Theorem for Irreversible Cycles. 527 
Moreover, such methods, even if valid, are unnecessarily 
complicated in principle, 
24. Again, as regards any useful deductions, Thermo- 
dynamics appears as yet to have gone no further than to 
compare one state of equilibrium with another, the two states 
corresponding to different values of the independent co- 
ordinates of the system, such as pressure, temperature, 
quantity of each ingredient, &. The methods referred to 
institute such comparisons not directly, but by the devious 
process of comparing each equilibrium state with all the 
non-equilibrium states possible for the same values of these 
independent coordinates. 
Such methods mathematically dificult. Carnot’s cycles 
preferable. 
25. I venture here to put in a plea for the treatment, at 
least in text-books, of such problems as those alluded to by 
the aid of Carnot’s cycles as applied, for example, by Van 
t Hoff *. I do so not so much on the ground of the above 
objections, as on that of the greater simplicity and directness 
of the latter method. Many readers who could readily follow 
reasoning based on reversible cycles are quite unable to 
understand the other methods. Elementary methods are 
conducive to clear thinking on tke part of all, teacher as 
well as student, the expert mathematician no less than the 
tyro. And in no other branch of mathematical physics are the 
difficulties which necessarily have to be faced, so greatly those 
which are incidental to clear thinking and so little those of 
mathematical analysis. The method of Thermodynamic 
Potential or of Available Hnergy appears as unnecessary for 
the discussion of many such problems as does the dynamical 
principle of Least Action for the investigation of the motion 
of a body sliding down an inclined plane. And the use of 
the more elementary method would obviate the pitfalls into 
which writers who are not trained mathematicians are 
occasionally betrayed in handling the other methods, 
* For an application of this method to the question of equilibrium in 
a gas-system, see Van t’Hoff, Vorlesungen tiber die Theoretische Chemie, 
or Lehfeldt’s translation, p. 105. The investigation of the effect of the 
change of temperature on the state of equilibrium, 7. c. p. 141, is, 
however, wanting in clearness. And the corresponding results in cases 
of solutions, in so far as that writer bases them on thermodynamic 
principles at all, appear to be merely written down from analogy, though 
it is stated that thev can be obtained by similar methods, as is of course 
the case, ) 
