Conditions favorable to Glaciation. 109 



the ellipse into which the parallel of X is projected, x and z 

 the coordinates of the point t at which the ellipse meets the 

 great circle in the projection, o the center of the ellipse, e the 

 center of the small circle which is projected into the ellipse. 

 Then it is easily seen from the diagram that 



a ■= a cos A ; b = a cos A sin S ; c = e cos d =z a sin A cos d : 



. a cos A / <,;,,'■* 



cosd r rW 



g _ a sin A _ ^ _ _____ „ cos A / sin_f 



cos 6" ' " * ' ' cos £ r cog 2 \ 



The area of the circular segment r, z, t is a? sin c_1] — — xz and 



that of the elliptical segment r, s, t is ab sin c_1] xz+xc. The 



difference is the area sought or 



W„ . r_n X 7 . r_ii 3^ 



— : at sin- J ab sin — — ccc ' 



a a 



The arc whose sine is x/a is the sun's semidiurnal arc. Its 

 value in the cold season may be called for brevity X. In the 

 warm season at the same point its value will be 7.-X. If & is 

 the earth's longitude in its orbit and if e is the apparent ob- 

 liquity of the ecliptic it is easy to see and perfectly well known 

 that 



sin d = sin t sin S - 



and this value is to be substituted in that of W. It is also 

 convenient to employ the abbreviation 



(*) = / l 



sin s • 2 cv 

 sin i>. 



cos 2 A 



so that the semidiurnal arc is 



zJ(S) 



X = arc sin 



Vl— sin 2 _ sin 2 5 

 The value of W may now be written 



W = or { sin [-1] (cos A sin X) — cos 2 A sin € sin S - . X 

 — sin A cos A __(3)}, 



and when X increases W decreases ; so that the rate of increase 

 of W, or the area on the circle of illumination occupied by 

 the projection of a zone of unit width in the cold season, is 

 represented by 



_ dW_ a . sin2 x | co t A__(3)-sinesin3. X}. 

 d A 



