INTENSITY OP SUN'S HEAT AND LIGHT. 55 



5 sin' 1 L {sin 2 m {sin 2 L + f ) + I) . 35 sin 4 " L sin 2 m(i + i sin* m) 



-—* — + f + 



315 sin 6 L sin 4, m) • „ -r-. ,„„ , 

 207 \^D-.... (62.) 



Here let us put cos N' = - ; and denoting the co-efficients of sin _D, sin 2 D 



cos L 



by — N u ' — N, we have, 



N - N' — N sin D — N 2 sin 2 D — N 3 sin* D — N, sin 4 D—.... 



Multiplying this by the former series for d x, and integrating the products, after 



substituting sin o sin T for sin D, and dividing by — , 



S2iV4 2411(:1 ~ e2)l \ N'[T{1 + 3e 2 cos 2 P) + 2 e sin PcosT—3 e 2 cos2P fo+....l 



9l7t ( J 



+ sinoN [(1 + 3 e 2 cos 2 P) cos T+2e sinP f+3e 2 cos2P f + ] 



_sm 2 6) iy . ^i + 2e i cos i P) C—2esinP f—3e 2 cos2P f — ....] 

 _smV^ 3 [(1 + 3 e , C0S 2 P) £ — 2e sinPf—% e 2 cos 2 P£ — . . . .] (63.) 

 _smy N, ^ + g e o cos2p , /• _ 2 e sinJp /- _ 3 e 2 cos 2 P f — . . . .] 



1.2.3.4 Ji J& Ja 



-r^.X5 [ ' 1+3eSrfp >i:- 2e ™ p X- 3 ^'' s2P X----] 

 -r^^^ +Se "- cm ' p V- 2esinP f- ;ie,c " s2P f--- -l-.+cl. 



The integral signs here designate the following quantities : — 



sin2T T 

 l- + 2- 

 cos 3 T_ 3 cos T 

 ~"l2~ 4 ' 



sin 4 T sin 2T,3T 



f = Csin 2 TdT=, - 

 f = fsiif TdT 



1 = j sm 1 d 1 = — . + (64 ) 



Ji J 32 4^8 { ■> 



r r ■ * ti j rn cos 5 T 5 cos ST 5 cos T 



i = I sin 1 d 1 — — 1 — . 



Ja J 80 ^ 48 8 



r sin 6 T 3sin4:T_15sin2T 5T 



Je~ ~T92~ + 64 64 + "16 



In the year 1850, 0= 23° 27*'; P= 280° 22£'; e = .01676; n=—^ t —; l + Se 2 cos 2 P 

 J 365.24' + 



1.000027; 2 esinP=— .032972; 3 e 2 cos2P= —.000788; 2 "^ (*— g8 ) g = 443.89. 



n.n 



For the lower and middle latitudes, where 2 iV and 2 (Jf + t) alternate in every 

 twenty-four hours through the year, we may integrate through an entire circum- 

 ference. In this case, equation (63) is materially simplified; and denoting by 

 brackets the common logarithms of the co-efficients, 

 2 2 N= [3.44564] N' — [1.26253] N x — [2.04360] iV 2 — [1.55940] iV 3 | 



— [0.03944] N, — [3.37930] N b — [3.68312] N«— ...}' ^ 65,) 



