so that the isometrics and isobars of a perfect gas are logarithmic curves on the (0, 4) diagram.

In the Carnot cycle aßyd for a perfect gas, for instance in an ideal gas engine, the work done in compressing the gas adiabatically from a to ß is

R(02 — 01)/(y — 1),

and this work is therefore given out again in the adia batic expansion from y to 8, so that the areas aabß and yedd are equal (fig. 107); and the above equations also

show that

= = ecp- Cv.

су

Oa ̄ ̄ds Ob

The work done, per lb or g of the gas, by the isothermal expansion from ẞ to y is

Re2log

V3 —

V2

R02

while the work consumed by the isothermal compression

from 8 to a is

J01(P2—P1) :

J(02 — 01)(Þ2 — P1).

the difference, as before, being

Examples.

(1) Prove that the orthogonal curves of the adiabatics on the (p, v) diagram are the similar hyperbolas

p2 - yv2 = constant.

Prove that the isothermals and adiabatics cut at a maximum angle cot-12/y on the line

Discuss the same problem for the isometrics and isobars on the (0, 4) diagram.

(2) Prove that, if a perfect gas expands along the curve puconstant, the work done by expansion is (y-1)/(y—k) of the mechanical equivalent of the heat absorbed.

(3) Prove that the specific heat of a perfect gas, expanding along the curve f(p, v) = 0, is

(5) Determine the heat equivalent of the kinetic energy of rotation of the Earth, supposed homogeneous and of S.H. c; and determine the number of degrees which this heat would raise the temperature of the Earth, taking c=0·2.

(6) Find what fraction of the coal raised from a mine 500 fathoms deep is used in the engine raising the coal, and 30 times its weight of water, supposing the heat of combustion of 1 lb of coal is 14,000 B.T.U., and the efficiency of the engine is

(7) Compare the work done and the work given out when V ft3 of atmospheric air is compressed adiabatically to n atmospheres, cooled down to the original temperature, and expanded adiabatically to atmospheric pressure; for instance, in a Whitehead torpedo.

TABLE I.-DENSITY OF WATER (MENDELEEF).

C.

s(g/cm3). D(lb/ft3).| v(cm3/g). C.

s(g/cm3). D(lb/ft3).| v(cm3/g).

0° 0-999873 62-4162 1.000127 40° 0.992334 61 9456 1007725

50° 0.988174 61 6860 1.011967 60° 0.983356 61 3852 1.016926

5° 0.999992 62.4237 1·000008 10° 0.999738 62-4078 1.000262 15° 0.999152 62:3712 1.000849 70° 0.977948 61 0476 1.022549 20 0.998272 62.3163 1.001731 80° 0-971996 60 6760 1·028811 25° 0 997128 62.2449 1.002881 90° 0.965537 60.2729 1.035693 30° 0.995743 62.1584 1.004275 100° 0·958595 59-8395 1.043193