INTENSITY OF SUN'S HEAT AND LIGHT. 33 



f(a) d T= „ — i cot 2 u I 2 \ — ' . Here substituting h — h cos 2 T 



J 2 sum / 1 — sin' u sin 2 T 



for its equal sin 2 T, the last term will take the known form of I , where 



^ p + q cos 6 



represents 2 T; and by the Calculus its value between the proper limits, reduces to 



n n C,\jrr n / l — cosco\ 



— == or . Hence I (a) a 1 = -^ { r-v /• 



v' ,y — « 2 cosu> J y 2 \ sin' o t 



Since / ¥ cos 2 TdT=-,v?e have f(b)dT=-(—i + 1 — cosco ) I . 



/ 4 J K 2 \ 2 sin 2 a ' 2 cos 2 L 



Since sin 2 T cos 2 T may take the form cos 2 T — cos* T; the formula of the Integral 



r , T i vi • r r \ i rp Tt I — sin 2 a i 1 — cosu\ 3 



Calculus readily gives j (c) d 1 = — ( — i-| ) . 



J 2 \ 8 " sin 2 a / 8 cos* L 



In like manner C(d) d T= — ( — j\ sin* q — i sin 2 a — }-\ ) — . 



J 2 \ " sin' cj / 1 6 cos b L 



35 



28 cos 8 L' 



And f(e) d T= - ( — j^-g sin 6 a — T V sin* a — J sin 2 a — -I- -\ r -- ) -- 



J 2 \ " sin' a 'i\ 



The general formula (33) may be written — 



Su- 4C \casL.-R+™*±™^( C(a)dT+f(b)dT + . ...)}. 



n-J\ — e 2 c cosL \J ^ /J 



Here — can take the form of — cos _L -) ; substituting this value, and 



cos L cos Li 



multiplying it into the series of terms denoted by C(a) dT + C(b) d T + . . . . , and 



adding the products to the series (34) for cos L . E, we at length obtain, 



2cit r 1 — cosa 1 — cosq — isin 2 o 1 — coso — isin 2 a- — ism 4 ca 

 %u= A coscocosL-\ — n ^-\ — — o — 5-? ^~~ 



n ^/\ e "\ 2cosL 8cos 3 L 16 cos' L 



5(1 — cosco — hsin 2 u> — Isin*^ — -^\ sin 6 a) 7 (N — j-^shfa) , } /q^ ^ 



+ 128™7Z 256 cos 9 L + J ' ^ ' ; 



In the last term, N denotes the series within the parenthesis of the preceding 

 numerator. Now, taking the obliquity of the ecliptic a at 23° 28', and representing 

 the particular value of 2 W on the equator by 365.24 thermal days as in the last 

 Section, the multiplier for converting other values into thermal days, will evidently 



be '"' or 365.24 -h — ~~ — , (.9590919); the latter being the value of 2 W 

 2 W n V 1 — e 2 



when L is 0, and cos L is 1. In this manner, and denoting the logarithms of the 



co-efficients by brackets, we find for the present century, 



2 u = [2.543225] cos L + [1.197235] sec L + [1.211695] sec" L + 



, . (36.) 

 [3.819015] sec 5 £ + [4.616548] sec' L + [5.509114] sec 9 L+ ' v 



Or in numbers, 



omono r 15.748 0.1628 0.00659 , 0.000414 , fVn , 



2 u = 349.322 cos L + -=- + — — r + =^=- + — r — + .... (37.) 



cos L cos L cos L cos L 



