552 PKOFESSOR WILLIAM THOMSON'S ACCOUNT OF 



quantity of the second order. During the second operation we may suppose 

 the volume to be increased by an infinitely small quantity f ; which will oc- 

 casion a diminution of pressure, and a diminution of temperature, denoted re- 

 respectively by ^ and t. During the fourth operation there will be a diminution 

 of volume, and an increase of pressure and temperature, which can only differ, by 

 infinitely small quantities of the second order, from the changes in the other di- 

 rection, which took place in the second operation, and they also may, therefore, 

 be denoted by ?, u, and t, respectively. The alteration of pressure, dm-ing the first 

 and third operations, may at once be determined by means of Mariotte's law, 

 since, in them, the temperature remains constant. Thus, if, at the commence- 

 ment of the cycle, the volume and pressure be v and p, they will have become 



v + dvandp — 3- at the end of the first operation. Hence the diminution of 

 pressure, during the first operation, isp-p ^_^^^ °^P v + dv' ^^^' tl^^refore, if we 



neglect infinitely small quantities of the second order, we have p — for the dimi- 

 nution of pressure during the first operation ; which, to the same degree of ap- 

 proximation, will be equal to the increase of pressure during the third. If t + r 

 and t be taken to denote the superior and inferior limits of temperature, we shall 

 thus have for the volume, the temperature, and the pressure at the commence- 

 ments of the four successive operations, and at the end of the cycle, the following 

 values respectively : — 



(1.) «, l + r, p; 



(2.) v + dv, t+r, ptl-il^; 



(3.) v + dv + (p, t, p(i — — )-<"; 



(4.) v+f), t, p-w; 



(5.) V, t+T, p. 



Taking the mean of the pressures at the beginning and end of each operation, we 

 find 



(1.) .(i-40 

 (2.) K^-v)-*" 

 (3.) p(i-40-^ 



(4.) p — Aw, 



which, as we are neglecting infinitely small quantities of the second order, wiD be 



