16 INTENSITY OF SUN'S HEAT AND LIGHT. 



SECTION IV. 

 DETERMINATION OF THE SUN'S HOURLY AND DIURNAL INTENSITY. 



In the last Section, the sun's vertical intensity upon a given point of the earth's 

 surface at any instant during the day, was proved to be measured by a perpendicular 

 drawn from the centre of the Sun to the plane of the horizon. If perpendiculars 

 be thus let fall at every instant during an hour, the sum of the perpendiculars will 

 evidently represent the sum of the vertical intensities received dming the hour, 

 which sum may be termed the Hourly Intensity. 



The Integral Calculus furnishes a ready means of obtaining this sum. For 

 during any one day, the sun's distance or apparent semi-diameter, and the meridian 

 Declination, may be regarded as constant, while H alone varies, and the deviations 

 from the implied time of the sun's rising and setting will compensate each other. 

 Therefore, multiplying the equation of instantaneous intensity (12) by d H, since 

 astronomy shows that H varies uniformly with the time, and integrating between 

 the limits of any two hour angles, H', H", we obtain an expression for the hourly 

 intensity. 



In like manner let H denote the semi-diurnal arc, and integrating between the 

 limits and H, we obtain the intensity for a half day, which, on cancelling the 

 constant multiplier 2, may be taken for the whole day, or Diurnal Intensity, as 

 follows : — 



J A 2 sin A d H = A 2 H sin L sin B + A 2 cos L cos D sin H. (13.) 

 The diurnal intensity is, therefore, proportional to the product of the square of 

 the sun's semi-diameter into the semi-diurnal arc, multiplied by the sine of the 

 latitude into the sine of the sun's declination, plus the like product of the square 

 of the sun's semi-diameter into the sine of the semi-diurnal arc multiplied by the 

 cosine of the latitude into the cosine of the declination. This aggregate obviously 

 changes from day to day, according to the sun's distance and declination. 



Introducing the astronomic equation, cosH=* — tan L tan D, or in another form, 



sin L sin D . 



cos L cos D= — jj — ; the expression reduces to the following: 



J A 2 sin AdH= A 2 sin L sin D (H — tan H). 



It only remains to adopt a unit of intensity, the choice of which is entirely arbi- 

 trary. For the present, and in reference to Brewster's formula hereafter noticed, 

 we will assume the intensity of a day on the equator at the time of the vernal 

 equinox to be 81.5 units. For this case, where D and L are each 0, formula (13) 

 reduces to A 2 , which is (965") 2 ; hence 81.5-r-(965") 2 , or k, will be the multiplier for 

 reducing all other values to the same scale ; where the common logarithm of k is 

 5.94210. Denoting the annual intensity by u, and taking A in seconds of arc, we 

 have in units of intensity, 



u = k A 2 sin L sin D (H — tan H). (14.) 



