A^■AILA]5LK SUXSIIIXE. 



229 



AVe may write 



ZS = 90°-/;, where h = altilude of the mean sun, 

 ZP = 90° -A, where A = S. hititiule of the phice, 

 PS — 90° + ^, where t> = angadar distance of the 



mean sun north of the 

 celestial equator, 

 and the formula reduces to 



sin /f = - sin A sin (^ + cos A cos o cos P. (1) 



In this formula sin A and cos A are constants for the given 

 place, sin S varies throughout the year exactly in proportion 

 to the sine of the angular distance of the mean sun from 

 the equinox of the 21st of March, and therefore completes 

 its fluctuation in a year. 

 In Fig. 2 we have 



sin SM = sin A sinJ/S". 

 I.e., sin 8 = sin 23° 27'. sin (>i x 30°), 



where it = number of months of equal length that have, 

 elapsed since the 21st of March. Hence 



sin A sin S = sin A sin 23° 27'. sin (>i x 30°), (2) 



and the value of this term can be represented by a curve 

 of sines throughout the year. In the second term on the 

 right-hand side of equation (1), cos A is, of course, constant, 

 and cos 8 is nearly so, its value varying only from unity at 

 the autumn and spring equinoxes to 0'917 in June and 

 ])ecember. Hence the value of the second term during any 

 day can be represented by a curve of cosines, the amplitude 

 of the curve changing slightly every three months. 



4. The left-hand side of the diagram, Fig. 3, gives the 

 values of the term sin A sin 8 througliout the year for the 

 latitude of Bloemfontein, while the right-hand side gives 

 that of cos A cos 8 cos F throughout an average day, the 

 continuous curve corresponding to a day at the equinoxes 

 and the broken one to the solstices. If, now, from the point 

 on the first curve corresponding to any given day a horizontal 

 line be drawn across the second cui-A^e, this line will represent 

 the horizon for the day. It will cut the second curve at points 

 corresponding to sunrise and sunset, and the hours of the 

 rise and setting of the mean sun can be read off at once in 

 tnie local time. The height of the second curve above the 

 horizontal line is the sine of the sun's altitude, and is there- 

 fore proportional to the intensity of available sunshine, and 

 the whole area above the horizontal line is the total available 

 sunshine for the day, save for the correction due to the 

 deviation of the " mean " from the true sun. 



For example, in the diagram which is drawn for latitude 

 29° ry S., the point E on the first curve corresponds io 2nd .July ; 

 AB is drawn through 7C parallel to the line of abscissae; 

 the dotted area ACBA is the total available sunshine on 

 that day in latitude 29° 5' S. The "mean" sun rises about 

 (;h .^4m"f,j-,i_^ local time, and sets about 5^^ Q^ p.m. Further, 

 if from the point Z>, corresponding to 9 a.m. local time, a 

 line of unit length DE be drawn to cut the horizontal line, 



