation common to all curves of no transmission ; and 
F= a given constant, ......... (14 a.) 
is the equation of a particular curve of no transmission. 
(11.) Proposition III. — Problem. Let it he supposed that for a given substance, 
the forms of all possible isothermal curves are known , hut of only one curve of no trans- 
mission ; it is required to describe, by the determination of points, another curve of no 
transmission, passing through a given point, situated anywhere out of the known curve. 
(Solution) (see fig. 7)- Let LM be the known curve of no transmission ; B 
the given point. Through B draw an isothermal curve QjABQ^ cutting LM in A. 
Qi being the quantity of heat to which this curve corresponds, draw, indefinitely near 
to it, the isothermal curve q x q 15 corresponding to the quantity of heat Q, — <5Q, where 
SQ is an indefinitely small quantity. Draw any other pair of indefinitely close isother- 
mal curves Q 2 Q 2 , q 2 q 2 , corresponding to the quantities of heat Q 2 , Q 2 —SQ ; <5Q being 
the same as before. Let D be the point where the isothermal curve Q 2 Q 2 cuts the 
known curve of no transmission. Draw the ordinates AV a , BV b parallel to OY, en- 
closing, with the isothermal curves of and Qj — <$Q, the small quadrilateral AB ba. 
Draw the ordinate DV d parallel to OY, intersecting the isothermal curve of Q 2 — <$Q 
