6430 
OPTICS. 
refra&ion of the mean rays is always attended with the 
fame difperfion, which may be deftroyed by an equal re- 
fradtion in the oppofite direction. But, if the latter re¬ 
fraction fall fhort of the former, the difperfion will not be 
'wholly corrected ; if it exceed the former, the difperfion 
will be the contrary way; that is, the order of the colours 
will be changed; and no refraCtion can finally be pro¬ 
duced by mediums of the fame kind, without colour. 
It has been already explained, (fee p. 54-9, 50.) that 
Mr. Dollond difcovered, about the year 1757, that dif¬ 
ferent fubftances have different difperfing powers; that 
the fame difperfion may be produced, or cprreCled, by a 
lefs refraCtion of the mean rays in one cafe than in ano¬ 
ther; and that thus refraCtion may, upon the whole, be 
produced without colour. 
Prop. IV. Having given the refraCting powers of two 
mediums, to find the ratio of the focal lengths of a 
convex and concave lens, formed of thefe fubftances, 
which, when united, produce images nearly free from 
colour. Fig. 4. 
Let i+hj : 1, and 1+« : 1, be the ratios of the fines 
of incidence and refraCtion of the red and violet rays out 
of air into the convex lens; 1 +p : i> and 1 + q : 1, the 
ratios of thofe fines out of air into the concave lens; F 
and/the focal lengths of the lenfes, for red rays. Then 
— : — F : the focal length of the convex lens for vio- 
m n 
let rays; therefore, the focal length of the convex lens for 
w?F . pf 
violet rays is ; in the fame manner it appears, that — 
n 9 
isthefocallengthoftheconcavelensforvioletrays. LetAC 
be the compound lens; Fq its foqal length for red rays; Ev its 
focal length for violet rays. Then/—F :/ :: F : E^; and 
Hi— — : Ev; hence E9= and Ei'= 
q n q n - J—i 
_ m l >L /— and, when Ev—Eo, the red and violet rays, 
npf—mqF ’ 1 \ 
after both refractions, are collected at q orv. In this 
Ff mpFf 
cafe, -- =-; or vpf— mqF~mpf— mpF ■, whence 
./—F vjiJ-mqY 
n—m . f—m . q—p 
n—m q—p 
. F; and F : / :: p. n—m : m . 
q — p :: —— : -—~. That is, the focal lengths are pro¬ 
wl p 
portional to the difperfing powers of the two mediums. 
If the intermediate rays be difperfed according to the fame 
law' by the two mediums, it is manifeft that the focal 
length of the compound lens, for thefe colours, will be 
Ei? or Ev ; and thus the image of a diltant objeCt will be 
formed in q or v, free from colour. 
When the rays of different colours proceed from a point 
at a finite diftance from this compound lens, after refrac¬ 
tion they will converge to, or diverge from, a common 
focus. For, the diftance of the focus of refraCted rays of 
any colour from the lens, depends upon the focal length 
of the lens, and the diftance of the focus of incident rays 
from it. And, fince the latter quantities, by the l'uppo- 
fition, are the fame for rays of all colours, the diftance of 
the focus of refradted rays from the lens is the fame; and 
thus, the image of an objeCt at any finite diftance from 
the compound lens will be free from colour. 
Example. In crown or common glafs, i1 • 54 ; and 
56. In flint glafs, i-\-p=i‘ 56 5 ; and 1 1*595 ; 
therefore. The difperfing power of common glafs: the 
. . *02 *03 
difperfing power of flint glafs :: ™ : - :: 2X565 ■ 3 
54 - '565 
X54°- To form a compound lens of thefe fubftances 
which fhall produce a real image of a diftant object, nearly 
free from colour, the convex lens null have the greater 
refracting power ; and therefore it mult be made t>f com¬ 
mon glafs, which has the lefs difperfing power. In this 
cafe, F : / :: 2 X 5 a 5 • 3 X 54 -° " 7 : 10, nearly. The 
Ff 10 F 
focal length of the compound lens, ———-. 
