ON THE CALCULATION OF SUN-HEAT COEFFICIENTS. 67 



When the absorption of heat by the atmosphere is neglected we have 

 merely to integrate 



A/cos z dh (1) 



from suurise to snnset, where 



A = the solar constant of radiation, 

 z = sun's zenith distance, 

 h = sun's hour angle ; 



an integration readily performed ; and then sum the results from day to 

 day, from the summer to the winter solstice ; a summation which presents 

 no serious difficulty. 



But when we attempt to compute the sun-heat received in a given 

 time and at a given latitude, allowing for the absorption of sun-heat by 

 the atmosphere, we are met by formidable mathematical difficulties 

 which have never yet been seriously acknowledged and attacked. 



It is, in fact, easy to see that we must now attempt the integration, 

 daily, from sunrise to sunset, of 



instead of 

 where 



A.jp u cos z dh, (2) 



A /cos z dh, 



p = the atmospheric constant of absorption ; 

 u = «y 2rh + h 2 -f r 2 cos 2 z — r cos z ; 

 h = height of homogeneous atmosphere ; 

 r = radius of earth. 



It is evident at sight that equation (2) is not integrable ; and if we 

 attempt to integrate it by series we fail completely, for the following 

 reason :— 



It will be seen, on trial, that the expansion of 



p u cos z 

 must be of the form 



A + A[ cos z + A, cos 2 z + &c. ; (3) 



+ Bj sec z + B 2 sec 2 z + &c. 



This series is to be multiplied by dh, and each term integrated from 

 sunrise to sunset. This is easily done for the cosine terms, but the secant 

 terms become infinite at the limits, because z = 90° at sunrise and sunset. 

 Hence any attempt to obtain the value of integral (2) by approximation 

 must be illusory, no matter how rapidly the coefficients 



B 1? B 2 , B 3 , &c, 

 may diminish. 



Under these circumstances it was proposed at the Dublin meeting of 

 the British Association (1878), to apply a small grant (30Z.) to a pre- 

 liminary quadrature of equation (2), at a few well-defined latitudes, 

 such as 0°, 30°, and 60°. 



The method used was the following : — 



1°. The values of p u cos z, for every value of z from 0° to 90°, were 

 first calculated, from which the values of p" cos z, for every zone of zenith 

 distance, one degree in width, were readily found. 



f2 



