INTENSITY OP SUN'S HEAT AND LIGHT. 49 



or the length of the arc is % — 2 sin~ 1 ( ). This divided by n, the mean 



\ sin cj / 



daily motion in longitude, and multiplied by 24, will give the number of hours of 



sunshine during the summer period, which is — ( n — 2 sin~ 1 ( — - — ) ). 



n \ \siug)// 



For the spring and autumn intervals, when day and night alternate, the values 



must be found by the general formula of summation. Here u = 2 H = n 



+ 2 sin ~ 1 (tan L tan D). (49.) 



, 2 tan L sin a cos T d T ,„ n ^ 



du = : (50) 



— sin- a sin- T) II — — ;-=■ sin" 1 

 \ cos- L 



1 ii (1 + e cos P cos T + e sin P sin T) 2 



da: = dT(l — e-)l 



Whence it will be seen that all the terms of i it + -rV - r - vanish between the 



doe 



Kmitsofr-«»- 1 (^,«»- 1 (- cw£ );andr-w»- 1 (-ig£^V«n- 1 (^J'); 

 \sinv>' ^ sin a ' \ smu I \smaJ 



sin o 



cosL 



4 e n sin P sin u tan L cos T sin T ( > 



except the following, ===== . Here make — — sin T 



3(1 — sin- o sin 2 T)\ 1 — (^y) sin 2 T 

 — sin Z, and the expression becomes 4 e n sin P sin L 



i COS 2 L . , ry. ry 



— sin- Z tan Z 



And 



3(1 — cos 2 L sin 2 Z) 

 this when taken between the proper limits of Z = + 90°, — 90°, evidently vanishes. 



It only remains to find C u d x between the same limits for the spring and the 



autumn interval. These limits show that the sun's longitude passed over in the 



two intervals is 4 sin ~ l (— — ) which divided by n gives the number of days. 

 \ sinu ' 



This multiplied by the first term of u which is n or 12\ and added to the like 



result for the summer period, gives 12 x — orxl2\ And if sinT=— sin Z, or 



L ° ii sin a 



sin D = cos L sin Z; then for the whole year, in hours, 



+ .2618 n J o \^\ — cos 2 Lsin 2 Z> (l + ecos(T—P))- 



It is here assumed that the integral will be taken successively between the limits 



of Z= — , — — : — — , + — ; that is through a whole circumference. 

 2 2 2 2 ° 



But d T = cos -LcosZ ( i% = cos L ^ — __ j ^ ^ ^ g tne w j 10 i e function 



sin acosT sin a I , /c os L \ 2 ^ ^ 



S \sino' 



of Z by which d Z is multiplied, would evidently develop in odd powers of sin Z, 

 it follows as in the former operation, that terms which would vanish by integration 

 1 



