Conditions favorable to Glaciaiion. Ill 



the plus sign giving the value for summer and the minus sign 

 that for winter. 



At the equator, or .for X = 0, (3) reduces to the term con- 

 taining E J (z). At the polar circle Z ceases to be doubly peri- 

 odic ; for then d{d) degenerates into cos # and cos X = sin e so 

 that 



dX _ cos £ 



d$ 1 — sin 2 € sin 2 S - ' 



These relations also reduce (1) to an integrable form and give 

 for this one latitude 



Z = 2 



3r . „ ( . , , ,7t/2 — & . 7t ■ n } 



sin 2 e -I tan s + cos £ In cot — ( ± — sin 2 y 



( 2 2) 



h 



Within the polar circle the limits of integration change, 

 because for a part of the time there is no illumination. 

 Furthermore sin e/cos X >1, and a transformation is needful to 

 reduce the functions to standard forms. To effect this let 



sin £ sin S - . -, cos A. ^ 



= sin cp, and = /(Cl. 



cos A sin £ 



Then 



H = 1 / /I; A (S) = cos cp ; cos 3" = ^J 1 — /r sin 2 cp = A (cp) ; 



A{cp) ' 



The superior limit of integration is given by 

 A (3) = or cp = 7t J 2. 



By these substitutions (2) becomes* 



d W H 7Cv Hot . n , \ sin £ A , , , cos £ 1 



— dz = sin 2 A \ Aim) + . 



dX h h ( sin A tan £ sin A A(cp) 



- sin A ^ 1 „ + JL (sin £ J(<p). X) I dcp, 



tan£ (1— cos 2 Asm 2 ^) z/ (cp) dcp ^' M 



and this when integrated from zero to tv is 



Q H« 2 . jsinfi—j, . cosfF^yu) sin A cos £ 



h i sin A ; ' tan £ sin A tan £ 



7t . 



± - sin £ 



Knowing the zonal receipt of solar energy for any and every 

 latitude, the heat per unit area is within reach ; for the length 

 of a parallel of latitude is 2 na cos X, and if u is the sunshine 

 per unit area for the interequinoctial period 



* In testing the truth of this formula it is convenient to remember that 

 cos 2 <j> _ A (cj>) - (1 - /i*) I \ (ft ). 

 A(ftJ fjfl 



Am. Jour. Sol— Third Series, Vol. XLVIII, No. 284.— August, 1894. 



