34 INTENSITY OF SUN'S HEAT AND LIGHT. 



These formulas of Annual Intensity are applicable to the Torrid and Temperate 

 Zones, and would have given those portions of the table in the last section with 

 nearly the same facility as elliptic functions, but for the slow convergence of the 

 series in the higher latitudes ; the elliptic expressions are also preferred for the 

 future case of secular values. 



Denoting the co-efficients of (36) by a, b, c, ... . and with reference to formula (32), 

 multiplying by cos L d L, and integrating, 



VT T -r OT 7 T 7 77" C . (I -Lj it . (I ±J € . Oj JLj 

 ^ u . cos Jb a L = a cos' L.dL + o.dL-\ j-^r-\ r-^- ^ r=r + • • • • 



cos" L cos L cos b L 



.(38.) 



f X u . cos L d L = a (I L + i sin L cos L) + bL + ctanL + d ( — --- + 2) 



J 3 ^cos 2 L ' 



i sinL 4 C_dL\- C( sinL 6 CdL_\ c 



\5cos 6 L^5 J cos'L) J \7cos 7 £ 7 J cos 6 _L/ + " - 

 The last two integrals are given in the respective preceding terms. To determine 

 the correction C, make L equal to ; in this case, the surface being 0, the left hand 

 member and all the other terms vanish, except C, which is, consequently, 0. 



The next process is to find a similar formula for the Frigid Zone. Accordingly 

 from (28), and the analysis preceding that equation, we have, 



COS 2 6) 



— . cos 2 L cos- Z d Z 

 4 c 



4c f 



sin co, E' + 



n^/l—e-i. I „ 2T . 2 ^ N |, cos 2 L ..„ 



_ — sin" Z 



( 1 — cos 2 L sitrZ) ) 



(39.) 



sin" co 



This equation has precisely the form of (33); but a there, corresponds to 90° — L 



here ; and L there, corresponds to 90° — co here; T there, corresponds to Z here, and 



has the same limits of integration. Hence, by making the proper substitutions in 



(35) we may pass at once to the series for the annual intensity in the Frigid Zonej 



as here subjoined. 



2c7t f . t- . 1 — sinL 1 — sinL — \cos 2 L 1 — sinL — icos 2 L — I cos i L 

 2 u = \ sin Lsinco-\ — -— ; 1 -— ; -= h 



n ^/\ e 2 I 2 sin co 8 sin 3 co 16 sin 6 co 



5(1 — sin L — i cos 2 L — i cos* L — T V cos 6 L) ) , , A ,. 



a ^ = 5J. 1 + .... v . (40.) 



+ 128 sin 1 co S } 



Multiplying this equation by cos L d L, or D sin L, and integrating, 



2 c ?t J sin co sin 2 L sin L — & sin 2 L 

 2 u . cos L d L = ^=g 1 o H- 



P 



' n </ 1 — e" <- 2 2 sin co 



sinL — hsin 2 L — \(— \-ZsinL\ N—^l—- |-f sin BL+ 10 sinL) 1 ,.* , 



+ 8 sin 3 co + f6 sin 6 co +..C] 



Here N denotes the numerator of the preceding fraction. Now, integrating 

 between the limits L = 90°, and L = 90° — co = 66° 32', also introducing the con- 

 stant multiplier described after (35), we shall find for the Frigid Zone, 



/f T 7 r rn iroftfiDi ^ sitfid \ COS CO + \ COS 2 CO \ t • i • 



2 u . cos L a L = [2.580718] < \- — (-•••• > •> which is 

 (2 2 sin co ) 



equal to 13.733. 



Again, by formula (38), taking L between the limits and 23° 28', we find the 

 like sum between the equator and tropic, for the Torrid Zone, to be 141.86. 



