to which constant ratio (122) Clausius gives the rather obscure name of "equivalence-value." It is evident that these different forms simply express that, not only for perfect gases, but for all substances, absolute temperatures vary proportionally to the quantities of heat absorbed and given out in cycles of Carnot, or in perfect engines. From equation (119) we obtain, for the heat necessarily lost in perfect engines, By the law of Joule, Edq is the total dynamical value, or equivalent, of the variation of heat dq; but by the theorem of Carnot the actual value, or proportion, of this heat which can be used in an engine is only If now in any cycle, q be the heat received and go that emitted, then will be the work, during the cycle, of the heat which can possibly be utilized. But if the engine be perfect, or the cycle be reversible, the first term denotes the amount of work; and the second term consequently becomes Generally, however, as engines are imperfect, and therefore not reversible, the work during a cycle is much less than the value of the first term, which expresses its amount in perfect engines only; consequently, (126) is the heat wasted, and neither converted into useful work, nor necessarily given to the refrigerator, as q, must always be. Hence the expression just found is called by Thomson, who first obtained it, that of the dissipation; and it measures the imperfection, which cannot possibly be a negative quantity. GENERAL EQUATION FOR ALL TRANSFORMATIONS. 118. We have proved that there is always a factor capable of rendering exact and integrable the partial differential equations of thermodynamic changes. We have also found that, for perfect gases, this factor is λ = a+ t = T, the absolute temperature, as defined and indicated by an air thermometer. So that our general equation of transformation for all substances, dQ=λdp, becomes for perfect gases dQ = - ταφ. (127) The theorem of Carnot serves to generalize this result, by proving that for all bodies λ is equal to 7, the absolute temperature; while is a determinate but unknown function for cach particular substance; the form of which can be obtained for perfect gases only. 119. To show that the factor λ is equal to 7 for all bodies, let an engine work in a cycle of Carnot, MNPQ, composed of two isothermal lines, T and ', and two adiabatic lines, and o'; which may be taken so near to each other that But the ratio of 7 to 7', or that of q to q', has been proved to be constant for all bodies; it follows, therefore, that such is necessarily the case for the equal ratio of 2 to 2'. Consequently, the factors and λ' must equal the same function of and ', multiplied by an arbitrary function u, dependent in each case upon the nature of the body, or algebraically, The function u being arbitrary, and therefore capable of an infinite number of values, see § 62, we may put u equal to unity, which Now it has been found, for perfect gases, that λ is equal to the absolute temperature, consequently for all bodies in which equation is a determinate function of the independent variables, but differing with the nature of the substance. This equation is perfectly general and applicable to all thermodynamic changes. Hence, it is called by Rankine the general thermodynamic function. 120. Integrating (129) between limits, and supposing constant, or the change to occur isothermally, we get or the quantity of heat requisite for any body to pass from an adiabatic line do to another p', by an isothermal change, is proportional to the temperature. From this result, we may readily get the expression for the theorem of Carnot. Let and be the temperatures of the and . be the adiabatic lines of a cycle isothermal lincs, and of Carnot, then 2 == ($ · Φυ), Yo= To (Po), which is Carnot's theorem of maximum efficiency, and true for all substances. 121. Integrating between the limits (1) and (2), we obtain the second member of which depends not only upon the initial and final states (1) and (2) but upon all the intermediate states. To render this evident, we have by equation (58), the dynamical result р N M If now in the annexed diagram we suppose a body to pass from the state M to the state N, it is perfectly clear that the second term of the second member, or the integral of pdr, will be represented by M'MNN', and that this area is a function of all the consecutive intermediate values of p between M and N. Moreover, its Ꮴ M' N' value for a complete cycle is evidently represented by the closed area included in the curve or diagram of energy; but that of the internal energy is zero. 122. To prove from equation (129) that the efficiency of the cycle of Carnot is the maximum; let that of any other cycle |