48 INTENSITY OF SUN'S HEAT AND LIGHT, 



2 u = it x + 2 ( l — e 'y C 2 *(h— -)dT \—2e sin P sin T— 4 e 3 (sin 3 P sin? T 

 n * / . o \ 2/ t 



+ 3 sin P cos" P sin Tcos 2 T) — I . (46.) 



Here II — — has been substituted for its equal from (43); multiplying the — - 



2 ■£ 



into the following term, and integrating between the limits and 2n, the result 

 vanishes. Also the terms multiplied by 4 e 3 being small ; it will be sufficient for 



them to develope from (43), to the first power, H — — = tan L sin u sin T — by 



which their' integral is immediately found to be — 4 e 3 tan L sin to x (sin 3 P . I it 



+ 3 sin P cos 2 P . —V or — 3 e % n sin a sin P tan L. Besides this, it only remains 



to integrate the first term depending on II sin T d T; but this corresponds to the 

 first term of the formula of annual intensity (23) ; and if & denote thermal days in 



the Table of Section V, then J ^sin TdT=([3M5±0]$secL—E')4:CotLcosecQ; 



whence finally, converting into hours, 



+ f e 2 Tt sin 2 a tan L +- . . . . > . (47.) 



On the equator, L is 0, and the last part vanishes, leaving for the annual dura- 

 tion of sunlight, x 12 h , where x denotes the number of clays 365.24. 



Therefore x 12 11 represents the mean value of the annual duration of sunlight, and 

 the following terms express the Annual Inequality. When the latitude is south, 

 both cot L and tan L change sign ; so that the inequality then becomes negative. 



For A. D. 1850, sin P is negative ; substituting the value of this and the other 

 elements for that epoch, 

 2w = xl2 ll + [2.16700]x{[3.61540]£sec.L— £'} co£i+ [3.88700] to Z, + ... . (48.)' 



Here brackets include the logarithms of the co-efficients. By this formula the 

 inequality may be readily computed for any latitude between the Equator and the 

 Polar Circle. 



In the frigid zone, the summer period of constant day will make another formula 

 necessary. As explained in Section V, the year in that zone may be divided into 

 four periods or intervals. At the beginning of the spring interval, H is 0, and 

 D = — (90° — L) ; at the end of the spring and beginning of the summer interval, 

 H is 12 h , and D = 90° — L ; at the end of the summer and beginning of the autumn 

 interval, also, D = 90° — L; and at the end of the autumn interval D = — (90° — L). 



With these data, the equation sin D = sin o sin T, or T = sin ~ l ( smJJ \ enables 



V sin u / 



us to define the lengths of the intervals. Thus the summer interval is measured 



by the sun's longitude passed over from T= sin- 1 (^-^),toT=7t—sin- 1 ( cosL ) 



\ sin o' \ sin u / 



