ZOOLOGY AND BOTANY, .MICROSCOPY, ETC. 



99 



reflexion at P 2 . is the common centre of the circles, a = o A, and 

 b - o B ; the inner and outer radii being r x R 2 respectively. By pro- 

 perties of a triangle, it follows that 



whence 



a sin a = ;• sin = R sin y = b sin /3 ; 



sin a 



« 



sin (B' 



If the origin be removed to infinity, then for a ray of incidence-height h, 



J " sin /?' 



where / is the focal distance of the zone in question. The condition 



for aplanatism is 



sin a 

 . — = a constant : 



Sill yS 



or, if A be at infinity, 



. ' = a constant, 

 sin p 



Moreover, it can be shown that a system of two concentric reflecting 

 circles is free from coma. Owing to the fact that one mirror reflects 

 convexly and the other concavely, the catacaustics will have opposite 



Pig. 8. 



sense, and will tend to neutralize one another. With medium aperture 

 the zone-aberration-values will change but slightly, and the system will 

 have its greatest advantage under such a condition. There will, again, 

 be many pairs of rays whose aplanatism will be perfect, and by suitable 

 choice of radii this property can be made to apply to any desired range 

 of aperture. The author also discusses the conditions under which the 

 brightness will be a maximum, and shows that theoretical values can 

 almost be attained in practice. 



Fig. 9 shows how the principle of the concentric condenser can be 

 actually realized for an aperture range of 0*97 to 1*35. 



It will be noticed that in Jentsch's design the two curved surfaces 

 are worked out of one and the same piece of glass, while in Ignatowsky's 

 and in Siedentopf's patterns two pieces are required, thus introducing 

 centring errors which are here absent. Jentsch's upper glass is a square 



H 2 



