AND THE OTHER PLANETS OP THE SOLAR SYSTEM. 641 



r 2 = - - = - 273° (C.) 

 a 



as in the previous cases. 



We may shew that p will not become negative for any value of z less than (5. We have 



from equation (3) 



P 

 P = «* (1 + aQ ' 



which, at the upper boundary of the atmosphere, assumes the indeterminate form $. To find 

 its real value we have, on substituting the assumed expression for 1 + et£, in the equation (3) 



p = a s p {l + ar - Ci% + c 2 x" + ... + c n x"}, 



1 dp 1 dp c x — 2c 2 x - &c. - nc n x n l \ 

 p dx p dx 1 + ar - c v x + ... +c n x n \ 



Also from (l) and (2), 



1 dp g r 2 



p dx a 2 c x (r + x)' z 1 + ar 



X . 



or omitting - , which may be done without risk of error, 

 r 



±dp = J[ I 



p dx afc, 1 + ai-, c„ 



--X+... + —X* 



c, c, 



(~ -l) +2-X+ +»-«•-' 



1 dp Xa'd / c, c, 



Hence — ■— — — ^ — — ^^— — ^^— . 

 p dx 1 + ot c 2 . c„ 



X + — 2T + ... — X" 



C l C x C t 



Since the numerator of this expression is of lower dimensions than the denominator, taking 

 /3 as before, we shall have 



. (-fl-i) +2 ^/3+ +«-"/3- 1 __, 



J dp _ Wd I Ci Ci M 



p dx~ /3 - x ^ (*) ' 



° J oi (VoV, 7 ci ci J & /3 Jo fW 



The same reasoning will apply to the last term of this expression as in the previous case, 

 and x = /3 will give p = provided 



(J- Cl ) +2c 2 /3 + »c„/3- 1 



be a positive quantity. If it were negative, log— would necessarily be positive when /3 - x 



should be sufficiently small, and consequently p would be greater than p v 



82 — 2 



