22 INTENSITY OF SUN'S HEAT AND LIGHT. 



SECTION V. 



FORMULA AND TABLE OF THE SUN'S ANNUAL INTENSITY UPON ANY LATITUDE OF THE 



EARTH. 



By the method explained in the last Section, the diurnal intensity, in a vertical 

 direction, might he computed for each and every day in the year, and the sum 

 total would evidently represent the Annual Intensity. 



The sum of the daily intensities for a month, or monthly intensities, might be 

 found in the same manner. But, instead of this slow process, we shall first find an 

 analytic expression for the aggregate intensity during any assigned portion of the 

 year, and then for the whole year. The summation is effected by an admirable 

 theorem, first given by Euler; a new investigation of which, with full examples by 

 the writer, may be found in the Astronomical Journal (Cambridge, Mass.), Vol. II, 

 p. 121. Thus, let u denote the ^th term of a series, where u is a function of x. 



Attributing to x the successive values 1, 2, 3, 4, x, and denoting the sum of 



the results by s u, it is shown that, 



n du d 3 u d°u 



Xu = J udx + i u + T V ^ - jh fa* + 3 oIto -fa -•■■• + C. (20.) 



Since this important formula has not yet been introduced into any American 

 treatise on the Calculus, I here insert one of the two demonstrations from the 

 Journal referred to, which indeed was suggested by the present research : — 



Imagine the several terms of the original series to be ordinates of a curve, and 

 erected at a unit's distance from each other, along an " axis of X;" then, by the well- 

 known formula of the Calculus, Cud x will represent the area of this curve. 



Again, connecting the upper adjacent extremities of the ordinates by straight 

 lines, there will be represented an inscribed semi-polygon made up of parallel 

 trapezoids whose bases are each equal to unity, and their areas equal to i (0 + _F (1) ) 



+ h (F {1) + Fp)) 4- i (F {x _ l) + F {x) ); adding the contiguous half terms, it 



becomes 2 F {x) — i F {x) , or 2 u — i u. 



Between each trapezoid and the curved line above it, is a small segment ; and if 

 fix) or u' denote the area of the last or tfth segment, then Xf{x) or 2 u' will denote 

 their collective area. The whole curve being made up of the inscribed semi-polygon 

 and these segments, we have 



Cu dx = S« — i u -f Sm', 

 or J 



2m= Cu d x + i u — 2 W. 



With respect to the last term, suppose u' to be referred to a new curve, as has 

 already been clone for u, and so on ; then, 



2 u' = Cu' d x -(- 2 v! — 2 u", 



2 u" = Cu" d x -\- i u" — 2 W", 



