OPT 
the fquare of the velocity. (Lemma I.) The fame may 
be faid of every fuch reflangle. And, if the number of 
the portions, fuch as FH, be increafed, and their magni¬ 
tude diminifhed, without end, the reftangles will ulti¬ 
mately occupy the whole curvilineal area, and the force 
will therefore be as the finite changes made on the fquare 
of the velocity, and the propofition is demonftrated. 
Cor. The whole change made on the fquare of the ve¬ 
locity, is equal to the fquare of that velocity which the 
accelerating force would communicate to the particle by 
impelling it along BC from a ftate of reft in B. For the 
area BCED will (till exprefs the fquare of this velocity; 
and it equally expreffes the change made on the fquare of 
any velocity wherewith the particle may pafs through the 
point B, and is independent on the magnitude of that 
velocity. 
The figure is adapted to the cafe where the forces all 
confpire with the initial motion of the particle, or all op- 
pofe it; and the area exprefies an augmentation or a dimi¬ 
nution of the fquare of the initial velocity. But the rea- 
foning would have been the fame, although, in fonte parts 
of the line BC, the forces had confpired with the initial 
motion, and in other parts had oppofed it. In fuch a 
cafe, the ordinates which exprefs the intenfity of the 
forces muft lie on different fides of the abfcifta BC 5 and 
that part of the area which lies on one fide muft be confi- 
dered as negative with refpect to the other, and be fub- 
trafled from it. Thus, if the forces be reprefented by the 
ordinates of the dotted curve line, DHe, which croffes the 
abfcifta in H, the figure will correfpond to the motion of 
a particle, which, after moving uniformly along AB, is 
fubjefled to the aftion of a variable accelerating force, 
during its motion along BH, and the fquare of its initial 
velocity is increafed by the ciuantity BHD; after which 
it is retarded during its motion along HC, and the fquare 
of its velocity in H is diminifhed by a quantity HCe. 
Therefore, the fquare of the initial velocity is changed by 
a quantity BHD—HCe, or HCe—BHD. 
This propofition, which is the 39th of the ift book of 
the Principia, is perhaps the moll important in the whole 
fcience of mechanics, being the foundation of every ap¬ 
plication of mechanical theory to the explanation of na¬ 
tural phenomena. No traces of it are to be found in the 
writings of philofophers before the publication of Newton’s 
Principia, though it is affumed by John Bernoulli and 
other foreign mathematicians, as an elementary truth, 
without any acknowledgment of their obligations to its 
author. It is ufually expreffed by the equation/’«=r v 
and ff sz=v 2 -, i. e. the fum of the momentary ablions is equal 
to the whole , or finite increment, of the fquare of the velocity. 
Proposition. When light pafles obliquely into, or out 
of, a tranfparent fubftance, it is refrafted fo that the 
fine of the angle of incidence is”to the fine of the angle 
of refraflion in the conftant ratio of the velocity of the 
refrafted light to that of the incident light. 
Let ST, KR, fig. 5, reprefent two planes (parallel to, 
and equidiftant from, the refrafling furface XY) which' 
bound the lpace in which the light, during its paflage, is 
afted on by the refrafling forces. The intenfity of the 
refrafling forces, being fuppoled equal at equal diftances 
from the bounding planes, though any-how different at 
different diftances from them, may be reprefented by the 
ordinates Ta, nq,pr, cR, &c. of the curve ahnpc, of which 
the form muft be determined from obfervation, and may 
remain for ever unknown. The phenomena of inflefted 
light fliow us that it is attracted by the refrafling fub¬ 
ftance at fome diftances, and repelled at others. 
Let the light, moving uniformly in the direftion AB, 
enter the refrafling ftratum at B. It will not proceed in 
that direftion, but its path will be incurvated upwards 
while afted on by a repulfive force, and downwards while 
impelled by an attraftive force. It will defcribe fome 
curvilineal path Bdo, CDE, which AB touches in B, and 
will finally emerge from the refrafling ftratum at E, and 
Vol. XVII. No. 1198. 
