422 H. 8. Uhler — Deviation Produced oy Prisms. 



do 

 the curve OY at the origin. At H, (0 = ^tt), — - = 3 and so 



tlie tangent to the curve makes an angle with the positive direc- 

 tion of the axis of abscissas equal to tan" 1 3 =71° 33' 54" '. When 



do 



=-- cos" 1 i (1 - 4/ 3) = 111° 28' 15", -^ = 0. It will be seen 



later that this stationary value pertains to a maximum of c. 

 The curve OY passes through the point Q, which was defined 

 as the intersection of the straight lines OA and PH, since the 

 coordinates = ir and c = \ir satisfy equation (50). At this 



dc 

 point — = -1-5, and tan" 1 (- 1*5) = 123° 41' 24". When 



the acute angle at H, the obtuse angle at Q, and the single sta- 

 tionary value of c between H and Q are taken into account, it 

 becomes evident that the arc OIH is the only portion of the 

 curve OY which lies within the triangle OQP, that is, within 

 the domain of physical significance. .Now 



d?c _ (2 - 2 cos - cos 2 - 4 cos 3 + 2 cos 4 0) sin /? 



dp 2 ~ (1 - 2 cos fl + 2 cos 3 Pf ~ 9 



d 2 c 

 hence for cos = - \ W 3 - 1), — = - ^ (5 + |/3)*/io8 



so that the existence of the maximum mentioned above is estab- 

 lished. At this point, c has the impossible value given by 



tan" 1 [ - 4-V12 (a/ 3 + 1) ]== 139° 43' 9*. The expression for 



d*c 



— vanishes for /3 = 0, j3 = cos" 1 (0-5709797), and = tt, 



thereby showing that there may be one point of inflection 

 at the origin, another at = 55 d 10 r 53" with c = 74° 48 r 8" 

 or n = 1-03624, and a third at Q. That these points are 



truly inflectional is easily seen without proving that ——^ does 



not vanish. The second point of inflection lies at the center 

 of the circle marked "I" in fig. 8. At this point the locus 

 OY makes an angle of 72° 49 x 7" with the axis of abscissas. 

 The equation 



tan 0=2 tan c 



has been shown to be the necessary and sufficient condition that 

 the discrete minima of deviation at grazing incidence or emer- 

 gence shall coincide with their respective maximum, which 

 occurs in a principal plane. [See (48).] Although the curve 

 representing the above equation has not been plotted in figure 

 8, nevertheless, its course can be readily followed by the aid of 

 the following data. In the first place 



