Encyclopaedia Britannica > Volume 15, NIC-PAR

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Theory.
Law of the accelerating forces F f, and let AC, a c, be
Refraction, spaces described in equal times; it is evident from
I'1 * ^ what has been saul under the articles Gravity and
acceleration, tliat because these spaces are describ¬
ed with motions uniformly accelerated, AC and a c are
respectively the halves of the spaces which would be
uniformly described during the same time with the ve¬
locities acquired at C and r, and are therefore measures
oi these velocities. And as these velocities are uni¬
formly acquired in equal times, they are measures of
the accelerating forces. Therefore, AC : «c=z:F :
Also, from the nature of uniformly accelerated motion*
the spaces are proportional to the squares of the acqui¬
red velocities. Therefore, (using the symbols v^C,
\/ c, &c. to express the squares of the velocities at
C r, &c.) we have
V1 ft : v/’ C-AB : AC
v/’C : -y/2 c—AC2 : at*
\/2 c : x/* c : ab
Therefore, by equality of compound ratios
V* B : AB X AC : x i* c, =:AB x F
And in like manner V/i D : v/2^ v/ AD x F
and v/* B—^ D : b—^Z2 (/—BD X F
Q. E. D.
Corollary. If the forces are as the spaces in¬
versely, the augmentations or diminutions of the squares
of the velocities are equal.
Remark. It D B, db, be taken extremely small, the
products BD X* and b dXf may be called the momen-
taiy actions of the forces, or the momentary increments
of the squares of the velocities. It is usually expressed,
by the writers on the higher mechanics, by the symbol
j s, ov f d s, wherey means the accelerating force, and
/ ?r ds means the indefinitely small space along which
u. is umfoimly exerted. And the proposition is expres¬
sed by the fluxionary equation fs=vv because vv is
half the increment oft;*, as is well known.
s.
20J
■ abxf.
ad X r;
: bdxf.
Lemma II.
wxxvn.^ a Part*c^e °f matter, moving with any velocity
fig. 6. along the line AC, be impelled by an accelerat¬
ing or retarding force, acting in the same or
in the opposite direction, and if the intensity
of the force in the different points B, F, H, C,
&c. be as the ordinates BD, FG, &c. to the
line DGE, the areas BFGD, BHKD, &c. will
be as the changes made on the square of the
velocity, at B, when the particle arrives at the
points F, H, &c.
changes made on the square of the velocity, and the Law of
proposition is demonstrated. ^ Itefraction.
Corollary. 1 he whole change made on the square '
of the velocity ,s equal to the square of that velocity
uhich the accelerating force would communicate to the
particle by impelling it along BC from a state of rest
in, * IoVh? area BCED wil1 still express the square
of this velocity, and it equally expresses the change
made on the square of any velocity wherewith the par¬
ticle may pass through the point B, and is independent
on the magnitude of that velocity.
Remark. The figure is adapted to the case where
the forces a conspire with the initial motion of the
particle, or all oppose it, and llie area expresses an aug¬
mentation or a diminution of the square of the Initial
velocity. But the reasoning would have been the same,
although, in some parts of the line B C, the forces had
conspired with the initial motion, and in other parts had
opposed it. In such a case, the ordinates which express
the intensity of the forces must lie on different sides of
the abscissa B C, and that part of the area which lies on
one side must be considered as negative with respect to
the other, and be subtracted from it. Thus, if the for-
cts be represented by the ordinates of the dotted curve
line DM e, which crosses the abscissa in H, the figure
will correspond to the motion of a particle, which, after
moving uniformly along AB, is subjected to the action
ot a variable accelerating force during its motion along
M and the square of its initial velocity is increased
by the quantity BHD ; after which it is retarded du¬
ll ng its motion along HC, and the square of its velo¬
city in H is diminished by a quantity HC e. Therefore
the square of the initial velocity is changed by a quan¬
tity BHB—HC e, or IIC e—BHD. y q
This proposition, which is the 39th of the 1st book
0} the I rincipia, is perhaps the most important in the
whole science of mechanics, being the foundation of
every application of mechanical theory to the explana¬
tion of natural phenomena. No traces of It are to be
found in the writings of philosophers before the publi¬
cation of Newton’s Principia, though it is assumed by
John Bernoulli and other foreign mathematicians, as an
elementary truth, without any acknowledgment of their
obligations to its author. It is usually expressed by the
equation /i'—nt; andyy"s—v*, i. e. the sum of the mo-
mentary actions is equal to the whole or finite inci'e-
ment of the square of the velocity.
Prqposition.
When light passes obliquely into or out of a trans-
r>ar/»ni- „„ ‘.l —c t i , . The ratio
o 1 --—1 ; u. a Li ana- IIJ
parent substance, it is refracted so that the sine The rat.f<
of the angle of incidence is to the sine of the ofVnddence >
angle of refraction in the constant ratio of thethe sine
velocity of the refracted light to that of the °-f refrac'
incident light. tl0n‘
. Eor let ft^ be divided into innumerable small por¬
tions, of which let FH be one, and let the force be sup¬
posed to act uniformly, or to be of invariable intensity
inring the motion along FH ; draw G[ perpendicular
to HK: It is evident that the rectangle FHIG will
- — mv/ j cmxji.z.vjr will
be as the product of the accelerating force by the space
a ong which it acts, and will therefore express the mo¬
mentary increment of the square of the velocity. (Letn-
ma I.). Ihe same may be said of every such rectangle.
And it the number of the portions, such as FH, be in¬
creased, and their magnitude diminished without end,
the rectangles will ultimately occupy the whole curvili-
flfia area, and thg force will therefore be as the finite
Let SF, KB, represent two planes (parallel to, andccclxxvii,
equidistant from, the refracting surface XY) which % 7*
bound the space in which the light, during its passage,
is acted on by the refracting forces.
I he intensity of the refracting forces being suppo¬
sed equal at equal distances from the bounding planes,
though anyhow different at different distances from
them, may be represented by the ordinates T o, n q, p r,
c Bj &c. ot the. curve ab np e, of which the form must
be.