570 
OPTICS. 
cutting the incident ray in I. About the centre P, with 
the radius PI, defcribe an arch of a circle IF, cutting 
VR in F ; draw PE tending from or towards F. We fay 
PE is the refradted ray, and F the point of dilperfion or 
convergence of the rays RV, RP, or the conjugate focus 
to R. For, fince GI and PV are parallel, and PF equal to 
PI, we have PF: PR=PI: PR,=VG: VR, =fin. incid.: 
fm. refr. But PF: PR=fin. PRV : fin. PFV, and RRV 
is equal to the angle of incidence at P; therefore PFV is 
the correfpondingangle of refradtion, FPEis the refradted 
ray, and F the conjugate focus to R. 
Car. i. If diverging or converging rays fall on the fur- 
face of a more refradfing medium, they will diverge or 
converge lefs after refradtion, F being farther from the 
furface than R. The contrary mult happen when the di¬ 
verging or converging rays fall on the furface of a lefs- 
refradling medium, becaufe, in this cafe, F is nearer to 
the furface than R. 
Cor. 2. Let Rp be another ray, more oblique than RP, 
the refradfing point p being farther from V, and let fp e 
be the refradted ray, determined by the fame conlfrudtion. 
Becaufe the arches FI,/ i, are perpendicular to their radii, 
it is evident that they will converge to fome point within 
the angle RIK, and therefore will not crofs each other 
between F and I; therefore R f will be greater than RF, 
as RF is greater than RG, for fimilar reafons. Hence it 
follows, that all the rays which tended from or towards 
R, and were incident on the whole of VPp, will not di¬ 
verge front or converge to F, but will be diffufed over the 
line GF f. This diffufion is called aberration from the 
focus, and is fo much greater as the rays are more oblique. 
No rays flowing from or towards R will have point of con- 
courl'e with RV nearer to R than F is : but, if the obliquity 
be inconfiderable, fo that the ratio of RP to FP does not 
differ fenlibly from that of RV to FV, the point of con- 
courfe will not be fenfibly removed from G. G is there¬ 
fore ul’ually called the conjugate focus to R. It is the con¬ 
jugate focus of an indefinitely-llender pencil of rays fall¬ 
ing perpendicularly on the furface. The conjugate focus 
of an oblique pencil, or even of two oblique rays, whofe 
difperflon on the furface is confiderable, is of more dif¬ 
ficult inveffigation. See Gravefande’s Natural Philolophy, 
for a very neat and elementary determination. 
We have thus pointed out, in an eafy and familiar 
manner, the nature of optical aberration. But, as this 
is thechief caufe of theimperfedtion of optical inftruments, 
and as the only method of removing this imperfection is 
to diminifh this aberration, or corredt it by a fubfequent 
aberration in the oppoiite diredtion, we tlrall here give a 
fundamental and very Ample propolition, which will (with 
obvious alterations) apply to all important cafes. This 
is the determination of the focus of an inflnitely-flender 
pencil of oblique rays RP, Rp. Retaining the former 
conftrudtion for the ray PF, fig. 7. fuppofe the other ray 
Rp, infinitely near to RP. Draw PS perpendicular to PV, 
and Rr perpendicular to RP, and make Pr: PSirrVR: 
VF. On Pr defcribe the femicircle rRP, and on PS the 
femicircle S(pR, cutting the refradted ray PF in (p, draw 
pr, p S, pep. It follows from the Lemma, that, if <p be the 
focus of refradted rays, the variation Repp of the angle of 
refradtion is to the correfponding variation RRp of the 
angle of incidence as the tangent of the angle of refrac¬ 
tion VFP to the tangent of the angle of incidence VRP. 
Now Rp may be contidered as coinciding with the arch of 
the femicircles. Therefore the angles RRp, P rp, are equal, 
as alfo the angles P (pp, PS p. But PS p is to Rrp as Pr 
to PS; that is, as VR toVF; that is, as the cotangent 
of the angle of incidence to the cotangent of the angle of 
refradtion ; that is, as the tangent ol the angle of refrac¬ 
tion to the tangent of the angle of incidence. Therefore 
the point (p is the focus. 
Of Refraction by Spherical Surfaces. 
Pros. To find the focus of refradted rays, the focus of 
incident rays being given. 
