80 



OF OPTICAL INSTRUMENTS. 



457. Theorem. In all refracting te le- 

 ficopes and microscopes^ the diameter of the 

 object glass is to that of its image formed 

 beyond the eyeglass, as the angle subtended 

 by the magnified image of the object at the 

 place of this image, is to the angle subtended 

 by the object at the object glass. 



Suppo«ng all the 

 rays to be <:ollected 

 in their foci, those 

 which proceed from the centre of the object glass will meet 

 in each of its images ; and those rays coincide iu direction 

 with the rays from different parts of the distant object 

 which cross in that centre, therefore these will also meet 

 in the same point, and with the same inclination, deter- 

 mining the angular magnitude of the ultimate image at an 

 infinite distance. But the inverse ratio of these angles is the 

 same as that of the magnitude of the object glass to its 

 image, and the successive images to each other : for the 

 images and objects are always as the distance from the cen- 

 tre, and the angles are inversely as the distances. 



458. Theorem. The field of view, or the 

 angular magnitude of the part of the object 

 of which the telescope forms an ultimate 

 image, is nearly equal, in the astronomical 

 telescope, to the angle subtended by the eye- 

 glass at the object glass ; the whole image 

 comprehending somewhat more, and its 

 brightest part somewhat less. 

 ^ B The extreme ray 



being AB, the angle 



CDB limits the whole 

 image : but no rays 

 coming to the eye- 

 glass from E fall above F, thetefore CDF limits the part 

 fully illuminated. 



Scholium. If a lens be added at the place of the first 

 image, it will ha\e no effect on the distance of any subse- 

 quent image, nor on the magnifying power, but it will en- 

 large the field of view, by throwing more rays on the ori- 

 ginal eyeglass. But, if the image fell exactly on such a 

 lens, a particle of dust attached to the lens, or any acci- 

 dental opacity, would intercept a portion of the image, since 

 all the rays belonging to each point of the object are col- 

 lected in the respective points of the image : the field glass 

 is therefore generally placed somewhat nearer to the ob- 



ject glass, both in telescopes, and in the common com- 

 pound microscopes. The best places for the vari6us lenses 

 in an eyepiece are partly determined from similar consi- 

 derations. 



459. Definition. Mr. Dollond's achro- 

 matic object glasses are composed of two or 

 more lenses, of different kinds of glass, which 

 produce equal dispersions of tlie rays of dif- 

 ferent colours, with different angular deviar 

 tions; the joint deviation being employed to 

 produce an image, while the equal disper- 

 sions are opposed to each other in such 

 a manner as to prevent a separation of 

 colours. 



460. Theorem. The focallengths of the 

 two lenses of an achromatic object glass 

 must be in the ratio of the dispersive powers 

 of the respective substances, at an equal de- 

 viation. 



If the ratios of the sines be for one glass 1 +m : 1, and 

 l+m+n: 1, and for the other l+p : I, and l+p+1 : 1 ; 



71 Q 



then the dispersive powers will be as — and — . Let the 

 '^ m p 



focal lengths of the lenses for the first kind of rays be there- 

 fore — and —, then for the second they will be ; — 



m p m^n. 



and —2— respectively (421) ; and the reciprocal of the 

 P+1 



m p ... 1 "<+" 



joint focus in the first case — , and m the second — — 



. n <j " 



_P_ii.(4a2)=- — — ; therefore the focal length will 



y " y 



be the same for rays of both kinds. 



Scholium. The chromatic aberration is also some- 

 times partially corrected in an eyepiece, by causing the 

 image formed by the least refrangible rays to subtend, at 

 the eyeglass, nearly the same angle with the image formed 

 by the most refrangible rays. 



4GI. Theorem, If the refractive den- 

 sity of a medium vary as a given power of 

 the distance from a certain central point, 

 the angular deviation of a ray of light will 

 be to the angle described round the centre 

 as the exponent of the power to unity. 



