704 
MR. A. McAULAT ON THE MATHEMATICAL 
B. Preliminary Dynamical and Thermodynamical Considerations. 
Ba. The “Modified Kinetic ” Energy and the “Free” Energy. 
32. It is not to be supposed that the coordinates we have assumed are sufficient to 
fix the position of all matter in space. The mathematical machinery we use cannot 
be supposed sufficiently fine to trace the motion of molecules. Such coordinates as 
would be required for that purpose are “ ignored.” Now (Larmor, £ Proc. London 
Math. Soc.,’ vol. 15, 1884, p. 173) in order that the principle expressed in eq. (1) of 
§ 13, above, may be true under these circumstances, L must be, not the true 
Lagrangian function, but what Bouth (‘ Elem. Big. Dyn.,’ 4th ed., § 420) has called 
a modified Lagrangian function. And that our principle may be true the particular 
type of modification is assigned, i.e., the ignored coordinates are those whose momenta 
appear explicitly. And a further restriction is necessary (Larmor, as above), viz., 
that the ignored coordinates must only appear through their momenta. That is, the 
ignored coordinates must be what Professor J. J. Thomson (‘ Applications,’ 1st ed., 
§ 7) has called kinosthenic or speed coordinates. This last restriction, however, is 
not absolutely necessary if we take L to be the average value of the modified 
Lagrangian function for a small time, sufficiently large to allow the molecules to go 
through all their types of motion many times. 
33. Whether these restrictions be imposed or not we have the following relation :— 
A = ZqdL/dq-L ......... (1), 
where A is the whole energy of the motion due to a modified function L, and q is a 
coordinate whose velocity appears explicity. (Notice that if A were supposed 
expressed, not as a function of the q s, but as a function of the d~L/dqs, it would be the 
reciprocal function of L with regard to the q s [Booth’s ‘ Elem. Big. Dyn.,’ 4th ed., 
§ 410.] It is not this reciprocal function only, because, for our purposes, it is more 
convenient to assume it an explicit function of the same quantities as L). To prove 
this, let" <f .>, <k be a coordinate, whose momentum appears, and its momentum respec¬ 
tively, and let L 0 be the Lagrangian function of which L is the modified form. 
Thus 
'Iq 0L /dq — L = %q 0L Jdq — (L 0 — S</><1>). 
[Bouth’s £ El. Big. Dyn.,’ 4th ed., §§ 410, 420]. 
= 2 (q 9L Jdq + (/>3L— L 0 . \ibidh] 
— 2© — (X — S3). [T = kinetic energy, © = potential energy.] 
— T -p © = A. 
* There is no danger of confusion of these meanings of 0, <t> with the stress meanings these symbols 
bear through the rest of this paper. 
