254 
Proceedings of Royal Society of Edinhurgh. [sess. 
the system considered by the letters a, c, &c. The change of 
energy in any process is 
0E , 0E 0E 
rtE = ^^aa-r 4- 7:— ac+ 
oa 00 do 
or c?E = Ada + Bdb + Cdc + 
where A, B, C, &c., are, in the ordinary language of dynamics, 
forces of the types which cause the changes da, dh, dc, &c. They 
are the quantities upon the magnitude of which the tendency to 
transference, or transformation, of energy depends. [In the dyna- 
mical view, transformation is merely transference to a dissimilar 
system.] When the system returns, at the end of the process, to 
its original condition, we get 
Ada-{-Bdh -^Cdc-^- ... =0 (1) 
This equation is the expression of the principle of conservation, for 
it asserts that any amount of energy of the type (A, a), which enters 
the system, passes out of it again either in the form (A, a) or in 
some other form into which it is changed by the medium of the 
system, and we get finally da = 0, dh = 0, &c. 
If now we apply the principle to a closed reversible cycle of 
operations, in which equilibrium is only disturbed to an infinitesimal 
extent at any stage, and in which, for example — 1st, A increases 
by dA, while a is constant: 2nd, a increases by da, while A is 
constant; 3rd, A decreases by dA, while a is constant; 4th, a 
decreases by da, while A is constant, we get 
d .dB = dA.da-\-dB .dh + dC .dc+ . . . =0 . . (2) 
In general, if the above cycle is strictly observed with regard to 
the energy (A, a), it cannot be simultaneously carried out for any 
other type. But the actual cycle for any other type may be sup- 
posed to be built up of an infinite series of such cycles on an 
infinitely smaller scale, so that the final effect is the same as if the 
cycle were carried out. 
Suppose that we are dealing with two types of energy only. In 
this case (2) becomes 
dAda - dBdh = 0 (3) 
[Here the sign has been changed for convenience. The question is 
