OPTICS, 
On the Aberrations produced by the Spherical Form 
of Reflecting and Refracting Surfaces. 
Prop VI. When a ray of light is incident obliquely 
upon a fpherical rcflcdor, to determine the infertion of 
the reflected rays and the axis of the pencil to which 
it belongs. 
Let ACB, fig. 6, be the fpherical refleCtor; E its cen¬ 
tre; QA a ray incidentobliquely upon it; QET the axis 
of the pencil to which the ray belongs. Draw AT, 
touching the arc ACB in A, and let it meet QT in T; 
bifeft ET, EC, in e and F; take q the geometrical focus, 
conjugate to Q; and let QA be reflected in the direction 
Ax; alfo, let Fe be fmall when compared with QF. Then, 
QF : FE :: QE : Eq; and QF±Fe : FE-j-Fe :: QE : Ex; 
QExFE . _ QExFE+Fe 
therefore, Eqz=.~ ^ 
by aCtual divifion, Err 
QEXFE,QE 
and Ex=. 
; whence, 
5F " r QF ! 
QE 2 x, 
— XFe, nearly. 
CT, or ET —EC 
QF±Fe 
.QExFE QE 2 xFe 
QF '’"QFXQFdtFe ~ 
XFe, nearly; therefore Ex— Eq=.qx— 
Alfo, fince ET2222EF, and EC2222EF, 
OF 2 CT 
=zFe ; confequently, qx ——A_X — 
nearly. 
This quantity, qx, is called the longitudinal aberration 
of the oblique ray QA. 
^ QE 2 
Cor. i. Since the exprefiion ^pjXFe has always the 
fame fign, qx is always meafured in the fame direction 
upon the line QT. 
Cor. z. Draw AD perpendicular to QC, and produce it 
till it meets the furface in B; join AC, CB. Then, the 
^/TAC — the^/CBA^the^/DAC ; and CD : CT :: AD 
: AT, (Euc. 3. vi.) and, w'hen the arc AC is evanefcent, 
AD is equal to AT ; therefore, CD—CT ; and the lon- 
QE 2 CD 
gitudinal aberration <7x2222——X-. 
& 1 QF 2 z 
Cor. 3. When QA is parallel to QC, QE becomes equal 
, CD 
to QF, and qx ——-. 
2 
Cor. 4. If Q, when in FE, or in FE produced, approach 
to E, the ratio of QE to QF decreafes; and therefore, if 
CD be given, the aberration decreafes. If Q be in FC, 
or FC produced, as QF decreafes, the aberration increafes. 
Cor. 5. If the diftances QE and QF be invariable, the 
aberration varies as CD. 
Cor. 6. Let CM be the diameter of the refleCtor; then, 
by the property of the circle, CD : DA :: DA : DM, 
DA 2 
. CM. 
The place of y, with refpeCt to q, will be affeCted by 
two caufes : Firlt, x is nearer to c than F, the geometri¬ 
cal focus after the firft reflection (Prop. VI. Cor. 2.) and 
therefore, on this account, yc is lefs than qc. Secondly, 
in confequence of the aberration arifing from the form of 
the refleCtor acb, yc is greater than qc; therefore thefe 
caufes counteract each other; and, by a properadjuftment 
of the reflectors, the aberration qy may in a great mea- 
fure be deftroyed. 
Cor. 1. Though the aberration qy, of the extreme ray 
DA, fhould be wholly deftroyed, the aberration of the in¬ 
termediate rays will not be entirely corrected ; and this 
feems to be an infuperable obftacle to the perfection of 
refleCting-telefcopes. 
Cor. 2. If the reflectors be both concave, as in Gregory’s 
telefcope, the aberrations produced by the two reflec¬ 
tions are in the fame direction ; that is, the fecond reflec¬ 
tion increafes the aberration produced by the firft. 
Prop. VIII. When a ray of homogeneal light is inci¬ 
dent obliquely upon a fpherical refraEling furface, to 
determine the interfeCtion of the refraCted ray and the 
axis of the pencil to which it belongs. 
