50 INTENSITY OF SUN'S HEAT AND LIGHT. 



may be neglected in advance, leaving for the last factor, precisely as in (46), 



— 2 e sin P sin T — 4 e 3 (sin 3 P sin 3 T + 3 sin P cos 2 P sin Tcos 2 T) — ... 



Substituting here for sin Tits equal sin Z; the first term of the product is 



sin w 



r> • r> • -i / sin L sin Z \ cos 2 L sin Z cos Z d Z 



— 2 e sin P . sin 



_! / sinJLsinZ \ 



\s/l—cos l Lsin 2 Z' sin* a \ 



Integrating this by parts, we have 



COS' L . o rr 



■ ; — sin' Z 



2 e sin P. sinr'f sinJLsmZ \ 1 1 C0 ^ L si ^z 

 V 1— cos 2 L siirZ'^ sin 2 o 



4 



K V 1 — cos 2 L siirZ' 

 — 2 e sin P I sinL \ 1 r -— sin 2 Z. dZ. 



^ 1 — cos 2 L sin 2 Z 



Multiplying both numerator and denominator of the last term by the radical, it takes 



the form of 



(sin L — sin L cos 2 L sin 2 Z) 



sin 2 u C sin L d Z 



J (1 — cos 2 L sin 2 Z) \l — C0S [ L sin 2 Z / sin 2 co \\ — ^Jksin 2 Z 

 J \ sin 2 G> *s S sm'o 



+ (sin L — sm . L ) n ; that is, s ^- . F— sin L cot 2 u. U ; where the letters F and II 

 \ sin" a)/ sin' to 



designate elliptic functions. When integrated between the above named limits, 



or an entire circumference, the former term of the integral vanishes, leaving only 



8 e sin P sin L /irt , 2 „,. 



— (F' — cos' to. IT). 



sin a 



Developing only to the first power, the next term to be integrated is 



4 e 3 sin 3 P cos* L 



sin a 



sin L sin 4, Z d Z, which between and 2 7t, gives 



— 3 e 3 7i sin 3 P sin L . -. — . As the remaining terms are still smaller they may 



sin to 



be omitted; whence 



,-, 10h 1 6 e (1 — e 2 )l sin P \ sin L -,-,, sin L cos 2 o „, 



2, u = x \.z — — ±— - — j - -; jb — n 



. 2 6 1 8 n sin g> [ sin at sin a 



...} (52.) 



3 e" it sin 2 P sin L cos* L 



The multiplier for converting 2 u in Section V into thermal days S, is - 



365.24 , , /OQN sin 2 L cos 2 coll' 1.5065 S cos 2 a t-,, . -r,, , 



x Y^na^ '' whence b y ( 28 )' — -■ = -o7fH-oT- ~~ - — F ~ sm "• E > and 



l.oUbo smu 365.24 sino 



substituting this, we have for the year 1850, 



S«=xl2"+[2.16700]^ \ ,-3.61540-1 JL— E'+(l — C -^-)f I 

 sin -Li 1 - J sin a \ sin" o' ) 



+ [1.87213] sin L cos* L + . . . (53.) 



Here the modulus or eccentricity of the elliptic quadrant is C ; and the brackets 



sin a 



denote logarithms of the co-efficients. Such is the formula for the Frigid Zone. 



