INTENSITY OF SUN'S HEAT AND LIGHT. 47 



epoch, the line of the apsides evidently coincided with the line of the equinoxes, 

 which is computed to have been about 4,000 years before the birth of Christ, at 

 which time chronologists have fixed the first residence of man upon the earth. The 

 luminous year was then of the same length, at all latitudes, from pole to pole. 



Though the annual Duration of sunlight thus varies from age to age, and in the 

 northern hemisphere differs from the southern ; yet such is the law of the planet's 

 elliptic motion, that the sun's annual Intensity at any latitude north, is precisely 

 the same as at an equal latitude south of the equator. This immediately follows 

 from formula (33), where the annual Intensity is developed in a series of powers of 

 cos L, which is always positive, whether the latitude L be south or north. 



Proceeding now to direct investigation, the half day with its augments, may be 

 represented under the general form, 



H 4. increase by Refraction + Twilight. 



The first term H is found from the astronomic equation, 



tan L sin a sin T 

 cosH= — tanLtanD = — — -^ =-1=; and this may also take the form 1 of 



V 1 — sin" co sin" 2 



Tr 7t • _i / tan L sin o sin T \ 



H = — + sin 1 1 \ (43) 



2 ^V 1 — sin 2 to siir T> 



Let u = 2 U, or twice the semi-diurnal arc ; then the sum of all the daily 

 values of u through the year, may be found by the method of summation described 



in Section V. By (22) we have dx = — — ^ -— J- — =^: whence, 



3 v ' n(l + ecos(T — P)) 2 



r , 2 (1 — e 2 f I sin- 1 (tan L tan D) d T ,... 



I u d x = n x -j 1 <— I — i J — . (44.) 



J n J 0- + ecos{T—P)f y ' 



The general formula of summation, Section V, has the terms J u + tV 1- ■ • 



dx 



which in the present case vanish between the limits T = 0, and T = 2 n ; as will 



appear from developing u or 2 H by (43), in terms of sin T. For the annual value 



'therefore, 2 u = Cudx. Developing the denominator of (44), and substituting for 



D in the numerator, 



Xu^nx+^—fL C ^A-i (t *L#°*££ \ dT \ 1 _2ecos(T-P) 

 n J Wl—sin 2 asiri i T / ( 



+ 3e 2 cos 2 (T—P) — ....l. (45.) 



It is evident that sin _1 here would develope in odd powers of sin T, which mul- 

 tiplied by d T, and integrated between the limits of and 2ti, will vanish, as 

 appears from the formulas of the Integral Calculus; when multiplied by d T. cos T, 

 or d sin T, and integrated between the same limits, they also will vanish, being 

 powers of sin T. Also developing cos (T — P) into cos TcosP + sin T sin P, and 

 neglecting terms, which would so vanish by integration, 



1 On this and the following pages, sin ~ l x denotes the arc whose sine is x ; where x represents any- 

 given quantity. 



