54 INTENSITY OF SUN'S HEAT AND LIGHT. 



Developing in powers of sin m by Maclaurin's Theorem, 



sinm i sin 2 m sin L sin D , , ., (cos 2 L + (3sin 2 L — l)sin 2 D)\ 



-, _ — — + s sin 6 m K )— - — ' 



Vcos'L — sin 2 D (cos-L — sm 2 _D)f (cos-L — sm 2 D)z 



3 sin L sin I) . 5 sin 6 L sin 6 D 



(59.) 



, . a 1 SsmL/SinU . bsin Jbsin U \ . 



— isitfm (- — — — — — + — r = - „ )+ 



\cos' L — sin Dp cos' L — sur D)l ' 



With respect to the yearly limits already assigned, (55), (56), (57), we know that 



in the lower latitudes, twilight recurs regularly, while the sun's longitude T varies 



from zero to an entire circumference; but in the Polar zone, this continuity is 



interrupted. Still, in integrating for the yearly duration of twilight between the 



proper limits, J a + J f - — being expressed in terms ofsinD or sinT will vanish, even 



in the Polar zone, leaving only C udx. And with respect to doc, since cos Td Tis 



d sin T, which multiplied into the development of r, would integrate in powers of 

 sin T which vanish, we may reject all such factors in advance, leaving, 



dx=S l ~~^—dT\\ — 2esinPsinT+ 3 e 2 (cos 2 P — cos 2 P sin* T) + . . . .]. (60.) 

 ii 



Were this multiplied into (59), making ti = 2t, and substituting for sin D its 

 equal sin a sin T, then integrating between the proper limits, and dividing by — in 



order to convert arc into hours of time, we should obtain the annual duration of 



twilight expressed in elliptic functions. It will be more convenient, however, to 



resort to circular functions. 



To obtain the duration of Twilight in another form, let N denote the interval of 



Night, from the end of the evening twilight to midnight, or from midnight to the 



morning twilight, computed by the sun's midnight declination. The duration of 



N will correspond to any assumed depression or elevation of the crepusculum circle, 



or to any compatible value of m. Then N = 12 h — (H + t) ; 



TV,- , Tri x sin L sin D 4- sin m 



cos JS == — cos (ii -\- 1) = — 



cos L cos D 



d N — sin L — sin m sin D — sin L — sin m sin D 



d sin D cos L cos 6 D sin N C os 2 D V cos 2 L— sin 2 m — sin 2 D — 2 sin L sin m sin D 

 Developing cos 2 D into the numerator under the form of ( 1 — sin 2 D) ~ l ; also 



resolving the radical into two factors, one of which is \/ cos 2 L — sin 2 m, and de- 

 veloping the other into the numerator to the fifth power of sin D ; then multiplying 

 the factors, and employing Maclaurin's Theorem, or integrating; also making- 

 cos 3 L — sin 2 m = s ; 



at _i/smm\ sin L sin D I sinm cos 2 m sin 2 D sinL / n r n , 



N=cos l { ) — _ — — -F-T- (cos2L + 2 + 



\cosL' V s s* 6 si \ 



3 sin 2 Lsin 2 m\ . 3 -,-. sinm j , n . 3 sin 2 L (I 4-sin 2 m) . 5 sm 4 L sin 2 m\ . iri 



■ ) sin 6 D — ( cos 2 m 4- 2 4- i ' '- 4- ) suvD 



s / 8 si \ s s~ ' 



sinL f - Q 3 sin 2 m (1 + sin 2 IS) + f 5 sin 2 L sin 2 m (sin" m + #) 



io7i \ C0SZlj + z + s + s 2 ~ + 



Y«tfL»*n ) sh f D _smm f ^ g ^ + g , 3 shr L (1 + sin 2 m) + f + 

 s' ) 1 2 s [ s 



