Page:Elementary Text-book of Physics (Anthony, 1897).djvu/264

250 infinitesimal increment of volume, the sum of all the terms $$pdv$$ will equal the total work done by the ezpansion of the body. Now let us consider the area $$bBCc$$ standing under the line $$BC$$ (Fig. 73). This area may be conceived of as made up of a series of infinitesimal rectangles, the heights of which are the ordinates of the successive points of the line $$BC,$$ and the bases of which are successive elements taken along the line $$bc.$$ If $$dv$$ represent the length of one of these elements, and $$p$$ the corresponding ordinate, the area of the infinitesimal rectangle determined by them is $$pdv.$$ The sum of such areas for the expansion indicated by the line $$BC$$ is the area $$bBCc;$$ and since $$\textstyle \sum \displaystyle pdv$$ represents the work done, the area $$bBCc$$ also represents the work done during the expansion of the body in the way indicated by the line $$BC.$$

Now to demonstrate the relation between the temperatures indicated by the perfect gas thermometer and those of the absolute scale, let us suppose an engine in which the working body is a perfect gas, and let us suppose that the changes in pressure and volume experienced by the working body during the cycle are so small that the portions of the isothermal and adiabatic lines which bound it are straight, and that the cycle is a parallelogram. This cycle is represented by the area $$ABCD$$ (Fig. 73). We may assume as the result of the experiments of Joule that when a gas expands at constant temperature, no internal work is done upon it, or that the heat which enters it is entirely spent in doing external work. Produce DA to e; then the parallelogram $$ABCD$$ is equal to the parallelogram $$eBCf$$ and this parallelogram represents the work done in the cycle by the gas acting as the working body.

The work done during the expansion from $$B$$ to $$C,$$ which is equal to the heat received during that expansion, is represented by the area $$bBCc.$$ Let $$g$$ be the middle point of the line $$BC;$$ the perpendicular $$gh$$ will bisect the line $$ef$$ at $$i.$$ The area $$bBCc = bc. gh,$$ and the area $$eBCf = bc. gi.$$ Therefore the efficiency of