1897-98.] Thermodynamics founded on Motivity and Energy. 127 
particular insulated pieces of metal in an electrical system ; or pro- 
portion of salt to solvent in an osmotic application. 
§ 3. Let m r (t, g v g 2 , g 3 , . . . .) and e(t, g v g 2 , g 3 , . . . .) 
denote the motivity and the energy of the system, the latter being 
absolute, the former being relative to a temperature T, the lowest 
available for carrying off heat. These expressions written in full 
denote that m T and e are functions of the independent variables 
t, 9 ii g^i e ^ c - > bid generally, except when it is convenient to be 
reminded that they are such functions, we shall for brevity denote 
their values simply by m and e. 
§ 4. First suppose the temperature of the apparatus kept con- 
stant at T, and let the other independent variables be augmented 
from g l - \dg v g 2 -\dg 2 , etc., to g^^dg^ g 2 + ^dg 2 , etc., through 
infinitesimal ranges dg v dg 2) etc. Let 
M 1 (T).c?^ 1 + M 2 (T).<i^ 2 + etc. (1) 
denote the quantity of heat (positive or negative) which must be 
taken in from without to keep the temperature constant at T, and 
let 
Pi(T ).dg l + P 2 (T )-dg 2 + etc. (2) 
denote the mechanical work required to produce the change. 
This work simply contributes its own amount to the motivity of 
the apparatus, because, as in Carnot’s theory, we have an infinite 
river or ocean at temperature T, always ready to give or take 
freely any heat to be taken in by, or rejected from, the apparatus 
to keep it at constant temperature T. Hence 
dm T (T, g 1 ,g 2 , . . •) = Tfl).dg 1 + P 2 (T).dg 2 + etc. (3). 
On the other hand, the energy is augmented, not only by the 
mechanical work done on it, but, in addition to this, by the 
dynamical equivalent of the quantity (positive or negative) of heat 
taken in. Hence, if J denote the dynamical equivalent of the 
thermal unit, we have 
d [ e (T,g 1} g 2 , • . .) ~ m T (T,g 1} g 2 , . . .)] = JM 1 (T).^ 1 + JM 2 (T).^ 2 + etc. (4). 
The second members of (3) and (4) are complete differentials of 
functions g v g 2 , etc., on the supposition that T is constant. Hence 
