452 PROFESSOR CHALLIS, ON ATMOSPHERIC TEMPERATURE 



but for difference of density would have the same temperature. Though 

 derived from the consideration of fluid in motion, it may be extended 

 to fluid at rest, if we take a case in which the effect of the motion is 

 insensible. Thus supposing the velocity at every part of the wave to 

 be exceedingly small, and consequently (p very smaU, and the density p 

 to be very little different from p, the density the fluid would have at 



rest, the ratio -y- approximates to — (1 + a0,) as its limit, which is of 



dp ap^ 



finite magnitude. This limiting value must therefore express the ratio 

 of the difference of temperature of two contiguous elements at rest, to 

 their difference of density, supposing the variation of temperature to 

 depend on nothing but variation of density. 



Hence, being the temperature of the atmosphere at any altitude «, 

 where the density is p, 



__= _(1 +ae), (5). 

 up ap 



We are thus conducted by reasoning, which, though indirect, appears 



do 

 to be exact, to an expression for -r- proper for substitution in equation 



(4), and containing constants of known numerical value. By making 

 the substitution, 



[dzl ~ 1+k'dx a'ail+k)' ^ ^' 



Neglecting for the present the first term on the right-hand side of 

 the equation, and taking g = 32i feet, a\/l + k = 1090 feet, k = ,4152, 

 and a - ,00375, it will be found that 



lde\ 1 



[dzl ~ 334' 

 Hence s; = - 3340, 



supposing 6 = where !k = 0. Hence if d=-l\ the height =334 feet: 

 that is, the centigrade thermometer falls 1" for an elevation of 334 feet 



