22 Transactions of the Society. 



or greater distance beyond the centre, according to the focal length 

 of the objective in question ; and the distance may be very con- 

 siderable with a low-power objective. The excentricity of the apex 

 will not permit a correct evaluation of the angle in every case ; 

 but if, as in Professor Smith's apparatus, the rays have to pass to 

 air from this excentrical point by a spherical surface, their 

 divergence of course must be considerably increased by the refrac- 

 tion at this surface. With the same apparatus Professor Smith 

 will obtain the true angular aperture of a low-power objective, 

 either by focussing the objective with an eye-piece on his 4-inch 

 tube, or observing the telescopic image by an eye-hole at a 

 considerable distance from the objective. 



In applying the method of telescopic observation by means of 

 an auxiliary Microscope (acting as a terrestrial eye-piece) for 

 enabling more accurate observation in the case of object-glasses of 

 short focal length, any 2-inch to 4-inch lens may be used. But if 

 the objective has been focussed previously to the centre of the 

 divided circle by an eye-piece on the 10-inch tube, and afterwards 

 an ordinary 2-inch or 4-inch lens is put to the draw-tube without 

 any further precautions, the result will be incorrect. The clear 

 aperture of the auxiliary lens representing now the critical 

 diaphragm s|)oken of above, and this diaphragm being at the 

 distance of a few inches only from the objective, the conjugate 

 focus to it, i. e. the apex of the aperture angle will be at a con- 

 siderable distance from the ordinary focus of the objective, and from 

 the centre of the measuring apparatus, at least in the case of an 

 objective of low or moderate power. Moreover, as a 2-inch or 

 4-inch lens generally has a clear aperture of • 5 inch or more, 

 the area of entrance of the image-forming pencils to the objective 

 (which is in this case the image of the auxiliary lens projected in 

 front of the objective) will be a considerable part of the focal 

 length of the objective, much too large for admitting a good 

 definition of objects situated so far from the aplanatic focus. For 

 these reasons a correct application of the method requires a 

 suitable stop behind the auxiliary lens, as expressly stated in the 

 original description of the apertometer. The position and the 

 diameter of this stop must be regulated in such a way, that its 

 clear area in respect to the auxiliary lens corresponds to, or is 

 optically conjugate with, a small central part of the field of the eye- 

 piece applied in the previous focussing of the objective. According 

 to this condition, the correct place of this stop may be easily 

 calculated from the focal length of the auxiliary lens and the 

 length of tube, for which apertures have to be measured. This 

 place will be always somewhat within the posterior principal focus 

 of the auxiliary lens. 



Pieducing the clear opening of the diaphragm, in the case of an 



