LOCAL CLLMATOLOGT. 71 



by nights of radiation and cooling, just as in winter there are no intervals 

 of sunshine to interrupt the process of cooling ; and it is probaljle that 

 the prolonged and uninterrupted radiation may produce an intensity of 



in all cases, except when CM = EC and the curve is a circle, will be found, for the first part of 

 the distance E'C, too long for an ellipse, and in the latter part a little too short, until, of course, 

 we come to C5I', which will be the half minor axis of an ellipse, EC being the half of the major 

 axis. Still, however, the iigure is nearly enough to an ellipse for all the purposes now before us. 

 Changing somewhat the ordinary notation for an ellipse, let us, for the convenience of the 

 notation, denote the half of the minor axis, which is, of course, sin. A, by A, and the half of the 

 major axis, which is half the time between sunrise and sunset, by D, and we have the heat of the 



day denoted by '-. 



But A, in this formula, denotes the greatest altitude of the sun, or the half of the minor axis 

 of the ellipse. If we recur to the process of obtaining the formula AD''' for the area of an ellipse, 

 we have 



<fs= A(D2-x2)*(/x; 



and by consequence, 



=/l 



{D"- - x'y dx. 



Whence it appears that if we divide '~ by D, we get the average value of A ; that is, 



represents a parallelogram whose base is D, and whose altitude is equal to the quotient of this 

 fraction, divided by D, or — , which is half the sine of the sun's altitude, into 3.141, &c. 



Hence I use the average thus obtained as a correction for the absorption spoken of in the text: 

 this correction, for the perpendicular rays of the sun, is .6. Hence, by multiplying the sine of 

 the sun's average altitude for midday by .6, and by the length of the da}', 12h., for the first term 

 in the proportion, and then multiplying the sine of the sun's altitude for any other day and latitude 

 by the correction for the day and altitude obtained as above, for a third term, or the antecedent 

 of the second ratio, we have, with 80° Fahrenheit, the average temperature at the equator, a 

 formula for obtaining the temperature in degrees of Fahrenheit for the day and latitude for which 

 the third term was made. 



When we reach the polar circle, however, a modification of the formula becomes necessary. 

 Within that circle the sun does not set or reach a zero of altitude at all ; and it becomes neces- 

 sary, in order to get our average for the correction for absorption, to integrate the values of A 

 between the limits of the maximum altitude for the 

 day, that is, the altitude for noon, and the minimum 

 of the altitude for midnight ; and to the average thus 

 obtained, we must add the value of the midnight 

 altitude. The figure that denotes the sun's heat for 

 the day under these circumstances, becomes a semi- 

 ellipse resting on a parallelogram, as in the annexed figure, in which AB represents the sun's 

 altitude at its minimum, and the distance HC its altitude at midday ; the base AE or BD having 

 now become constant, an.i equal to double its length at the equator, where the days are only and 

 constantly twelve hours long. 



But in winter, December 21st, the sun never rises within the Polar circle. Hence a new expe- 

 dient must be resorted to. I have taken the angular depression of the sun at midday as the 

 minimum, and its depression at midnight as the maximum or superior limit within which to 

 integrate for the average to be used as a correction. This implies that radiation of heat from the 

 earth, or the cooling process, goes on at the same rate as the reception of heat from the sun, 

 or the warming process, other things being equal. This is proved to be the case by two con- 

 siderations : 



1. Otherwise, that is, in case either heating or cooling were in excess, the earth would be 

 growing cooler or warmer, not from season to season as it now does from summer to winter, but 



