32 INTENSITY OF SUN'S HEAT AND LIGHT. 



the intensity upon a single point multiplied by the circumference 2 n cos L, will 

 express the sum of the intensities received upon the whole parallel of latitude ; 

 consequently 2 u . 2 7t cos L .d L integrated between the limits of L and U will 

 denote the sum of the intensities upon the zone or surface between the latitudes L 

 and U . By Geometry, the surface of this zone is proved to be equal to (sin L — 

 sin L') 2 it. Therefore the sum of the annual intensities divided by the surface, 

 will evidently give u, the average annual intensity of the Sun upon a unit of surface 

 in that zone, as follows : — 



J L'%u .cos LdL zoo \ 



u = J — _ — . (32.) 



sin Ju — sin U 



To find the value of this integral, 2 u must first be developed in terms of cos L. 



It is shown in (23), and in the analysis following that equation, that the annual 



intensity, exclusive of terms cancelled by the integration, is 



4 c r sin 2 L sin 2 a I 2 cos 2 TdT 

 2u = — ====,] cos L.E' + — —j — I -====== .(33.) 



/ „ (1—surusnrl) 1 — sin I > 



*J ° v J S cos 2 L 



From the well known formula for the rectification of the ellipse, we have in the 



first place, 



^i.£'=^(co^-i^- A ^-^^- ( HI )2 ^-....). (34.) 

 2\ cosL cos s L ' cos-'L cos 1 L > 



Next, to find the value of the last integral, let the radical of the denominator 



be first developed, and its terms multiplied into the other factors separately; then, 



preparatory to integration, let each numerator be divided by its denominator, as 



follows : — 



cos 2 T l—sin?T 



(1 — sin 2 a sin 2 T) 11 sin 2 T 



\ cos~L 



1 — sin 2 asin 2 T 



/-. , i sura . orn n sin a . irr \ 

 ( 1 + i — — sm-T+ f — sin*T+ ...). 

 \ cos- L cos L I 



(a) (b) (c) 



1_ 1 

 (a) - 1 — sm 2 T = 1 sin 2 a 



W — 1 — s i n z Q s i n i T sin 2 a 1 — sin 2 a sin 2 T 

 , n 1 sin* a sin 2 T cos 2 T 1 , „ m , „ 



(h) - WZ X 1-sin'osi nW = Wl ( ~ C ° S ~ T + ^ 

 , >. 3 sin* cd sin 4, T cos 2 1 3 , . , . „ „, , ™ „~ . .. 



( c ) = o TT X "1 — ^Tm = a Tt ( — Sm " " ml T C0S ~ T ~ C0S ~ T + ( a ))- 



8 cos* L 1 — sm 2 a sm 2 T 8 cos* L 



5 



(d) = — — ( — sin* co sin* T cos 2 T — sin 2 a sin 2 T cos 2 T — cos 2 T + (a)). 



v 16 cos 6 L v T v " 



35 



0)= -.^n sr (—sin 6 a sin 6 Tcos 2 T— sin* a sin* Tcos 2 T— sin 2 a sin 2 Tcos 2 T— cos 2 T+ (a)). 

 128 cos L 



Multiplying now each term by d T, and integrating between the limits of T = 

 — , and T = 0, we obtain the following results: — 