Let ACB, fig. 8, be the refraCtor; E its centre; QA 
a ray incident obliquely upon it; QCq the axis of the 
pencil to which QA belongs ; Ax the refraCted ray ; q the 
geometrical focus, conjugate to Q. Draw AD perpen¬ 
dicular to the axis; and from the centres Q, x, with the 
radii QA, xA, defcribe the circular arcs AV, AP, cut¬ 
ting the axis in V and P. Take m- f-r : 1 :: fin. inci¬ 
dence : fin. refraCtion. 
Then, in the triangle QAE, QE : QA :: fin. inci¬ 
dence : fin. ^/AEQ ; alfo, in the triangle AEx, Ax : 
Ex :: fin. ^/AEX (fin. xjAEQ) : fin. refraCtion ; there¬ 
fore, by compounding thefe two proportions, QExAx : 
QAxEx :: fin. incidence : fin. refraCtion :: m 4-1 : 1 ; 
hence wi-j-i . QA : QE :: Ax : Ex; by divifion, 7/i-J-i . 
QA—QE : QE :: Ax—Ex : Ex; that is, . QV— 
QE : QE :: EP : Ex; or tn+i . QC-fwi+ i' . CV —QE : 
QE :: EC—CP : Ex; or 
QE :: EC—CP : Ex. 
Now, DC : DV :: QV : EC :: QC : EC, nearly; and 
by compolition, DC : CV QC : QE; therefore, when 
QExDC 
the arc AC is fmall, CW= * . Alfo, DC : DP 
y l 
by divifion, DC : CP :: Ax : Ax—EC :: 
QE, nearly; therefore CP = 
QC—EC+m+i . CV 
Ax : EC 
Ax : Ex 
QExDC 
m +1 
9H-j-I . QC 
, nearly. And, by fubftituting thefe values of 
Q C 
and CD222 — —; therefore, when CD i*s very fmall, or DM 
nearly equal to the diameter of the given refleCtor, CD 
varies as DA 2 nearly; and confequently, when QE, QF, 
are given, and the arc AC is very fmall, the longitudinal 
aberration varies as DA 2 . 
Cor. 7. When parallel rays 3re incident upon the re¬ 
flector, the longitudinal aberration is ultimately equal 
CD DA 2 DA 2 DA 2 , , 
to-= 7,7.rrp;=-rTTr2 i and therefore it vanes as 
2 2DM 4E C 8EF 
DA 2 
E F " , 
Prop. VII. If parallel rays be reflected at a concave, and 
afterwards fall upon a convex, fpherical refleCtor, con¬ 
verging to a point between its furface and principal 
focus, as in Caflegrain’s telefcope, the aberration of the 
lateral rays produced by the firft reflection will, infome 
meafure, be corrected by the latter. 
Let ecC, fig. 7, be the axis of the telefcope ; x the inter¬ 
feCtion of the axis and lateral ray after the firft reflection ; 
y their interfeClion after the fecond reflection; q the geo¬ 
metrical focus after both reflections. 
Vol. XVII. No. 1206. 
CV and CP in the former proportion, we obtain 
—EC-1- 
wi-j-i . QExDC 
QC 
hence, Ex2=QEx 
QExDC 
EC- - - 
Wi+I . QC 
QE :: EC— 
QEXDC 
m-t-i . QC 
. QC 
Ex ; 
QC—EC -1 
w +7 . QExDC ' and b y dually divid- 
QC 
ing, and taking the remainder, Ex=- 
QExEC 
QE 2 QC+im+2 . EC 
- x 
m +1 . QC m . 
vanifhes, Ex—Eq\ 
tion qx—' 
XDC, nearly. 
QC—EC 
When DC 
therefore the aberra- 
EC 
XDC. 
. QC" ni . QC—ECl 2 
Cor. 1. If the refraCtor be given, and the fituation of 
the focus of incident rays, the aberration varies as DC, 
the verfed fine of the arc AC. 
8 F Cor. 
