DYNAMICAL PRINCIPLES TO PHYSICAL PHENOMENA. 
473 
In the first place, there is the mental satisfaction to be got by explaining things on 
dynamical principles ; and, again, there is the certainty that the method is capable of 
completely solving the question (whether we can make it do so is another matter), 
while we have no certainty that all possible information is given by the two laws of 
Thermodynamics, or that some unknown third law might not enable us to arrive at 
results beyond the powers of the first and second. 
§ 2. The researches of Clausius, Szily, and Boltzmann have shown that the 
Second Law of Thermodynamics is closely connected with the principle of Least 
f 
Action, and it might therefore be thought that the Second Law was only a more con¬ 
venient way of stating this principle, so that no advantage could be gained by the 
direct use of dynamical principles. In the investigations on the connection between 
the Second Law and the principle of Least Action there are, as I shall endeavour to 
show later on, a good many assumptions implicitly made, so that it seems to be much 
preferable to proceed, if possible, in any special case by the direct use of dynamical 
principles. 
Again, there can, I think, be no question that the principle of Least Action and 
the Second Law of Thermodynamics are not equivalent; for, in the first place, as is 
well known (see Routh’s ‘Advanced Rigid Dynamics/ p. 257), the principle of Least 
Action includes that of the Conservation of Energy, so that, if the Second Law of 
Thermodynamics included all that could be got from the principle of Least Action, it 
ought to include the First Law as a particular case. 
Again, in the most general case, the principle of Least Action will for a system 
fixed by n coordinates give n equations ; but the Second Law of Thermodynamics, 
which asserts that a certain function is a perfect differential, would, in the most 
general case, give rise to \ n (n — 1) equations, as that is the number of conditions 
to be satisfied if 
D-pbc' i —f- P Abu -p 
is a perfect differential. 
§ 3. The dynamical methods we shall most frequently use in the following paper 
are the Hamiltonian principles expressed by the equations 
28 [ Tclt — iS (T + V) + 
J o 
• • (i) 
28 f (T - V) dt = (T + V) Si + 
Jo 
V dT £ ‘ 
where T and V are respectively the kinetic and potential energies of a system, q a 
typical coordinate helping to fix the configuration of the system, and t the time. 
The first of these equations has been used to show the connection between the 
principle of Least Action which it expresses and the Second Law of Thermodynamics ; 
MDCCCLXXXVII.—A. 3 P 
