336 METEOROLOGY. 



The sum of the daily intensities for a moiitli, or monthly intensities, 

 might be found in the same manner. But, instead of this slow pro- 

 cess, we 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. 2, and in the Smithsonian memoir. By this general summation 

 the following remarkable principle was rigorously demonstrated : 



The sun's annual intensity upon any latitude of the earth is propor- 

 tional to the sum of tioo elliptic circumferences of the first and the second 

 order, diminished by an elliptic circumference of the third order. 



On the equator, the sun's annual intensity reduces to the circumference 

 of an ellipse, tvhose ratio of eccentricity is equal to the sine of the ohli- 

 quity 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 duration of constant night 

 at mid-winter. The second interval at mid-summer is constant day ; 

 the third and fourth are intermediate sjjring and autumnal intervals, 

 when the sun rises and sets in every twenty-four hours. 



With respect to the unit of measure for annual intensity, the mean 

 tropical year contains 365.24 days; let this represent the annual 

 number of vertical rays impinging on the equator ; that is, let the 

 sun's intensity during a mean equatorial day he taken as the thermal 

 day,. and let the values for all the latitudes be converted in that pro- 

 portion.* Also denoting the annual intensity on the equator by 12, 

 the mean equatorial month may be used as another thermal unit. 

 And taking the annual intensity on the equator as 81.5 units, with 

 reference to Brewster's formula, the intensity on other latitudes may 

 be expressed in that proportion. It may here be observed that the 

 diurnal value of the last section will be changed to this scale by in- 

 creasing them in the ratio of 1 to 1.049. 



With the aid of Legendre's elliptical tables the computation of 

 annual intensities is entirely practicable. The results converted into 

 units, with differences for every five degrees of latitude, have been 

 carefully verified and tabulated as follows : 



* The three species of circumferences, each representing four equal and similar, quad- 

 rants, are discussed at great length by Legeudre in his TraM des Fondions ElUpiiques. 

 Let L denote the latitude of the place, and developing in thermal days for the 

 torrid and temperate zones, we find for any year in the present century: 



A 1 • X -4. OAF, ooo T I 15.748 , 0.1628 , 0.0066 , 



Annual mtensityz^ 349.322 cos. L A -\- — \- 



•' ' oos.L^^ COS." L ^^ COS.' L "^ 



