177, 178] EQUATIONS OF HYDROSTATICS. 465 



Also, since, by 8) V differs from P only by a constant, the sur- 

 faces of equal pressure are equipotential or level surfaces. If the 

 fluid is incompressible y is constant, so that we have 



11) -V= const. + -' 



For gravity we have, if the axis of Z is measured vertically upward 



so that 



12) P~tt(C 



If, neglecting the atmospheric pressure, we measure z from the level 

 surface of no pressure 



13) P = -9Q*, 



which is the fundamental theorem for liquids, namely, that the pressure 

 is proportional to the depth. 



178. Height of the Atmosphere. If we consider a gas whose 

 temperature is constant throughout, the relation between the pressure 

 and volume is given by the law of. Boyle and Mariotte 



p = a$, 

 accordingly 



14) P = f*2 = r!L = a log e + const. 

 and 



15) V=gs = c a 



16) ? = 9o ", 



where p is the density when = 0. 



Thus as we ascend to heights which are in arithmetical pro- 

 gression, the density decreases in geometrical progression, vanishing 

 only for # = oo. If on the other hand, we consider the relation 

 between pressure and density to be that pertaining to adiabatic 

 compression, that is compression in such a manner that the heat 

 generated remains in the portion of the gas where it is generated, 

 we shall obtain a law of equilibrium corresponding to what is known 

 as convective equilibrium. The temperature then varies as we go 

 upward in such a way that, if a portion of air is hotter than the 

 stratum in which it lies, it will rise expanding and cooling at the 

 same time until its temperature and density are the same as those 

 of a higher layer. When there is no tendency for any portion of 



WEBSTER, Dynamics. 30 



