vol.. XXXIX.3 l-HILOSOPHICAL TRANSACTIONS. 55. 



which Mr. Hadley formerly laid before this Society, (See N° 420) it gave oc- 

 casion to consider the effects of combining several kinds of telescopes with re- 

 flecting planes, and, among others, led to the following proposition : 



That if two lenses, of equal focal length, be put together in the form of a 

 telescope, and a plane speculum be placed before one of them, so that the axis 

 of the telescope make any angle with its surface, and a ray of light, the line of 

 whose direction lies in a plane perpendicular to that surface, and passing through 

 the axis of the telescope, fall on it, and be reflected from it, so as to pass 

 through the telescope ; then the line of its last direction, after passing the 

 telescope, will make an angle with that of its first direction, before its incidence 

 on the speculum, very nearly equal to double the angle made between the axis 

 of the telescope, and the surface of the speculum. 



Lemma. — Let the line fg be the common axis of the two lenses id and ke, 

 of equal focal lengths, fig. 6, pi. 1 ; to which let the lines ad, db and be, be 

 each equal ; and let a ray of light, issuing from a point in the axis p, fall on 

 the lens id at i, and be there refracted into the line ig, cutting the axis in g, 

 and meeting the lens ke in k ; where let the ray be again refracted into the line 

 KH, cutting the aforesaid axis in h : the angles ipd and khe are very nearly 

 equal, 



Demonstr. — It is known from dioptrics, that the lines pi, I6, kh, and fg, are 

 all in the same plane ; and by the construction the lines ad, db, and be are 

 equal ; and by prop. 20 of Huygens's dioptrics, the lines fa, pd, and fg are 

 continually proportional ; consequently fa is to ad as pd to dg ; and dividing, 

 PA is to AD as PD — pa (= ad) is to DG — AD (= bg). Therefore ad is to 

 bg as FD to dg. By the same prop, the lines bg, eg, hg are also continually 

 proportional, and be (=: ad) is to bg, as eh is to eg. Hence it follows, that 

 the lines pd, dg, and eh, eg, are proportionals. But pd is to dg, as the tangent 

 of the angle igd or kge, to the tangent of the angle ipd ; and eh is to eg, as 

 the tangent of the angle kge to the tangent of the angle khe. The tangent 

 of the angle kge therefore has the same ratio to the tangents of each of the 

 angles ifd and khe, and consequently those angles are equal, a. e. d. 



In the demonstration of the above-cited proposition of Hiiygens, the thick- 

 ness of the lenses are neglected, and the distance of the points i and k, from 

 the line fg, supposed very small ; so that if either of those are too great, there 

 may arise a sensible difference between the angles ifd and khe. 



Let DP and cg, fig. 7> represent the two lenses, put together as before, 

 having their common axis in the line bl ; and bn a plane speculum, to which 

 that line is inclined in the angle ghn ; and let ab be a ray of light falling on 

 the speculum at b, as is before expressed, and let it be there reflected towards 



