20 Transactions of the Society. 



than greater quantity of light — if there were no specific difference 

 of the rays which can be utilized by different apertures — the whole 

 question would be only of somewhat subordinate interest. More 

 light from an object can always be gained when more is thrown 

 ui^on the object by means of a brighter source of illumination. 



Inasmuch, however, as the determination of the photometrical 

 equivalents of different apertures affords an additional illustration 

 of numerical aperture, it will be useful — for the sake of complete- 

 ness merely — to add a brief outline of the photometrical principles 

 relating to the matter, though nothing can be said here which has 

 not been established long ago. 



(1) In the last century Bouguerf and Lambert | established the 

 important fact that with any surface of uniform radiation (so called) 

 the intensity of the emitted rays is not the same in all directions. 

 The poicer of emission and the intensity of the rays (i. e. the 

 quantity of light emanating from a given surface-element within a 

 cone of a given narrow angle) varies in the proportion of the 

 cosine of the angle of obliquity under which the ray is emitted. 

 This proposition is nothing more than the expression of the simple 

 fact, that a surface of uniform radiation shows the same visual 

 brightness in all directions ; and that such a surface, if curved (for 

 instance the sun, or the porcelain shade of a lamp, &c.), is always 

 seen projected as a surface of uniform hriglitness. 



This theorem, which at a later period was confirmed by Fourier, 

 Melloni, and other physicists, shows at once that the quantities 

 of Hght emitted from one and the same object within solid cones 

 of different angles are not in the ratio of these solid cones, but 

 in the ratio of the squares of the sines of their semi-angles. Thus 

 the whole emitted light (embraced by the entire hemisphere of 

 radiation), and that portion which is emitted within a cone of 30° 

 around the perpendicular (or 60° angle) are not, as is so con- 

 stantly assumed, in the ratio of 7 • 46 : 1 (as the soHd cones in fact 

 are), but in that of 4 : 1 only. 



As in one and the same medium the number of rays conveyed 

 by a pencil and the photometrical quantity of light are propor- 

 tional, this old-established Lambert theorem is sufficient of itself 

 for overthrowing the very basis of the angular expression of aper- 

 ture, and for proving that even when we are dealing with one 

 and the same medium only, the angle is not the sufficient ex- 

 pression, but that it is the sine of the semi-angle which must be 

 taken. 



(2) In more modern times, but still seventeen years ago, a 

 distinguished physicist, well known in Englard, E. Clausius, 

 estabhshed by a famous research " On the Concentration of 



t ' Traite d'Optique sur la Gradation de la Lumiere,' 1760. 

 : ' Photometria,' 1760. 



