64 



THE AMERICAN MONTHLY 



[April, 



the law of conjugate foci this fact 

 may be stated thus: the angle of aper- 

 ture is the angle at the principal focus 

 for the divergent rays from the im- 

 age / d' , incident upon the lens / /' 

 and refracted to o. I do not doubt 

 that this is a proper mode of deter- 

 mining that angle in all objectives 

 which form the telescopic image be- 

 hind the back-lens. It shows the 

 varying size of the field under vary- 

 ing lengths of tube, and marks the 

 only significant phenomena which can 

 properly be called the aperture of the 

 telescopic instrument. It does this 

 by the examination of the image it- 

 self, so that there is no question as to 

 what are image-forming rays. 



It seems to me that the same meth- 

 od should be applied to the instru- 

 ment used as a microscope. But it 

 then produces different results. 



The relation of the image to the 

 object is the same in the microscope 

 as in the telescope. They stand to 

 each other in exactly the relation of 

 the " two limited areas in plains per- 

 pendicular to the axis," at conjugate 

 points upon it, as stated in the propo- 

 sition already quoted. In both cases 

 the examination of the image may be 

 made by a properly constructed eye- 

 piece, or by the naked eye; for though 

 the microscope, as constructed, is not 

 well adapted to the latter mode of 

 viewing the image, there is no serious 

 difficulty in adapting it to that meth- 

 od. In the one case we have a com- 

 paratively small image of a large, dis- 

 tant object; in the other we have a 

 large image of a small, but near ob- 

 ject. In one case the image is com- 

 paratively near the back of the lens, 

 in the other it is distant from it. It 

 is evident that these variations of 

 condition do not aifect any principle, 

 and that the rays which determine 

 the angle of the field in the one case, 

 must be exactly homologous to those 

 which determine it in the other. 



As we have already seen in the case 

 of the telescopic image, the rays which 

 determine the aperture are those pro- 

 ceeding from the extremities of the 



image / d' and by refraction by the 

 lens are made to cross at o and reach 

 the extremities of the object at c d. 

 Trace the homologous rays from a' b' 

 fig. 24, the microscopic image, and we 

 find that they pass the lens and cross 

 at 0" reaching the extremities of the 

 , object a b. If the former is the mea- 

 sure of the aperture of the telescope, 

 the latter is that of the microscope. In 

 both cases they are the limitation of 

 the field by the rays crossing at the 

 principal focus for rays having the di- 

 vergence or convergence of / /' and d' 

 I' and d' I in the one case, and of a*^ 

 I' and b' I in the other. 



The crossing of the telescopic rays 

 at in the centre of the microscopic 

 field, is due to the fact that 0' , 

 taken as the point of view, is in the 

 centre of the microscopic image 

 a' b', and as « <5 is the conjugate 

 of a' b', must also be that of 0' . 

 It thus affords an easy way of mea- 

 suring the telescopic angle of field, 

 or of aperture; but that this is in 

 any proper sense the measure of 

 the microscopic angle is questionable. 

 To apply the method to the micro- 

 scope, the natural way would seem to 

 be to place a minute hole in a stop 

 at o" , and, by some proper device, to 

 bring this point over the axis of motion 

 or the optical centre of the apparatus. 

 The distance from to o" would then 

 be the distance of the object from the 

 principal focus for rays of the con- 

 vergence a' I' and b' /, according to 

 the second proposition laid down at 

 the beginning; and the angle thus 

 measured is the angle made at the 

 principal focus by rays of the given 

 convergence on the other side of the 

 lens. If there be any significant angle 

 in optics, it would seem to be this, 

 and it would seem also to be the 

 true angle of aperture of the lens, 

 if the term has any definite mean- 

 ing. 



The results of the practical experi- 

 ments with such a pin-hole diaphragm 

 will be given presently; but I must 

 call attention to the theoretical dif- 

 ference between the telescopic and 



