OPTICS. 
574 
Let BDK (fig. 21.) be the sl-iipfe, by the revolution of 
which, about its major axis DK, the fpheroid is gene¬ 
rated ; H jind I its foci ; then, by the fuppofition, DK : 
HI :: fin. incidence : fin. refraftion. Let AB, which is 
parallel to DK, be a ray of light incident upon the fphe¬ 
roid ; join LIB, IB ; draw EEC touching the generating 
ellipfe in B ; through B draw GEL at right angles to EBC, 
meeting DK in N. 
Then, fince the /HBC is equal to the ,/IBE, by the 
nature of the ellipfe, and the ^/NBC to the ^NBE, the 
angles HBN, NBI, are equal ; therefore, IB : BH :: IN 
: NH (Euc. 3. vi.) comp. IB : BI-f-BH (DK) :: IN ; IH ; 
alt. IB : IN :: DK : IH :: fin. incidence : fin. refrac¬ 
tion ; alfo, IB : IN :: fin. INB : fin. IBN :: fin. BNH, 
or fin. ABG : fin. IBL; therefore, fin. ABG : fin. IBL :: 
fin. incidence : fin. refraftion; and, fince fin.-AEG is the 
fine of incidence, fin. IBL is the fine of refraftion ; and, 
beeaufe the angle LBI is lefs than a right angle, BI is the 
ref rafted ray. In the fame manner it may be fiiown, that 
every other ray in the pencil will be refrafted to I. 
Cor. x. If from the centre I. (fig. 22.) with any radius 
lefs than ID, a circular arc PQ be defcribed, the folid ge¬ 
nerated by the revolution of PDQ about the axis DI will 
refraft all the rays, incident parallel to DI, accurately to I. 
For, after refraftion at the furface PDQ, the rays converge 
to I; and they fuller no refraftion at the furface PQ, be- 
caufe they are incident perpendicularly upon it. 
Cor. 2, Rays diverging from I will be refrafted parallel 
to ID. 
Prop. V. If any hyperboloid, whofe major axis is to the 
diftance between the foci as the fine of incidence to the 
fine of refraftion out of the folid into the ambient me¬ 
dium, be generated in a fimilar manner, parallel rays, 
incident in the direftion of the axis, and refrafted out 
of the hyperboloid, will converge to the farther focus. 
—The proof is nearly the fame as in the former cafe. 
See fig. 23. 
Cor. 1. If PQ, fig. 24.. be drawn perpendicular to the 
axis of the hyperbola, and meet the curve in Pand Q, the 
folid generated by the revolution of PDQ, about the 
axis MDI, will refraft; all the rdys, incident parallel to MI, 
accurately to I. For the rays will fuifer no refraftion at 
the plane furface PQ. 
Cor. 2. Rays diverging from I, and incident upon the 
furface PDQ, will be refrafted parallel to ID. 
Pro p. VI. The image of a ftraight line, formed by a lens 
or fphere, is the arc of a conic feftion. 
Let AB, fig. 25, be a lens, or fphere, whofe centre is E; 
PD a ftraight line placed before it ; through E draw QE7, 
at right angles to PQ; in PD take any point P; join PE, 
and produce it. Let F be the principal focus of rays in¬ 
cident in the direftion qE ; with the centre E, and ra¬ 
dius EF, defcribe the circular arc FV, cutting PE in V. 
In PEp, take PV : PE :: PE : Pp, meafuring PV and Pp 
in the fame direftion from P; thertp is the image of P. 
Draw pD parallel to qQ. Then, fince the triangles PEQ, 
PpD, are fimilar, PE : Pp :: QE : Dp; confequently, 
PV : PE :i QE : Dp. Alfo, PV : VE :: PE : Ep; al¬ 
ternately, PV : PE :: VE (FE) : Ep; therefore QE : 
Dp :: FE : E p; and alternately QE i FE :: Dp : _Ep; 
confequently, the locus of the point p is a conic feftion, 
vrhofe focus is E, and direftrix PD. 
Cor. 1. The curve is an ellipfe, parabola* or hyperbola, 
according as QE is greater than, equal to, or lefs than, FE. 
Cor. 2. When Ep coincides with EL, that ordinate to 
the axis which palfes through the focus Dp becomes equal 
to QE, and therefore ELmEF ; that is, half the latrus 
reftum of the conic feftion is equal to the focal length of 
the glafs. 
