22 Transactions of the Society. 



densities of any two media in respect to the propagation of luminous 

 waves is expressed by the squares of the refractive indices of these 

 media, f 



(3) Further, in 1874 another well-known and distinguished 

 physicist, Helmholtz, confirming certain propositions of the author 

 which were directed to the same subject, demonstrated J a similar 

 principle pertaining to the photometrical equivalent of the pencils 

 of light which travel from a luminous object through different media 

 successively. In this case the quantity of light conveyed by equal 

 solid cones is also in the ratio of the squares of the refractive 

 indices of the media. 



From these established theories of photometrical optics it is 

 seen that the quantity of light emitted from an object under a 

 given illumination is not measured by the angle of the emitted 

 cone at the radiant, nor can it be measured in any way by means 

 of the angle alone. The quantity depends under all circumstances 

 on the product of the sine of the semi-angle and the refractive 

 index of the medium in which the object emits, and is expressed 

 by the square of this product, or by the square of the " numerical " 

 aperture of the pencil. 



Thus it is shown that the general aperture-equivalent, which is 

 defined by the value of a, indicates at the same time the ]^hoto- 

 metrical equivalent of different apertures. 



The practical outcome, as regards microscopical vision, of this 

 photometrical inference is the general proposition of the illumin- 

 ating poiver of the Microscope, or the brightness of the microscopical 

 image, first propounded in the author's paper of 1873,§ and in that 

 of Professor Helmholtz quoted above : — 



If the losses of light by reflection and absorption in an optical 

 system are disregarded, the brightness of the microscopical image 

 under a given illumination of the object depends solely on the linear 

 diameter of the transmitted pencils of light at their emergence 

 from the ocular, and is always the same when this diameter is the 

 same, whatever may be the composition of the Microscope (objec- 

 tive, eye-piece, &c.) and the amplification of the image. The 

 diameter of the ultimate emergent pencil, or the cross-section of this 

 pencil, is visible within the so-called " Bamsden circle " above the 



t The supposition of cold and hot air would render the accordance of the 

 circumstances of the acoustical and the optical phenomena still more complete. 

 But as, under the point of view in consideration, the ccmsa efficiens is the density 

 of the medium, and not the velocity of propagation, the difference is immaterial. 



The above popular elucidation of the principle is not, of course, intended as 

 a scientific demonstration. It is only given for the purpose of showing that 

 common sense is by no means on the side of opposite opinions. The demon- 

 stration of Clausius, moreover, does not depend on the hypothesis of Fresnel 

 nor on any other assumption which can be a matter of dispute among physicists. 



X ," Die theoretische Grenze fiir die Leistungsfahigkeit der Mikroskope," 

 Pogg. Annalen d. Phy^ik, Jubelband 1874, p. 564. 



§ Arch. f. Mikr. Anat., ix. (1873) p. 438. 



