718 
MR. A. McAULAY ON THE MATHEMATICAL 
same independent variables as l) equal to the function which is reciprocal to l with 
regard to p, C and H. These three vectors may be [§ 35] called velocities, and thus 
X is in value the reciprocal of l with regard to all the velocities involved in the latter. 
Adopting now the notation of § 33 and its assumption (end of § 33), we have 
l + X = 2t = - (D m p ' 2 + SC c VZ + SBH/4tt).(11) 
K = t s = 0 .( 12 ). 
That l, the Lagrangian function (per unit volume), and X, the free energy, should be 
reciprocal functions (in value only) with regard to the velocities they contain, is 
in accord with the fact (but not deducible from it) that a similar statement is true 
for an ordinary dynamical system [§ 33, eq. (1) above]. 
Let now A stand for the part of the free energy due to a finite portion of matter. 
Required A. To find this, first obtain the rate of increase of A that would occur if 
all the circumstances were such as actually occur, except that the temperature of each 
element of matter is kept constant, and then add the part due to the rate of variation 
of temperature. To get the first of these we have at first to find the corresponding 
part of L by changing all the S’s of eq. (9) into differentiations with regard to the 
time. Then we have to subtract the result from A -f- L, which is given by eq. (11). 
Thus we get 
A = j"jj [6 d\/dO — S@ 0 VX) ds 
- ||| {S p' [D,„ (p‘ + V'W) - ro^'V,'] - Sc (l V/ + SC ( d c Yl/dt+ A - B Vl)}ds >- (13). 
— || Sp<f>'d%' — ( 47t) — 1 j]yAHc£2 J 
It should be shown perhaps how the last integral appears. It comes from the term 
— SBH/47T in l -fi- X and from the two terms f 
-IIII sasd *-4vIL sa8hcK 
in SL. These three terms contribute to A 
I III ( SA0 - SBH /4ir) * +^\l « AH dt. 
But, since, 4vC = YVH, 47tSAC — SBH = SAAH, so that the volume integral can be 
