INTENSITY OP SUN'S HEAT AND LIGHT. 29 



4 C COS 2 co ( 1 



2 W = — : \ F' + tan 2 co . E' — sin 2 L.W } . (28.) 



n sin coV\ — e' (. J 



Here the eccentricity or common modulus of the three elliptic integrals is — — , 



sin co 



being the reciprocal of the modulus in (26); but the intensity is still denoted by 



three entire elliptic circumferences. 



At the Poles, where L is 90°, and cos L is 0, the expression of annual intensity 



2 c n sin a 



reduces to — — : . (29.) 



nV\ — e~ v ' 



The three species of elliptic functions are known to represent four equal and simi- 

 lar quadrants, as in the ellipse. Extensive tables have been published by Legendre, 

 of the numeric values of E and F; and in his Traite des Fonctions Elliptiques, Vol. 



I. p. 141, the value of the quadrant II' is given in terms of E and F. Thus, if x 



denote an axillary arc, such that c 2 sin 2 x = n, a negative quantity, less than 1 ; then, 



old) ,_ tan x 



] 7i ~ o , /T = f=^=> W = F' +—===== (F'E, T) — E'F, x) y 



(1+n sin' (p)</l — <r sin' <p V 1 — c sm" x K ' 



Comparing with (24), sin' x = cos 2 L; IT = F' + cot L (F> E M — E' F laA ); 



\ COS CO 



substituting this value into (26), we find for the Annual Intensity in the Torrid and 

 Temperate Zones, 



4c \ 



/yi T^ ^ E ' ( sm Lcosa - F (»(p-l) + cos L) + ( , go ,. 

 F'. sin L {sin 2 co tan L — cosco.E (90 o _ i} )] ) 

 We have heretofore denoted whole circumferences by the double accent, thus 

 E" = 4 E'. In (30) E', and F' denote quadrants; F {W _ L) and E {S0O _ L) elliptic 



functions whose amplitude in Legendre's system is 90° — L; is the common 



modulus ; L denoting the latitude of the place, and co the obliquity of the ecliptic. 

 The interpolation of Legendre's tables for second differences, is described in Vol. 



II. p. 202 of the Fonctions Elliptiques. From the Polar Circle to the Pole, 



sin co 



k-St 



will denote the common modulus, x becomes co, and </ 1 — c' 2 sin 2 x = sin L; hence 

 by (28), the Annual Intensity in the Frigid Zone is, 



4-c ~\ 



2 u' = ===== \_E' (sin L cos co. Ft*,) + sin co) + I ,oi n 



n V \ — e" r- \ 01 -) ' 



F'. cos co cosL(cot co cos L — tan L . -E («„))] J 



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 imping- 

 ing on the equator ; that is, let the sun's intensity during a mean Equatorial day be 

 taken as the thermal Unit, and let the values for all the latitudes be converted in 

 that proportion. 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. 