ICS. 53.9 
move uniformly in a ftraight line EF, which touches the 
curve in E. If through b, the interfeflion of the curve 
of forces with its abfcifta, we draw bo, cutting the path of 
the light in o, it is evident that this path will be concave 
upwards between B and o, and concave downwards be¬ 
tween o and E. Alfo, if the initial velocity of the light 
has been fufficiently fmall, its path may be lo much bent 
upwards, that in fome point, d, its direftion may be pa¬ 
rallel to the bounding planes. In this cafe it is evident, 
that, being under the influence of a repulfive force, it 
will be more bent upwards ; and it will defcribe df, equal 
and fimilar to dB, and emerge in an angle gfs, equal 
to ABG. In this cafe it is reflefted, making the angle 
of refleftion equal to that of incidence. By which it ap¬ 
pears how refleftion, refraflion, and inflection, are pro¬ 
duced by t^ie fame forces, and performed by the fame laws. 
But let the velocity be fuppofed fufficiently great to 
enable the light to penetrate through the refrafling ftra¬ 
tum, and emerge from it in the direftion EF ; let AB and 
EF be fuppofed to be defcribed in equal times -. they will 
be proportional to the initial and final velocities of the 
light. Now, becaufe the refrafling forces muft aft in a 
direftion perpendicular to the refrafling furface (lince 
they arife from the joint aftion of all the particles of a 
homogeneous fubftance which are within the fphere of 
mutual aflion), they cannot affeft the motion of the light 
eftimated in the direftion of the refrafling furface. If, 
therefore, AG be drawn perpendicular to ST, and FK to 
KR, the lines GB, EK, mult be equal, becaufe they are 
the motions AB, EF, eftimated in the direftion of the 
planes. Draw now EL parallel to AB. It is alfo equal 
to it. Therefore, EL, EF, are as the initial and final ve¬ 
locities of the light. But EF is to EL as the fine of the 
angle ELK to the fine of the angle EFK ; that is, as the 
fine of the angle ABH to the fine of the angle FEI ; that 
is, as the fine of the angle of incidence to the fine of the 
angle of refraflion. 
Of the Focal Diftance of Rays rtf cabled by puffing out of 
one Medium into another of different Den/it a, and through, 
a Plane Surface. 
Lemma. The indefinitely-fmall variation of the angle of 
incidence is to the fimultaneous variation of the angle 
of refraflion, as the tangent of incidence is to the tan-* 
gent of refraflion ; or the cotemporaneous variations 
of the angles of incidence and refraflion are propor¬ 
tional to the tangents of thefe angles. 
Let RVF, rVf, fig. 6, be the progrefs of the rays re- 
frafted at V (the angle rVR being coniidered in its naf- 
cent or evanefeent ftate), and VC perpendicular to the 
refrafling furface VA. From C drawCD, CB, perpendicular 
to the incident and refrafted rays RV, VF, cutting rV, Vf, 
in i? and (3, and let C d, Cb, be perpendicular to rV, Vf. 
Becaufe the fines of incidence and refraflion are in a con¬ 
ftant ratio, their fimultaneous variations are in the fame 
conftant ratio. Now the angle RV/- is to the angle F Vf 
B 0 D S' , . BC DC , . 
in the ratio of — to —; that is, of — to —; that is^ 
fin. incid. fin. refr. 
of——:—— to---; that is, of tan. incid. to tan. 
cor. incid. cof. refr. 
refr. 
Cor. The difference of thefe variations is to the greatest 
or leaft of them as the difference of the tangents to the 
greateft or leaft tangent. 
Problem. Let two rays- RV, RP, (fig. 7, 8, 9, 10.) di¬ 
verge from, or converge to, a point R, and pafs through 
the plane furface PV, feparating two refrafling me¬ 
diums AB, of which let B be the moll refrafting, and 
let RV be perpendicular to the furface. It is required 
to determine the point of difperfion or convergence, F, 
of the refrafted rays VD, PE. 
Make VR to VG as the fine of refraflion to the fine 
of incidence, and draw GIK parallel to the furface, 
7 F cutting 
