OPTICS. 
571 
tion of the focal diftances, thofe lines are to be accounted 
pofitive which lie from their beginnings, that is, from the 
vertex, or the centre, or the radiant point, in the direc¬ 
tion of the incident rays. Thus, when rays diverge 
from R on the convex furface of a medium, VR is ac¬ 
counted negative and VC pofitive. If the light paflesout 
of air into glafs, m is greater than n; but, it it pali'es out 
of glafs into air, m is lefs than n. If, therefore, parallel 
rays fall on the convex furface of glafs out of air, in 
■which cafe m : n— 3 : 2 very nearly, we have for the 
principal focal diftance_i_VC, or+3VC. But, if it pafs 
2 3 
out of glafs into the convex furface of air, we have VO= 
2 ypj or —2VC; that is, the focus O will be in the 
g,-3 
fame fide of the furface with the incident light. In like 
manner, we lhall have for thefe two cafes CO= + 2VC 
and—3VC. 
Cor. 4. By conftrudion we have RK : RP—m n 
by fimilarity of triangles PF : RK=CF : CR 
therefore PF s PReramCF : nCR. 
and wPR X CF=«CR X PF 
therefore wiPR : wCR—PF : CF 
and »iPR—nCR : jhPR=PF-—CF : PF 
ultimately mVR—«CR : wVR=VC : VF 
This is a very general optical theorem, and affords an eafy 
method' for computing the focal diftance of refraded rays. 
For this purpofe let VR, the diftance of the radiant poknt, 
be expreffed by the fymbol r, the diftance of the focus of 
refracted rays by the fymbol f, and the radius of the fphe¬ 
lical furface by a; we have mr—nr—a : mr—a 1 f, and 
f— mar } __ -. In its application due 
mr — nr — a m — nr + nd 
attention rnuft be paid to the qualities of r and a, whether 
they be pofitive or negative, according to the conditions 
of laft corollary. 
Cor. 5. If Q, fig. 14, be the focus of parallel rays com¬ 
ing from the oppofite lide, we {hall have RQ : QC = 
RV : VF. For draw Cq parallel to PF, cutting RP in q; 
then R q : t/C=RP : PF. Now' q is the focus of the pa¬ 
rallel rays FP, C q. And, when the point P ultimately 
coincides with the point V, q muft coincide with Q, and 
we have RQ : QC=RV : VF. This is the mod general 
optical theorem, and is equally applicable to lenfes, or 
even to a combination of them, as to fimple furfaces. It 
isalfo applicable to reflections, with this difference, that 
Q is to be affumed the focus of parallel rays coming the 
fame way with the incident rays. It affords us the moft 
compendious methods of computing fymbolicaliy and 
arithmetically the focal diftances in all cafes. 
Cor. 6. We have alio Rq : RP=RV : RF, and ulti¬ 
mately for central rays RQ : RV=RV : RF, and RF= 
^ 2 This propofition is true in lenfes and mirrors, but 
RQ 
not in Angle refrading furfaces. 
Cor. 7. Alfo Rq : RCc=RP : RF, and ultmately RQ .• 
RV=RC : RF, and RF= RV X RC .. N. B. Thefe four 
RQ 
points Q, V, C, F, either lie all one way from P, or two of 
them forward and two backward. 
Cor. 8. Alfo, making O the principal focus of rays 
coming the fame way, we have R^ : qC=.Co : o F, and 
ultimately RQ : Qc=cO : OF, and OF= S , S . x . CQ .,and 
RQ 
therefore reciprocally proportional to RQ, becaufe QC X 
co is a conftant quantity. 
Thefe corollaries, or theorems, give us a variety of 
methods for finding the focus of refracted rays, or the 
other points related to them j and each formula contains 
four points, of which, any three being given, the fourth 
may be found. Perhaps the laft is the moft Ample, as the 
quantity oe+rQ is always negative, becaufe o and Q are 
on different fides. 
Cor. 9. From this conftrudion w-e may alfo derive a very 
eafy and expeditious method of drawing many refraded 
rays. Draw through the centre C (fig. 16, 17.) a line to 
the point of incidence P, and a line C A parallel to the in¬ 
cident ray RP. Take VO to VC as the fine of incidence 
to the fine of refradion ; and about A, with the radius 
VO, defcribe an-arch of a circle cutting PC produced in 
B. Join AB; and PF parallel to AB is the refraded ray. 
When the incident light is parallel to RC, the point A 
coincides with V; and a circle defcribed round V with the 
diftance VO will cut the lines PC, pC, &c. in the points 
B, h. The demonftration is evident. 
Having thus determined the focal diftance of refradted 
rays, it will be proper to point out a little more parti¬ 
cularly its relation to its conjugate focus of incident 
rays. We fnall confider the four cafes of light incident 
on the convex or concave furface of a denfer or a rarer 
medium. 
1. Let light moving in air fall on the convex furface of 
glafs, (fig. 11 to 15.) Let us fuppofe it tending to a point 
beyond the glafs infinitely diftant. It will be colleded 
to its principal focus o beyond the vertex V. Now let 
the incident light converge a little, fo that R is at a great 
diftance beyond the furface. The focus of refraded rays 
F, will be a little within O, or nearer to V. As the in¬ 
cident rays are made to converge more and more, the point 
R comes nearer to V, and the point F alfo approaches it, 
but with a much flower motion, being always fituated 
between O and C till it is overtaken by R at the centre 
C, when the incident light is perpendicular to the fur¬ 
face in every point, and therefore fuffers no refradion. 
As R has overtaken F at C, it now paffes it, and is 
again overtaken by it at V. Now the point R is on the 
fide from which the light comes ; that is, the rays di¬ 
verge from R. After refradion they will diverge from 
F a little without R ; and, as R recedes farther from 
V, F recedes fti!l farther, and with an accelerated mo¬ 
tion, till, when R comes to Q, F has gone to an infinite 
diftance, or the refraded rays are parallel. When R 
ftill recedes, F now appears on the other fide, or be¬ 
yond V; and, as R recedes back to an infinite dif¬ 
tance, F has come to O : and this completes the feries 
of variations, the motion of F during the whole changes 
of fituation being in the fame direction with the mo¬ 
tion of R. 
2. Let the light moving in air fall on the concave fur¬ 
face of glafs; and let us begin with parallel incident rays, 
conceiving, as before, R to lie beyond the glafs at an infi¬ 
nite diftance. The refraded rays will move as if they 
came from the principal focus O, lying on that fide of the 
glafs from which the light comes. As the incident rays 
are made gradually more converging, and the point of 
convergence R comes toward the glafs, the conjugate fo¬ 
cus F moves backward from O ; the refraded rays grow¬ 
ing lefs and lefs diverging, till the point R comes to Q, 
the principal focus on the other fide ; the refraded rays 
growing parallel, orF has retreated to an infinite diftance. 
The incident light converging ftill more, or R coming 
between Q and V, F will appear on the other fide, or be* 
yond the furface, or within the glafs, and will approach 
it with a retarded motion, and finally overtake R at the 
furface of the glafs. Let R continue its motion back¬ 
wards, (for it has all the while been moving backwards, 
or in a diredion contrary to that of the light;) that is, 
let R now be a radiant point, moving backw-ards from the 
furface of the glafs. F will at firft be without it, but will 
be overtaken by it at the centre C, when the rays will 
fuffer no refradion. R, ftill receding, will get without F; 
and, while R recedes to an infinite diftance, F will recede 
to O, and the feries will be completed. 
3. Let the light moving in glafs fall on the convex fur¬ 
face of air; that is, let it come out of the concave furface 
of. 
