640 Mr W. HOPKINS, ON THE EXTERNAL TEMPERATURE OF THE EARTH, 



and therefore 



1 + a£ = 1 + ar - c^Z + c^x 2 + c n z n . 

 Let /3 be a positive root of the equation 



1 + ut -c 1 z + ... +c„#" = (6); 



then we may write 



l+«£-(/3-*)0(*). 



<p (it) being a rational function of fe ; and we shall have 



^ cfy> gr* 1 



p~ite~ ~ ~tf (/3 -*)(»• + *)* ^ (*) ( rt " 12 ^ ; 



a 2 1/3 - * + (r + %)* <p (z) I 



where L is independent of z, and M a function #, of lower dimensions in z than the denomi- 

 nator. Hence 



logg-g'ziogg-r'-gV' ^f . 



J/ 



Now — can always be expressed in partial fractions of the following forms, 



(r + *)<£(*) 3 V ' 8 ' 



L In L 3 L t P + Qz 



y — z ' & + * (V + *)' r + * ' «" + X* + i^ 



where the quantities denoted by the large letters are independent of z, -y being a positive root 

 of equation (6), 5 a negative root, and a 8 + \z + v 2 a quadratic factor corresponding to a pair 

 of impossible roots of the equation. Now it is easily seen that none of the integrals of the 

 last four of the above fractions could become infinite for any finite positive value of z, or for 

 such, therefore, as could be available for satisfying in the last equation the condition p = 0, 

 which must hold at the upper atmospheric boundary. The only admissible value of * for 

 that purpose must be a positive root of equation (6). And this, moreover, must be the least 

 positive root of that equation ; for /3 and -y being positive roots, if y be taken for the height 



P 

 of the atmosphere and be greater than /3, 1 + a£, and therefore, from equation (3), — must 



have different signs according as z is greater or less than /3 ; i.e. either p or p must become 



negative at certain heights, which is absurd. Consequently if /3 be the least positive root of 



p 

 equation (6), it is the value of z admissible for satisfying the condition p = 0, or log — = — oo in 



the last equation. Hence the least positive root of (6) will give the height of the atmosphere 

 provided the increment of temperature at any height z can be expressed in a series proceeding 

 according to positive and integral powers of z, and provided, also, the value of p does not 

 become negative for any value of * less than j8. 



At the upper limit of the atmosphere, where * = /3, 1 + a£ is reduced to zero; 



i.e. 1 + ar 2 = 0, 



