28 INTENSITY OF SUN'S HEAT AND LIGHT. 



ning of the spring, and end of the autumn interval, when H is 0, I u becomes 0; 



and u being a zero factor, T V — =— in (25) reduces to 0. Then exclusive of the three 



ax 



elliptic functions, the intensity of the spring interval will be 2- u + J* — ; 



" dx 



that of the autumnal interval, — i u' — T V . — . ; and for the summer interval, 



ax 



h u' + T V -= i u — A- -=— ; the sum of which is evidently 0. 



"ax "ax 



The expression of annual intensity thus reduces to the three elliptic functions in 



(26) integrated between the limits of the spring and autumnal intervals. Their 



collective differential in (23) and the analysis subjoined to it, will give, by making 



sin co sin T 



sin Z 



cos L 



cd T f s i n j, f an £ s i n Xj tan L cos 2 a 



' /'/ic / . /vie S. -4- 



j cai sin L, tan -L sin Li tan h cos" a ) 



u a x = — — I cos L cos Z + — v, 



n V 1 — e" [ cosZ (1 — cos" L sin" Z) cos Z J ' 



sin L tan L cos 2 L cos 2 Z + sin 2 L 



Here cos L cos Z + -^ — may take the form r ^ , oi', 



cos Z ■> cos L cos Z 



1 — cos 2 L sin 2 Z sin 2 o p 1 , / , cos 2 L 



or 



Lsm a \ sin" o /J 



cos L cos Z cos L cos Z 



Again, differentiating the above value of sin Z, 

 , T cos L cos Z , y _ cos L cos Z d Z 



ai=z ii^-cWr- ~ ^r ■ Tr^oTz^Tz^ whence ' 



. 1 — ( ) sin" Z 



~ V smcd I 



cd Z C COS 2 CO • 2 I 1 /COS L\ 2 . 2 ry 



udx = == \ —=======. + sin" a 1 — I — — ) sin 2 Z — 



nsina\/ — e 2 ( , /cosL\ 2 . „ „ 



N \ A" 



- ) sin 2 Z 

 sina' 



sin 2 L cos 2 id ) 



; or, 



(1 — cos 2 L sin 2 Z) I , /cos L\ 2 . » „ 



\ \ sin a i 



f 



u dx = o < jP + tan 2 a . E — sin 2 L . II 



71 sin a \/ 1 — e" ( 



As before remarked, these three integrals are to be taken between the limits of 

 the spring and autumnal intervals. At the beginning of the former and end of the 

 latter, H is ; whence cos H or 1 = — tan L tan D ; and D = 90° — L taken with 

 an opposite sign for south Declination. In this case, 



. rr Sill O Sill T Sill D -, ry ni-mo 



sin Z = — == — = — 1, or Z = 270°. 



cos L cos L 



At the end of the spring, and at the beginning of the autumnal interval H is 



12 h ; cos Hox — 1 = — tan L tan D ; whence D is 90° — L, sin Z = 1, or Z = 90°. 



Now the elliptic functions integrated between the limits, Z = 270°, Z = 90° give 



semi-circumferences for the spring interval, and the same for the autumnal interval, 



the sum of which will be entire circumferences. We have, therefore, for the Annual 



Intensity in the Frigid Zones, 



