VOL. LXIX.] PHILOSOPHICAL TRANSACTIONS. 56 1 



This will be easier understood by attending to the positions of the 1st and 2d 

 images ; when a curve is of sucii a form that lines drawn from each image, and 

 meeting in any part of the curve, make equal angles with the tangent to the curve 

 at that point, it is evident, that such curve will be free from aberration. This is 

 the property of a circle when the radiant and image are in the same place; but 

 when they recede from each other, of an ellipse, of such form that the radiant 

 and image are in the two foci, till, one distance becoming infinite, the ellipse 

 changes into a parabola, and to an hyperbola when the focus is negative; that is, 

 when reflected rays diverge, and the focus is on the opposite side of the miiror. 

 These principles made me prefer Cassegrain's construction of the reflecting 

 telescope to either the Gregorian or Newtonian. In the former, errors caused 

 by one speculum are diminished by those in the other. 



From a property of the reflecting telescope, which has not been attended to, 

 that the apertures of the 2 specula are to each other very nearly in the propor- 

 tion of their focal lengths, it follows, that their aberrations will be to each other 

 in the same proportion, and these aberrations are in the same direction, if the 

 2 specula are both concave ; or in contrary directions, if one speculum is con- 

 cave, and the other convex. In the Gregorian construction, both specula being 

 concave, the aberration at the 2d image will be the sum of the aberrations of the 

 2 mirrors; but in the Cassegrain construction, one mirror being concave, and 

 the other convex, the aberration at the 2d image will be the difference between 

 their aberrations. By assuming such proportions for the foci of the specula as 

 are generally used in the reflecting telescope, which is about as I to 4, the 

 aberration in the Cassegrain construction will be to that in the Gregorian as 3 

 to 5. I have mentioned these circumstances in hopes of recommending the 

 demonstration of curves, suited to the purposes of optics, to the attention of 

 mathematicians, which would be of great use to artists. 



I shall conclude this paper with the description of a new micrometer suiteil to 

 the principle of refraction; being sensible that both principles have their peculiar 

 advantages. Though the former part of this paper proves my partiality to the 

 principle of reflection applied to micrometers, yet the very favourable opinion I 

 have of the refracting telescope made me attentively consider some means of 

 applying a micrometer to it, which might obviate the errors complained of in 

 the former part of this paper. The application of any lens or medium be- 

 tween the object-glass and its focus must inevitably destroy the distinctness of 

 the image; I therefore have employed, for the micrometer-glass, one of the 

 eye-glasses requisite in the common construction of the telescope; but if it 

 should be found necessary to apply an additional eye-glass, for the conveniency 

 of enlarging the scale, I am able by it to correct both the colours and spherical 

 aberration of the first eye-glass. 



VOL. XIV. 4 C 



