520 



Prof. J. D. Everett, 



[Jan. 22, 



around it. This property is of great assistance in locating the ends of 

 the focal lines. Physically interpreted, it indicates close aggregation 

 of rays, and consequent increase of luminous intensity, at the ends 

 of the two focal lines. 



31. To find at what distant z from the plane of the annulus the 

 values of x and y for the ray at 0° are stationary for changes of 6. 



We have in general 



V 



m 



sin 6 -\ — - z ; y = cos + 



Hence, for constant z, 



dx n z I , dV 7 , dn'\ 



d0 = ™ + n>i[ n M - l M )> 



dy . . z / , dm' ,dn \ 

 dd n 2 \ dO dO I 



Differentiating the harmonic expressions (§ 20) for /' m' n' } we see 

 that the expressions for dm'/dO and dn'/dd contain only sines, and 

 consequently vanish at 0°. Hence dyjdO vanishes identically at 0°, 



and dx/dO reduces to cos 6 + ~, , which is to vanish. 



n au 



mn • • 560 



Ihis gives 1 - yggg z = 0, z = 13'57, agreeing fairly with the direct 



determination z = 13'43. 



In like manner, for the ray at 180°, the conditions reduce to 

 i 688 



~ 6308 = ^ = 9*1 / ; to compare with the direct determina- 

 tion 9-04. 



These two cusps are shown (on ten times the scale of Plate 9) 

 at the top of Plate 10. 



32. In the case of the cusps at the ends of the primary line, which 

 are known to be on the rays from ±79° 10', direct application of 

 formula? (3) to the values 78°, 79°, 80°, shows that, for x and y to 

 have the same values approximately for all three, z must be approxi- 

 mately 6. More exactly, the equations of the ray at 79° 10' are — 



x = 0-98218 -0-083713, y = 0-18795 + 0-97178; ; 



and those of the 79° ray, 



x = 0*98163 -0-08359.2, y = 0-19081 + 0-9713U 



The value z = 6 05 gives, for both rays, 



x = 0-476, y = 6-0672. 



Transforming from y and z to and f, we have 



q = 0-012, f = 8-568. 