J F 3 
Cor. 1. If the greater refraCtion be produced by the 
concave leris, Its focal length : the focal length of the 
convex lens :: 7 : 10, nearly; and the refraCting power 
of the compound lens correfponds to that of a Angle con¬ 
cave glafs. 
Cor. 2. It is found by experience, that the extreme and 
intermediate rays are not difperfed by crown and flint 
glafs according to the fame law; therefore, though the 
red and violet rays are united by the compound lens above 
defcribed, yet the intermediate rays are not collected at 
the fame point; and, confequently, the images formed are 
not entirely free from colour. 
The difcovery of two forts of glafs, which (hall difperfe 
the extreme and intermediate rays in the fame proportion, 
is ftill a defideratum in optics. Seep. 551. and the Edin¬ 
burgh Tranf. vol. iii. 
To form the mod diftinft image, the lenfes ought to be 
fo adjufted as to colleCt the brighteft and ftrongeft colours, 
the yellow and orange. 
Cor. 3. By a method fimilar to that employed in the 
Propofition, two compound lenfes, which colleCt the ex¬ 
treme rays, but difperfe the intermediate rays in differ- 
ent proportions, might be fo adj ufted as to colleCt rays 
of three different colours, exactly ; but the advantage 
thus gained would probably not compenfate for the lot's 
of light. 
Cor. 4. Inftead of a Angle convex lens, two are fre¬ 
quently employed, one on each fide of the concave lens, 
which,-w’hen combined, have the fame focal length with 
the tingle lens for which they are fubftituted. This con- 
ItruCtion leffens the aberration arifing from the fpherical 
form of the refraCting furfaces. 
Prop. V. Having given the aperture of any lens, Angle 
or compound, and the foci to which rays of different 
colours, belonging to the fame pencil, converge; to 
find the leaft circle of aberration through which thefe 
rays pafs. 
Let QCv, fig. 5, be the axis of the lens; AB, or 2AC, 
its linear aperture ; q and v the foci of differently-coloured 
rays. Draw Av, Bv ; A qb, B<?«; join a, b, the points of 
their interfeftion, and let ah cut the axis QCv in c. 
Then, in the fimilar and equal triangles ACv, BCV, 
Av=Bv ; and the,/ AvC—t\xe£ CvB. In the fame man¬ 
ner, the/; AqC—the/^ YqC, or the/; bqc~t\\e/_aqc ■, there¬ 
fore, in the triangles aqv, bqv, the angles avq, aqv, are 
refpeitively equal to the angles bvq, bqv, and qv is com¬ 
mon to both triangles, confequently aq is equal to bq. 
Hence it follow’s, that the triangles A^B, aqb, as alfo the 
triangles A qC, bqc, are fimilar, and that ab is perpendi¬ 
cular to Qcv ; therefore ab is the diameter of the leaft 
circle of aberration, into which the rays converging to 
q and v are collected. 
Now, from the fimilar triangles A<;B, aqb, AB : ab :: 
Aq : qb ; and from the fimilar triangles ACq, bqc, Aq : 
bq :: Cq : cq ; therefore AB : ab :: C q : cq. In the fame 
manner, AB : ab :: Cv : cv ; therefore AB : ab :: Co 
-j-C<2 : cv-\-cq (Cv—Cq). 
Cor. 1. When the ratio of Cv to Cq is given, ab varies 
as AB ; and the area of the leaft circle of aberration va¬ 
ries as AB 2 . 
Cor. 2. Let parallel rays fall upon a finglelensof crown- 
glafs to compare the linear aperture of the lens; with 
the diameter of the leaft circle into which all the rays, of 
different colours, are colledted. Here i-|-m : 1 :: i'54 
: I ; and ift-w : 1 :: 1 -56 : 1, (fee the Ex. to Prop. IV.) 
and Cv : Cq :: 56 : 54; therefore, Cv-j-C<? : Cv — Cq :: 
110 : 2 :: 55 : i_; that is, AB : ab :: 55 : 1 ; or the 
diameter of the leaft aberration into which the extreme 
rays, and confequently all the intermediate rays, are col- 
ledfed, is -5% part of the linear aperture. 
On 
