ZOOLOGY AND BOTANY, .MICROSCOPY, ETC. 515 



Ultra -microscopic Image.* — II . Siedentopf, in this treatise, goes 

 very fully into the conditions underlying the formation of the ultra- 

 microscopic image. The scope of his article will be gathered from 

 the follo\ving list of its sub-divisions : 1. Limits of applicability of the 

 ultra-microscope, 2. Dark-ground illumination by means of central 

 diaphragms in the objective, and their disadvantages. 3. Dark-ground 

 illumination by unilateral oblique light and the consequent azimuth- 

 errors. 4. Dark-ground illumination by central diaphragms in the con- 

 denser, the cardioid condenser. 5. Variation of the diffraction disks by 

 diaphragming the rear focal plane, by wrong use of the Microscope 

 objective, and by obliquely placed cover-glasses. 6. Polarization of the 

 light by diffraction of ultra-microns, and the corresponding appearances 

 in the rear-plane of the Microscope objective. 7. Ultra-microscopic 

 photography by rapid exposures. One of the plates illustrating the 

 paper shows the Brownian movement of fine silver particles of about 

 20 n in a colloidal solution, taken with an exposure of -^ of a second. 

 The other plate illustrates (1) the variation of the diffraction-disks by 

 diaphragming the rear focal plane ; and (2) gives a valuable criterion 

 for cover-glass correction with dark-ground illumination. A bibliography 

 of 30 references is appended. 



Aplanatic or Cardioid Condenser for Dark-ground Illumination.f 



In section 4 of the article described in the fore^oincr abstract the author 

 discusses the cardioid curve, and shows that its properties may be advan- 

 tageously applied to the construction of a new form of dark-ground 

 condenser. The cardioid, which derives its name from its heart-shape, 

 is the curve traced out by a point on the circumference of a circle rolling 

 on another circle of equal radius. It is the caustic curve assumed by 

 rays reflected from a circular surface when the luminous origin is a point 

 on the circumference of the surface. The polar equation to the curve 

 is R = r (1 + cos u), when r is a constant. If the cardioid be combined 

 with a circle of radius r, the circle being placed in a certain manner, 

 then all rays incident parallel to the axis will, after reflections at the 

 convex surface of the circle and at the concave surface of the cardioid, 

 pass through the cusp of the cardioid. It follows, therefore, that the 

 resulting ray -combination will be free from aberration. This will be 

 understood from fig. 71, where Z Z is the common axis of cardioid and 

 circle. P is a ray parallel to the axis, reflected at P and P'. M is the 

 centre of the circle. 



If the ray is to converge to C, the final branch of the ray will be 

 parallel to the radius P M, and the angle P'CM will be equal to the 

 angle of incidence u. Xow it is a property of the cardioid that the 

 angle i at P', between the radius vector and normal, is equal to half the 

 angle rotated by the radius vector, i.e., i = \ P' C M. 



This is proved thus (P' C M being called u) : — 



^R 7 sin u u 



tan i = - -=—. a u = = tan - 



It 1 + cos a 2 



Therefore . u 



l = 2 



* Zeitschr. wiss. Mikrosk.,xxvi. (1910)pp. 391-410 (2 pis. and 6 figs.), t Loc. cit. 



2 M 2 



