552 PROFESSOR 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 9; which will oc- 
casion a diminution of pressure, and a diminution of temperature, denoted re- 
respectively by » and 7. 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 9, », and 7, respectively. The alteration of pressure, during the first 
and third operations, may at once be determined by means of MariorTe’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+dvand p — at the end of the first operation. Hence the diminution of 
‘ {pate e d : 
pressure, during the first operation, is p—p ae or p ae and, therefore, if we 
neglect infinitely small quantities of the second order, we have p a 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. Ifé+7 
and ¢ 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 :— 
Go v, t+r, Pp; 
(2.) v+dv, t+r, pa-@ ; 
wo 
(3.) v+dvu+o, tt, pGQ-@)-s: 
o 
(4.) o+9, & p-—®; 
(5.) v, t+r, p- 
Taking the mean of the pressures at the beginning and end of each operation, we 
find 
dv 
(1, p(i-:2) 
dv . 
(2.) p(i—“2)-48 
(3.) p (1-3 )-« 
(A.) | P— 24, 
which, as we are neglecting infinitely small quantities of the second order, will be 


