174 Lord Hayleigh on the 



obeys Boyle's law, viz. such that the pressure is always pro- 

 portional to the density. And in the first instance we shall 

 neglect the curvature and rotation of the earth, supposing 

 that the strata of equal density are parallel planes perpen- 

 dicular to the direction in which gravity acts. 



If p, a be the equilibrium pressure and density at the 

 height z, then 



£—>'- a 



and by Boyle's law, 



P=d?<r, (2) 



where a is the velocity of sound. Hence 



_^L-_£ (HI 



adz~ a 2 ' {) 



and 



a = (7oe - gZ /a^ (4) 



where a is the density at £ = 0. According to this law, as is 

 well known, there is no limit to the height of the atmosphere. 

 Before proceeding further, let us pause for a moment to 

 consider how the density at various heights would be affected 

 by a small change of temperature, altering a to a' ', the whole 

 quantity of air and therefore the pressure p at the surface 

 remaining unchanged. If the dashes relate to the second 

 state of things, we have 



cr = (T e -z*/" 2 3 a f = a- / e-^ a '% 



p =p e~^ a \ p' = p e~^ a ' 2 } 

 while 



a 2 cr = a /2 aQ. 



If a /2 — a 2 = 8a 2 , we may write approximately 



p'-p _ha 2 gz /a2 

 p <r or 



The alteration of pressure vanishes when z=0, and also when 

 2 = go. The maximum occurs when gzja 2 —l, that is when 

 p=Po/e- But relatively to a, (p'—po) increases continually 

 with z. 



Again, if p denote the proportional variation of density, 



o= <T ^- = 4 (e-^ a ' 2 +^ a2 - 1). 



If a'' J >a 2 , p is negative when z = 0, and becomes +00 when 

 2 = c© . The transition p = occurs when gz/a 2 =l, that is at 

 the smme place where p' —p reaches a maximum. 



