456 Mr. A. W. Eiicker on the [June 18, 



represented by the curve, at the point of contact ceases to emit and 

 begins to absorb heat, or vice versa ; and that therefore every closed 

 cycle, if a continuous curve, must have 2n points of contact of an odd 

 order with adiabatics, and if a discontinuous curve, 2n r such points of 

 contact and r points of discontinuity at which the curve does not cut the 

 adiabatics passing through them. 



This, however, evidently does not hold for a curve which meets the 

 curve in the plane of pv , denned by 



\ w "/ 



which is the projection of the curve in space, whose equations are 



/r7A 



/*/ f\ r\ i '*'/ \ r\ 



/(^).0,(^0. 



For since it does not follow that the curves in space have contact because 

 their projections touch, we see that the curve in the plane of pv may 

 touch an adiabatic without any change taking place in the absorption or 

 emission of heat ; and such a curve may, even if continuous, have contact 

 of an odd order with an odd number of adiabatics. The point of contact, 

 for instance, of a curve which touches but does not intersect the limiting 



curve at all points on which [ J- )=0, projects into a point of contact of 



\dtj 



the third order at least ; and therefore the projected curve must lie 

 -entirely between the adiabatic and projection of the limiting curve, which 

 only have contact of the first order i. e. it has a single point of contact 

 of an odd order, with an adiabatic which does not correspond to a change 

 in the absorption or emission of heat, and therefore on the whole it has 

 an odd number of points of contact of an odd order with adiabatics. 

 Let us now suppose that ABB'A' and a/3/3'a' are two adiabatics (fig. 4) 



which meet the curve ( ~ ) = 0,andlettwoisothermals,Aaa'A'andB/3/3'B', 

 \at / 



meet the first in A A' and BB' and the second in a a' and /3/3' respectively. 

 "We can now make the water go through Carnot's cycle of operations 

 between the same temperatures in four different ways, of which we need 

 only consider the cycles a'A'B'/3' and a'A'B/3. In each of these the quan- 

 tities of heat received along a' A' are the same, therefore the quantities of 

 work done must be the same, i. e. 



area a'A'B'/3'a = area a'A'B/V 



= area a'A'B'/3'a'+area /3'B'B/3/3' ; 

 .-. area/3'B'B/3/3'=0. 



But this area is composed of the two B'/3'M and B/3M, and they are of 

 opposite signs ; for in going round the closed curve M/3'B'M, the work 

 done on the body is greater than the work done by it, while in the loop 

 MB/3M the contrary is the case ; whence we conclude that the areas 

 B'/3'M and B/3M are equal. 



