256 Proceedings of Boyal Society of Edinhurgli. [sess. 
particular case, the usual analytical expression of the Second Law 
of Thermodynamics. It is preferable, however, to work with the 
equivalent equation (3). 
To find whether the positive or negative sign must be used in that 
equation, consider A and B as external ‘‘forces.” If A tends to 
cause increase (or decrease) of a while B tends to cause increase (or 
decrease) of h, the negative sign is to be used ; if otherwise, the 
positive sign must be used. Again, if we wrote (3) thus, 
dAda±dhd^ = 0^ we must take the opposite sign to that which 
we take when we write the second term in the order d!Bc?6. For 
the cycle represented by dhd^ is performed in the opposite order to 
that represented by d^dh. 
Collecting the results we get, when the + sign is used in (3), 
A few special examples are appended. 
I. Heat and Mechanical Work . — Let 6, <f>, p, v represent respec- 
tively temperature, entropy, pressure, and volume. The quantity 
d0dcj> represents energy supplied to the system, while dpdv repre- 
sents work done by it. Hence 
dOd(^ -1- dpdv = 0 , 
giving = 
the four well- 
dO dp dO dv d4> dv d<f> dp 
V dcfi’ dp ’ dp dd ’ dv dO ’ 
known thermodynamical relations. 
II. Latent Heat and Work during Evaporation . — Let I be the 
amount of liquid, and let p be its density, while cr is the density of 
the vapour, and A is the latent heat. From dOd(j> + dpdv = 0 we 
get by (6) 
d6 = dp^ 
How 
d<f> 
n _ 1 
p 
dv= E-E]dl 
\cr 
d8 = d/—^ ^ • 
crp dxj) 
whence 
