14 INTENSITY OF SUN'S HEAT AND LIGHT. 



Let L = the 'apparent' Latitude of the place, 

 D = the sun's meridian Declination, 

 A = the sun's apparent semi-diameter, 

 A = the sun's Altitude, and 

 H = the Hour-angle from noon. 

 Also in reference to future applications, let 

 T= the sun's true Longitude, and 

 a = the obliquity of the Ecliptic. 

 The horizontal section of a cylindrical beam of rays from the Sun's disk upon 

 a plain on the Earth's surface, is well known to be an ellipse ; and if 1 denote the 

 sun's radius, 1 will likewise denote the semi-conjugate axis of this projected ellipse; 



while the horizontal projection, — — -j, will be the semi-transverse axis. The area 



1 



of the elliptic projection is, therefore, 1 x -- — -j- x %. But the intensity of the same 



quantity of heat being inversely as the space over which it is diffused, the reciprocal 

 of this area, or sin A, on rejecting the constant %, will express the sun's heating 

 effect, supposing the distance to be constant for the same clay. But, on comparing 

 one day with another, the intensity further varies inversely as the square of the 

 distance, that is, directly as the square of the apparent diameter or semi-diameter 

 of the disk. Hence, generally, A 2 sin A, expresses the sun's intensity at any given 

 instant during the day. 



To determine the value of sin A, by spherical trigonometry, the sun's angular 

 distance from the pole, or co-declination, the arc from the pole to the zenith, or 

 co-latitude, and the included hour-angle from noon are given to find the third side 

 or co-altitude. Writing, therefore, sines instead of the cosines of their com- 

 plements, 



sin A = sin L sin D + cos L cos D cosH. 

 A 2 sin A — A 2 sin L sin D + A 2 cos L cos D cos H. ^ '' 



At the time of the equinoxes, D becomes 0, and the expression of the sun's 

 intensity reduces to A 2 cos L cos H. That is, the degree of intensity then decreases 

 from the equator to each pole, and is proportional to the cosine of the latitude. At 

 other times of the year, however, a different law of distribution prevails, as indicated 

 by the formula. 



The intensity at a fixed distance being as the sine of the altitude, it follows that 

 the sun shining for sixteen hours from an altitude of 30°, would exert the same 

 heating effect upon a plain, as when it shines during eight hours from the zenith ; 

 since sin 30° is 0.5, and sin 90° is 1. At least, such were the result independently 

 of radiation. 



By some writers, the measure of vertical intensity, as the sine of the sun's alti- 

 tude, has been stated without limitation. Approximately it may apply at the 

 habitable surface of the earth, when the influence of the atmosphere is neglected; 

 yet it is strictly true only at the exterior of the atmospheric envelope which encom- 

 passes the globe, or at the outer limit where matter exerts its initial change upon 

 the incident rays. 



