104 Professor Airy on the 



V. Taking the general expression for the brightness, and 



2k , 



making tan ^ = 1 ^ tan a > we have 



ire , 2k . ttG »/ " | 4~F ~ t6 



COS a . COS — + ,„, Sin a . Sin — = V COS a + /, , My; sm " • COS — >/> 



and the general expression for the brightness is 



„/i-AY , — — . ,7re „/ , 4 A 2 . . \ „^e " 



( i P/ ' « + 2 • sm 2 — + c " ^ cos " a + /, . t>,i sm" « ) • cos — - - yf/. 



Supposing that k is not much altered by a small alteration of 6, 

 this is a maximum or minimum for a given value of <f> if 



a,re . /rTTfl* —*- . ibjjj „:„27re 



= (1 -/5-°) 5 . cos- a + 20. sin— (1 + A'p. cos 5 a + 4&' sin c a) . sin ^-T -2\J/, 



* 2ttO 

 or tan — ^ 



2/5- 1 + £*| c .cos 5 a + 4Fsnra + 1 - A°l e .cos 2 a + 2d) 



tan a x - 



1 + ft 2 1 + A* |* . cos s a + 4 ¥ sin 2 a - 1 - A 5 ] 5 . COS 2 a + 20 



Therefore ::: ^— will be greater than ^ (or ^ + w, or ^ + 2 ir, or 



2A 

 ^ + 3tt, &c for each of these satisfies the equation tan ty = ^ — r*tan«) 



by the angle u> whose tangent is the second side of the expression. 

 1. If k he nearly equal to 1 (which we suppose to be true when 6 

 is small) and a less than 90°, the expression for tan w is always positive : 

 its greatest value, when 



0=90°-?, or 180°-?, or 270°-?, or 360°- I, 



2k 2 + 2A 4 -l-A ! 



is = — ; — n tan a x 



sura 



1+A S 4 A 2 - 1 - #f.shra 



and its least, when = 45°--, or 135°--', or 225°-? or 315° -", 



Sa m 2i 3S 



2k 



is ^ 1% tan a. 



1 + Ir 



