variable body, it can only depend on the temperature of the two bodies K, and K,, which act as heat reservoirs. The same will of course be true of the sum

This last ratio, which is that between the whole heat received and the heat transferred, we shall select for further consideration; and shall express the result obtained in this section as

in which (TT) is some function of the two temperatures, which is independent of the nature of the variable body.

§ 7. Determination of the Function (TT).

The circumstance that the function given in equation (2) is independent of the nature of the variable body, offers a ready means of determining this function, since as soon as we have found its form for any single body it is known for all bodies whatsoever.

Of all classes of bodies the perfect gases are best adapted for such a determination, since their laws are the most accurately known. We will therefore consider the case of a pergas subjected to a cyclical process, similar to that graphically expressed in Fig. 8, § 1; which figure may be here reproduced (Fig. 11), inasmuch as a perfect gas was there taken as

fect

an example of the variable
body. In this process the
gas takes up a quantity of
heat Q during its expansion
ab, and gives out a quantity
of heat Q, during its com-
pression cd. These quanti-
ties we shall calculate, and
then
compare with each

other.

For this purpose we must

first turn our attention to the volumes represented by the

abscissæ oe, oh, of, og, and denoted by v1, v2, V, V,, in order that we may ascertain the relation between them. Now the volumes vv, (represented by oe, oh) form the limits of that change of volume to which the isentropic curve ad refers, and which may be considered at pleasure as an expansion or a compression. Such a change of volume, during which the gas neither takes in nor gives out any heat, has been treated of in § 8 of the last chapter, in which we arrived at the following equation (43), p. 62:

where T and v are the temperature and volume at any point in the curve. Substituting for these in the present case the final values T, and we have:

2

In exactly the same way we obtain for the change of volume represented by the isentropic curve be (of which the initial and final temperatures are also T1T):

1°

We must now turn to the change of volume represented by the isothermal curve ab, which takes place at the constant temperature T,, and between the limits of volume v, and V, The quantity of heat received or given off during such a change of volume has been determined in § 8 of the last chapter, and by the equation (41) there given, p. 61, we may put in the present case:

Q1 = RT, log

V

. (6).

Similarly for the change of volume represented by the isothermal curve dc, which takes place at temperature T, between the limits of volume v and V, we have:

The function occurring in equation (2) is now determined, since to bring this equation into unison with the last equation (8) we must have:

We can now use in place of equation (2) the more determinate equation (8), which may also be written as follows:

The form of this equation may be yet further changed, by affixing positive and negative signs to Q, Q. Hitherto these have been treated as absolute quantities, and the distinction that the one represents heat taken in, the other heat given out, has been always expressed in words. Let us now for convenience agree to speak of heat taken in only, and to treat heat given out as a negative quantity of heat taken in. If accordingly we say that the variable body has taken in during the cyclical process the quantities of heat Q, and Q2, we must here conceive Q, as a negative quantity, i.e. the same quantity which has hitherto been expressed by - Q2 On this supposition equation (10) becomes:

2

§ 8. Cyclical processes of a more complicated character.

Hitherto we have confined ourselves to cyclical processes in which the taking in of quantities of heat, positive or negative, takes place at two temperatures only. Such processes we shall in future call for brevity's sake Simple Cyclical Processes. But it is now time to treat of cyclical processes, in which the taking in of positive and negative quantities of heat takes place at more than two temperatures.

We may first consider a cyclical process with heat taken in at three temperatures. This is represented graphically by the figure abcdefa (Fig. 12), which, as in the former cases, consists of isentropic and isothermal curves only. These curves are again drawn, by way of example, in the form which they would take in the case of a perfect gas, but this

ૐ

Fig. 12.

1

is not essential. The curve ab represents an expansion at constant temperature T; be an expansion without taking in heat, during which the temperature falls from T1 to T2; cd an expansion at constant temperature T,; de an expansion without taking in heat, during which the temperature falls from T to T; ef a compression at constant temperature T,; and lastly fa a compression without (taking in) heat, during which the temperature rises from T, to T,, and which brings back the variable body to its exact original volume. In the

expansions ab and cd the variable body takes in positive quantities of heat Q, and Q,, and in the compression of the negative quantity of heat Q. It now remains to, find a relation between these three quantities.

For this purpose let us suppose the isentropic curve be produced in the dotted line cg. The whole process is thereby divided into two Simple Processes abgfa and cdegc. In the first the body starts from the condition a and returns to the same again. In the second we may suppose a body of the same nature to start from the condition e, and to return to the same again. The negative quantity of heat Q,, which is taken in during the compression ef, we may suppose divided into two parts q, and q,', of which the first is taken in during the compression gf, and the second during the compression eg. We can now form the two equations, corresponding to equation (11), which will hold for the two simple processes. These equations are, for the process abgfa,

In exactly the same way we may treat a process in which heat is taken in at four temperatures, as represented by the annexed figure abcdefgha, Fig. 13, which again consists solely of isentropic and isothermal lines. The expansions ab and cd, and the compressions ef and gh, take place at temperatures T1, T2, T,, T1, and during these times the quantities of heat Q1, Q2 Q3 Q4 are taken in respectively; the two former being