76 



OF OPTICAI. IXSTRUMENTS. 



^- , , , , " . 260 130 ^ , ^. , 



Which uddeil to — '/, makes , or of the thickness. 



«i 81 81 



SciiOLiUM. In a similar manner it maybe shown, that 

 If the radii are a and ?•, and the versed sines y and z, the 



aberration will be .y-^ .( ~ -H'-\ 



qr*[a-\-lij' i^a-^0)' \ rra r 



l-\ ].;. llcncc, by proper substitutions, the aberration 



may be expressed in terms of the focal length and one of the 

 radii, and by making its fluxion vanish, the form of the lens 

 of least aberration may be determined. The aberration of a 

 system of lenses may be found in a similar manner, and 

 their proportions may be so determined that the whole aber- 

 ration may be destroyed. 



43G. Theorem. The radial image of 

 an object infinitely distant, formed by a 

 double convex lens of equal radii, is a por- 

 tion of a spherical surface of whic-li the radius 

 is to the focal length of the lens as r to r + 1 ; 

 and the peripheric image coincides at the 

 axis with a surface of which the radius is to 

 the focal length as r to Sr + 1 . 



The focal length for oblique 



rays is AB=:- 



-;but if CD 



■l(Ui—t) 



:z.r.AC, C being the centre of cur- 

 vature, AD is (rit — t)a (43i), and 

 drawing the circle DE, AE^ 



AF.AG (rn + q^na a 



— —— = — — ^-^=rq+tj = 2(n7-f9).AB. The 



AJJ (ru — Ija 'ru — t i. ' t /; 



point B will therefore always be situated in a figure similar 



to DE, that is, in a circle, and the radius of this circle will 



CD ra 



; but the focal length of the 



be 



•irq + iq 



'iiq+1q 



tens being — , the radius will be to the focal length as 

 2j 



to 1, or asr to r-J-l. Now the distance of the peri- 

 att 



pheric focus is 



-zrAB . It, and the curvature of 



'2{ru—t) 



the image may be found by adding the sagitta of any small 

 arc jf in the circle BH to the difference of AB and AB./f. 



The sagitta belonging to' BH is —^ and ultimately 



AB.(1— ((;=:AH.(2— 2()=: — , and thesum is — i — ^^—, 



the radius of curiaturc, which is to the focal length e as r 

 to 3r-f 1. 



Scholium. Hence the mean radius of curvature of the 



image at the axis may be called- 



'ir+i 



•, which istoc, when 



rzz^, or as 3 to 8. It has hitherto been usual to neglect the 

 effect of the obliquity, and to consider the focal length as 

 the radius of curvature of the image ; but it is obvious that 

 this estimation is extremely erroneous. By similar calcu- 

 lations it may be found that the radius of curvature of the 

 image of a right line, formed by a single spherical surface, 

 with a diaphragm placed at its centre, so as to exclude all 

 oblique rays, is equal to the principal focal length of the 

 surface, whatever roay be the distaace of the line. 



SECTION VIII. OF OPTICAL INSTRUMENTS. 



437. Theorem. When an angle is mea- 

 sured by means of Hadley's quadrant, and 

 the ray proceeding from one of the objects 

 is made to coincide, after two reflections, 

 with the ray coming immediately from the 

 other, the inclination of the reflecting sur- 

 faces is half the angular distance of the ob- 

 jects. ^ 



*,^ A The angle ABC=2CBD, and 



~P"~>4b BCE=:2BCF; therefore BAC=: 



X 2BCF— 2CBD=2CBD (108). 



and XX divided by this becomes 



3r-|-i 



•, therefore 



3r+i 



438. Theorem. When an imaoe of an 

 actual object is formed by anj- lens or spe- 

 culum, it is inverted if the rays become con- 

 vergent to an actual focus, but erect if they 

 diverge from a virtual focus; and the object 

 and image subtend equal angles at the cen- 

 tre of the lens ; so that a convex lens and a 

 concave mirror form an image smaller than 

 the object, when the object is at a greater dis- 

 tance than twice the principal focal length; 

 but larger, when the object is within this dis- 



