(25.) 



INTENSITY OP SUN'S HEAT AND LIGHT. 27 



/c. cos L ( „ T „ "I 



udx= < E — tan L sin a cos T. 11 + tan'E . F — tan-Lcos-co.Tl > 

 ny/l — e 2 \ j 



+ \ u = h A 2 (sin L sin D . H + cos L cos D sin H) 



du u A 2 n V 1 — e 2 / — 2 e sm to H cot T \ 



^^dx~ TJ ' c \l + ecosd (H — tan H) cos 2 D' 



.... + C=.... + C. 



Having thus obtained 2 u, we may regard it as an implicit function, varying 



continuously with the longitude T, which returns to the same value at the end of 



a tropical year. Taking, then, the sum of the above terms as an integral between 



the limits, T = 360°, and T= 0; the purely trigonometric terms and constant having 



the same values at the beginning and end of the year, will vanish, leaving only the 



three elliptic functions, multiplied as follows: — 



c cos L r 1 



2«' = — : < E" + tail 2 L.F"— tan 2 L cos" a . IT' } . (26.) 



n v 1 — <r { J 



Here the eccentricity or common modulus is y ■ 



J cosh 



The Sun's Annual Intensity upon any latitude of the Earth is thus proportional to 

 the sum of two Elliptic circumferences of the first and the second order, diminished 

 by an Elliptic circumference of the third order. 



On the Equator, L and tan L are 0, cos L is 1, and the expression reduces to 



cE" 

 2 u' = = ■ -• (27.) This proves that the Sun's annual Intensity on the 



Equator is represented by the circumference of an ellipse, whose ratio of eccentricity 

 is equal to the sine of the obliquity of the ecliptic. 



In the Frigid Zones, where the regular interchange of day and night in every 

 twenty-four hours, is interrupted, the formula will require modification, though the 

 general enunciation of the elliptic functions remains the same. The year in the 

 Polar regions is naturally divided into four intervals, the first of which is the dura- 

 tion of constant night at mid-winter. The second interval at mid-summer is 

 constant day; the third and fourth are intermediate spring and autumnal intervals, 

 when the sun rises and sets in every twenty-four hours. For a criterion of the 

 beginning and end of the winter interval, Ave evidently have H = ; and for the 

 limits of the summer interval H = 1 2 h . 



During the winter interval, there is of course no solar intensity. The intensity 

 of the spring and autumn intervals will be found by integrating (25) between the 

 including limits, which results, added to that of the summer interval, give the 

 annual intensity. First, then, to examine the summer interval; H is 12 hours or 

 7t, sin H is 0, and consequently by (23) , 



/r cdT c sin L sin u cos T . n , . 



u d x = I ==.-== sin L sin o sin T .n = — — , which is pre- 

 " / n y/ 1 — e' n V 1 — e" 



cisely equal to the second term of (25), at the end of the spring, and at the 



beginning of the autumnal interval; so that on integrating between these limits, it 



will entirely disappear; and the same will apply to * \ u + -^ -% — For at the begin- 