Let PVtt (figs. 11, 12, 13, 14,15,) be a fpherical furface 
whofe centre is C, and let the incident light diverge from, 
or converge to, R. Draw the ray RC through the centre, 
cutting the furface in the point V, which we ihall deno¬ 
minate the vertex, while RC is called the axis. This ray 
pafi’es on without refradtion, becaufe it 'coincides with 
the perpendicular to the furface. Let RP be another 
incident ray, which is refradted at P ; draw the radius 
PC. In RP make RE to RP as the fine of incidence m 
to the line of refradtion n; and about the centre R, with 
the diftance RE, defcribe the circle EK, cutting PC in K; 
draw RK, and RF parallel to it, cutting tire axis in F. 
PF is the refradted ray, and F is the focus. For the tri¬ 
angles PCF, KCR, are fimilar, and the angles at P andK 
are equal. Alfo RK is equal to RE, andRPD is the an¬ 
gle of incidence. Now m •• »=RK : RP, =fin.DPR ; fin. 
RKP, =fin. DPR : fin. CPF. Therefore CPF is the angle 
of refradtion correfponding to the angle of incidence RPD, 
and PF is the refradted ray, and F the focus. Q.E.D. 
Cor. 1. CK : CP = CR : CF, and CF= SI* > U: R .. 
CK 
Now CP XCR is a conftant quantity ; and therefore CF 
is reciprocally as CK, which evidently varies with a va¬ 
riation of the arch VP. Hence it follows, that all the 
rays flowing from R are not colledted at the conjugate 
focus F. The ultimate fituation of the point F, as the 
point P gradually approaches to, and at laft coincides 
with, V, is called the conjugate focus of central rays ; and 
the diftance between this focus and the focus of a lateral 
ray is called the aberration of that ray, anting from the 
fpherical figure. 
There are, however, two fituations of the point R 
fuch, that all the rays which flow from it are made to 
diverge from one point. One of thofe is C, fig. 11. be¬ 
caufe they all pals through without refradiion, and there¬ 
fore ftill diverge from C ; the other is, when rays in the 
rare medium with a convex furface flow from a point R, 
fo fituated beyond the centre that CV is to CR as the line 
of incidence in the rare medium is to the fine of refradtion 
in the denfer, or, when rays in the rare medium fall on 
the convex furface of the denfer, converging to F, fo 
fituated that CF : CV—m : n. In this cafe they will 
all be difperfed from F, fo fituated that CV : CF~« : m, 
—CR : CV; for fine RPC : fine PKC=» : m, =CR : 
CP, =line RPC : fine PRC. Therefore the angle PRC 
is equal to PKC, or to FPC (by conftrudtion of the pro¬ 
blem), and the angle C is common to the triangles PRC, 
FPC ; they are therefore fimilar, and the angles PRC, 
FPC, are equal; and n : m—CR : CF,=CK : CR,=:CR 
: CP; therefore CP : CK=CP 2 : CR 2 : but CP and 
CR are conftant quantities, and therefore CK is a con¬ 
ftant quantity, and (by the corollary) CF is a conftant 
quantity, and all the rays flowing from R are difperfed 
from F by refradtion. In like manner, rays converging to 
F, will, by refradtion, converge to R. This was firft ob- 
ferved by Huygens. 
Cor. 2. If the incident ray R'P, fig. 11, is parallel 
to the axis RC, we have PO to CO as the line of inci¬ 
dence to the line of refradtion. For the triangles 
R'PIC PCO are fimilar, and PO : CO=R'K' : R'P ; 
—m : n. 
Cor. 3. In this cafe, too, we have the focal diftance 
of central parallel rays reckoned from the vertex = 
n - V VC. For, fince PO is ultimately VO, we have 
m—11 
m : n —\ 7 O : CO, and m — n : »!=VO—CO : VO, = 
VC : VO, and VO=-^~ X VC. This is called the 
m —w 
principal focal diftance, or focal diftance of parallel rays. 
Alfo CO, the principal focal diftance reckoned from the 
centre, =_!L_ x VC. 
m—n 
When m is lefs than n, m — n is a negative quantity. 
Alfo obferve, that, in applying fymbols to this computa- 
1 tion 