Cor. 3. The curvature of the image, at its vertex, is the 
fame, wherever the objeft is placed. 
Cor. 4. If xq be the major axis of the conic feftion, 
Qq : E q :i QE ; FE j and by divifion, or compofition, 
QE : E q -.i QF : FE; therefore E<p= 
QEXFE QEXFE 
QF QE-£FE 
t , r QEXFE 
In the fame manner, Ex= -^-— ; confequently, xq=z 
QEXFE QEXFE 2 Q E 2 x~F E ... ^ ^ 
QE-^FE“QE-j-FE QE 2 ^fK 2 ‘ A fo ’ xExE 1 > the 
OF 2 y 
fquare of the femi-axis minor, =z— -1-- 
QE 2 -~FE 2 - 
Cor. 5. If the focal length of the refraftorbe finite, and 
pr be drawn perpendicular to the axis, the evanefcent arc 
pq is equal to pr; and QP : qp EQ : E q. Alfo, whilft 
the angle QEP, which QP fubtends at the centre of the 
glafs, is fmall, though finite, the image pq, when formed 
at a finite diftance from the refraftor, will, as to lenfe, be 
a right line; and QP : qp EQ : E q. 
Prop. VII. The fun’s image, formed by a fpherical re¬ 
frafting furface, lens, or fphere, is a circle, and nearly 
in the principal focus of the refraftor. 
Let E, fig. 2,6, be the centre of the refraftor; F and/ 
its principal fdfci ; PQ a radius of the fun’s difc. Then, 
fince FEis inconfiderable with refpeft to QF, the image 
of_Q may, for all practical purpofes, be confulered as 
coincident with the principal focus/of the refraftor; alfo, 
fince QP fubtends a fmall angle (about x6') atE, its image, 
fp, may be coniidered as a ftraight line (by Cor. 5. above). 
Now let the figure revolve about Qf as an axis; and, 
whilft QP generates the circle which reprefents the fun’s 
difc, fp will generate its image, which is, therefore, a 
circle. And in the fame manner it may be fhown, that 
the fun’s image, formed by a fpherical refieftor, is a circle, 
and in the principal focus of the refieftor. 
Cor. 1. Since the angle /Ep is given fp, the radius of 
the image, is proportional to E/, the focal length of the 
glafs. 
Cor. 2. The area of the image varies as the fquare of its 
radius; and, therefore, as the fquare of the focal length 
of the refieftor, or refraftor. 
By the reflefting or refrafting powers of different fub- 
ftances, we underhand the ratio of the number of rays ra- 
flefted or tranfmitted by them, if the number of incident 
rays be the lame. Thus, if one furface refleft tvvo thirds 
and another one third of the incident rays* the reflefting 
powers are faid to be as 2 : x. 
Cor. The number of rays refiefted, or tranfmitted, 
varies as the number incident, and the reflefting or re¬ 
frafting power, jointly. For, if the number of incident 
rays be given, the number refiefted, or tranfmitted, varies 
as the power; if the power be the fame, the number of 
rays refiefted, or tranfmitted, varies as the number inci¬ 
dent ; therefore, when both vary, the number of rays re- 
flefted, or tranfmitted, varies as the number incident, 
and the reflefting or refrafting power, jointly. 
Prop. VIII. The denfity of rays in the fun’s image 
varies direftly as the area of the aperture of the refieftor 
or refraftor by which it is formed, and the reflefting 
or refrafting power, jointly; and inverfely as the fquare 
of the focal length of the refieftor, or refraftor. 
The denfity of rays in the image varies direftly as their 
number, and inverfely as the ipace over which they are 
diftufed ; that is, direftly as the number, and inverfely 
as the fquare of the focal length of the refieftor, or re¬ 
fraftor. Alfo, the number of rays refiefted, or tranf¬ 
mitted, varies as the number incident, and the reflefting 
or refrafting power, jointly; that is, as the area of the 
aperture through which the incident rays pafs, and the 
power, jointly; confequently, the denfity of rays in the 
image varies, direftly in die compound ratio of the 
aperture and power, and inverfely as the fquare of the 
focal length of the refieftor, or refraftor. 
Cor. 1. When the apertures are circular, the denfity 
varies, direftly in the compound ratio of the fquare of 
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