GG2 
OPTICS. 
Cor. 2. When the incident rays are parallel, QC be- 
DC 
comes equal to QE; and gx— -- ---- -- 
1 m.m + 1 
Cor. 3. When diverging rays are incident upon a con¬ 
cave fpherical refrafting furface of a denier medium, (fee 
fio-. q.) the conftrudtion being made as before, EC and DC 
° _ QExEC mi . QE 2 
are negative ; whence, E.r=-7 - .- :; --]-r- 
__ m . QC+EC mi+ 1 . QC 
EC m . QE 2 
^2— XDC; and q.r— — —-77-77 X 
Q C — mi + 2 
m . QC+EC) 
QC— mi+ 2 . EC 
7)1 —h 1 . Q C 
XDC; this aberration, therefore, is to 
7 n . QC+EC] 
be mealured in an oppolite diredtion to the former. 
Cor 4. In this laft cafe, jf QC=mi+2 . EC, the aber¬ 
ration vanilhes ; that is, if QC : EC m-f-2 : 1, or 
QE : EC :: mi+i : 1 :: fin. incidence : fin. refradtion. 
Cor. 5. When converging rays are incident upon 
a concave fpherical furface of a rarer medium, (fee 
fig. 10.) QC, QE, EC, and DC, are negative. Alfo, 
if 1— y. : 1 :: fin. incidence : fin. refraction, — jj, mull 
, , „ QEXEC «, . QE 2 
be fubftituted for m, and E.r=- a —=—— 
[A . 1 — j ^ 
I*. . QE 2 
QC+2 — y. . EC 
y. . QC+EC] 
QC+2 —fx . EC 
XDC; hence, gx- 
1 — . Q C 
X 
X DC. 
y. . QC+EC| : 
Cor. 6. When diverging rays are incident upon a con¬ 
vex fpherical furface of a rarer medium, (fee fig. n.) 
E x: 
QEXEC 
d- 
QC+EC 
QE 2 QC+2— y. . EC 
:X 
d 
-f* • QC 
DC ; therefore the aberration gx =— 
QC+2— y. ■ EC 
QC+EC 1 
/a . QE 
X 
-XDC. 
y. . QC + EcT 
aberration may be found in the other cafes. 
DA 2 
Cor. 7. Since DC— nearly, (Prop. VI. Cor. 6.) by 
fubftituting this value of DC in the foregoing- exprefiions, 
we obtain the aberrations in terms of the femi-aperture. 
Cor. 8. Since Eq—-- ^ ^^^^ ,,if QE be diminilhed by 
the fmall quantity x, Eg will beincreafed by the quantity 
mi+i . EC 2 X.r „ 
——rr-:-, For, on this fuppofition, E n becomes 
771 . QC—EC) 2 * 
■ EC 
QE- 
QE —x . EC 
QEXEC 
7 )> . QC— x —EC 
»«+ 1 . EC 2 X>r 
QC—EC—mw in 
nearly; and therefore 
»« +1 
QC-EC T 
E C 2 X x 
coincide with the geometrical focus g, (Prop. VIII.) and, 
finceu is determined on fuppofition that g is the focus of 
rays incident upon the fecond furface,"an aberration will 
be produced, which may be determined by Cor. 8. Se¬ 
condly, Ax is incident obliquely upon the latter furface, 
and the aberration arifing from this caufe may be deter¬ 
mined by Cor. 5. Therefore the whole aberration vij may 
be found. 
Cor. 11. If the lens and place of the focus of incident 
rays be given, the aberration arifing from each of tiiefe 
caufes will vary nearly as AD 2 (Cor. 7, 9.) and therefore 
the final aberration v>j, which is the fum or difference or 
the former, will alfo vary nearly as AD 2 . 
Prop. IX. To find the lead circle of aberration into 
which all the homogeneal rays of the fame pencil, re¬ 
fracted by a lens or 1'mgle furface, are collected. 
Let AB, fig. 13, be the refractor; QC q its axis; Q the 
focus of incident rays; T the interleClion of the extreme 
ray QAT, and the axis ; t the interfeftion of any other 
ray Q at on the other fide of QC, and the axis; y the in- 
terfeftion of AT j and at. Draw AD, ad, and yx, at right 
angles to QC q; then, if the point a move from C towards 
B, the perpendicular xy will vary on two accounts ; the 
increafeof the angle C ta, and the decreafe of the diftance 
T t; and, when xy is a maximum, all the rays incident 
upon the fame fide of QC with Q a, will pafs through it ; 
and, if the figure revolve about the axis Qg, all the rays 
incident upon the lens will pafs through the circle gene¬ 
rated by xy. It is alfo manifeft, that the circle thus ge¬ 
nerated is lefs than any other circle through which ail the 
refracted rays pafs. To find when xy is the greatest pofii- 
ble, let Ta=.r ; ad—v; A E)—a; DT —f; T g—h. Then, 
fince AD, ad, when the lens is thin, are the femi-aper- 
tures through which the rays QAT, Qat, pafs, AD 2 : 
2 X ad 2 :: g'E : qt, (Prop. VIII. Cor. 11.) or, 
f* • QC 
And in the fame manner the 
bv 2 b 
qt; whence qt =—; therefore Tg — qt—'Tt—b -- 
-Xa 2 -» 2 . Again, DT : DA 
xy; confequently, xy——; alfo, da 
■ty; or, / : a : 
dt :: xy : tx . 
hence, T.r + vx— 
7 H. QC—EC y‘ ' . 7)1 . QC-u] 5 ’ 
where x is the decrement of QE, is the increment of Eg, 
nearly. 
Cor. 9. If x vary as DC, the increment of E q, when 
the radius of the refractor, and the fituation of the focus 
of incident rays, are given, will alfo vary as CD. 
Cor. 10. By a proper application of the foregoing rules, 
the longitudinal aberration, arifing from the fpherical 
form of refradiing furfaces, may be found in all cafes 
where the aperture are fmall. 
Example. Let Qg, fig. 12, be the axis of a lens; Q 
the focus of incident rays ; q the geometrical focus after 
the firft refradtion ; v the geometrical focus of emergent 
rays. Alfo, let QA be refradted at the firll furface in the 
diredlion A.r, and emergent in the diredtion Ay. Then, 
the aberration vy arifes from two caufes. Firft, x does not 
and x—— XvXa—v ; confequently, .ris the greeted pofii- 
ble, and therefore xy is the greatefi: poifible, when uX«— u 
is the greatefi; pofiible ; or when v=\a. Flence it follows, 
.6 
that the greeted value of x is —; and the correfponding 
4 
, „ ab D AX?T 
value or xyzz ;——- - - 
J 4/' 4DI' 
Cor. 1. If the focal length of the refradlor, and the focus 
of incidence, be given, DT is given, and x'yQUfT X DAOl 
DA 3 . 
Cor. 2. On the fame fuppofition, the area of the lead 
circle of aberration varies as DA 6 . 
Cor. 3. Exadtly in the fame manner, we may find the 
lead circle into which a pencil of rays refledled by a fphe¬ 
rical furface is colledted. 
Cor. 4. When parallel rays are incident upon a fpherical 
refledtor, the longitudinal aberration varies directly as 
the fquare of the femi-aperture, and inverfely as the focal 
length, (Prop. VI. Cor. 7.) therefore, xy, the radius of 
the lead circle of aberration, varies diredtly as the cube 
of the femi-aperture, and inverfely as the fquare of the 
focal length of the refledtor. 
Of 
