416 H. 8. Uhler — Deviation Produced by Prisms. 



c , Z> 15 and D 2 have already been given in Table I. The 

 points A and H indicate the intersections of the carve for 

 Z> = 22° 13' 2" with the left boundary locus. The abscissae of 

 these points are 7^55° 48' 52" and 7 X =84° 11' 8" respectively. 

 The corresponding ordinates, which also pertain to the inter- 

 sections with the right boundary locus, are ?; 1 = 77 30' 5" and 

 t 7] = 78° 35' 59", in the order named. This locus of constant 

 deviation has been plotted beyond the right boundary. The 

 equation of the locus of the algebraic maxima and minima of rj^ 

 on the curves of constant deviation, has been designated as (44). 

 This locus is marked "\=0" in fig. 7. It crosses the 7,-axis 

 at the abscissa 7, = 15° 5 7 53". When the numerical value of 

 the deviation is given, the coordinates of the points of inter- 

 section of the constant deviation locus with the curve \ = can 

 be obtained from equations (30) and (31). As soon as ??/ 

 has been evaluated, rj 1 can be found at once by using (4). Then 

 7 X may be obtained from (5) since, for \=0, 7 1 '=J/8. In gen- 

 eral, at the intersections of the left boundary locus with the 

 axis of rj 1 



dy l 



— — = ± sec c cot f$ Vsin (c + fi) sin (c — (3) , 



which requires /3 < c, of course. For the special case of fig. 7 

 the acute angle obtained from the above derivative equals 

 64° 38' 8". The deviation at these points of intersection may 

 be obtained from (42), and it equals 26^ 59' 43". At the ori- 

 gin of coordinates of the y 1 tj 1 diagram the deviation is given by 

 (43). For /3 = 20° and n = l-5 the deviation equals 10° 51' 57". 

 Returning to the loci of constant deviation, and inspecting 

 fig. 7, it is seen that the curves which pertain to the smaller 

 values of D only have algebraic maxima and minima of y t for 

 ??i = 0, that is, in a principal plane. On the contrary, for rela- 

 tively large deviations the curves have additional maxima 

 and minima of 7^ More specifically, each complete locus 

 of large constant deviation has six stationary values of 7, , 

 namely, one maximum and one minimum on the 7,-axis, two 

 minima for numerically equal and finite ordinates, and two 

 maxima for still greater (arithmetical) values of 77^ Stated in 

 a slightly different manner, it is evident that a straight line 

 parallel to the axis of ordinates, and lying wholly within one 

 quadrant, may intersect a curve of constant deviation in one 

 real point, or in two separate points, or in two coincident points 

 and one other point, or in three discrete points, etc. The 

 quantitative investigation of these points of intersection is 

 equivalent to the discussion of the roots of the equation ob- 

 tained by expressing r) x as an explicit function of y t and D. 

 Rationalization of formula (35) leads to the following cubic in 

 cos 2 77 l5 namely, 



