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



taining Q , Q /u , &c, are very small on account of the smallness of those factors ; so that 

 the maximum and minimum values of u for any assigned value of x will be very approx- 

 imately determined by those of the second term, provided /ul be not too small. Let H be 

 the difference between these values; then 



H = — . h . ir sin ix sin ye'z^. 



Again, let W be the value of H at a depth x, at the same place. Then 



D 



> b ■ ■ '-j- 



H = — . h . tr sin n sin ye <• *«. 



Therefore, 



H _x'-x 



whence a is determined by means of observed values of H, H ', x and so'. 



The value of b is determined by means of observed values of the times at which the 

 above maximum and minimum values of u occur; and a and b being thus determined, 

 the value of h is known by the preceding expression for H. 

 The values of these quantities are 



a = 5,11655, 

 b = 1,05719, 

 h = 35°,925 (C). 



These determinations were obtained by means of careful observations made for many 

 years at Paris, on the temperature of the Earth at different depths. It must not, 

 however, be supposed that these values are as accurately determined as the number of 

 decimal places might seem to imply. The probable errors of observation render such 

 accuracy unattainable. The exact times of the maxima and minima values of u are 

 difficult to obtain, and hence arises a somewhat greater uncertainty in the value of b 

 than in those of a and h. The above value is the mean of two determinations which 

 differed from each other by nearly ,1. There can be no reason, however, to doubt the 

 approximate accuracy of the above values as determined from the observations at Paris. 

 Poisson remarks that h would probably be somewhat greater in lower, and somewhat less 

 in higher latitudes, on account of greater general purity of the atmosphere in the former, 

 and less purity in the latter case. Its value also depends on the ratio of the absorbing 

 and radiating powers of the Earth's surface ; and this ratio may possibly differ materially 

 at different points not remote from each other. Its mean value, however, throughout a 

 whole district, will probably differ but little from its mean value in another district ; 

 and this is the principal consideration in reference to its influence, which it is our object 

 to ascertain, on the mean temperature of the air in any proposed latitude. 



19. If u denote the value of u at the Earth's surface, we have (putting x = in 

 the preceding expression for u) 



u = I + Q.h 



{b ir ir 9 b 1 

 — .- sin fx sin y cos (2-n-t - — - S) + — Q, cos (4-irt - S 1 ) + &c. > (10). 



