William Kingdon Clifford. # Elements of dynamic; an introduction to the study of motion and rest in solid and fluid bodies (Volume 2) online

. **(page 7 of 9)**

Online Library → William Kingdon Clifford → Elements of dynamic; an introduction to the study of motion and rest in solid and fluid bodies (Volume 2) → online text (page 7 of 9)

Font size

and Dynamic or Energetic of elastic bodies are grouped together as the

science of Elasticity.

These divisions may be represented by the following scheme :

Science of Motion.

Kinematic, -v /- Particles (Translations),

Dynamic, or | J Static, L of -1 Kigid Bodies (Twists),

Energetic, viz. J ( Kinetic, J ( Elastic Bodies (Strains).

Translations.

DBF. If two bodies A and B are in motion, the motion of B is said

to be compounded of the motion of B relative to A, and the motion of A.

PKOP. Translations represented by the sides of a parallelogram

compound together into a translation represented by the diagonal.

DBF. A vector is a quantity having magnitude and direction. A

translation is a particular kind of vector, and the composition of

translations is equivalent to their addition as vectors ; it satisfies the law

DBF. Uniform rectilinear motion is that in which equal spaces are

traversed in equal times.

Its equation is

p = a + pt.

PROP. Two uniform rectilinear motions compound into a uniform

rectilinear motion.

Harmonic Motion.

DBF. Uniform motion in a circle is that in which equal arcs are

traversed in equal times.

DBF. If a point P move uniformly in a circle, and a perpendicular

PM be always drawn from it to a fixed diameter A A' of the circle, the foot

M of the perpendicular will oscillate to and fro in the diameter; this

motion of the point M is called a Simple Harmonic Motion.

Its equation is

p = acos(nt-e).

DEF. The radius of the circle is called the amplitude of the s. H. M.

DBF. The time which P takes to go once round the circle is called

the period of the s. H. M.

DBF. The circular measure of the arc described by P from the era of

reckoning till it came to the positive end of the diameter A A' is called the

epoch of the s. H. M.

SYLLABUS OF LECTURES ON MOTION. 87

DEF. The portion of the whole period which has elapsed since

the point M last passed through its middle position in the positive

direction is called the phase of the s. H. M.

PEOP. Two s. H. M. of the same period compound into a s. H. M. of

that period.

The construction here made use of for compounding two s. H. M.

is exemplified in the Tidal Clock of Sir W. Thomson. The clock has

two hands whose lengths are proportional to the solar and lunar tides

respectively, while their periods of revolution are made equal to the

periods of these tides. A jointed parallelogram is constructed, having the

hands of the clock for two sides; the height of that extremity of the

parallelogram which is furthest from the centre will then be proportional

to the height of the compound tide. For this purpose a series of

horizontal strings at equal distances are stretched across the face of

the clock, and the height is read off by running the eye along these to a

vertical scale of feet in the middle.

DEF. The curve described by a point which has a uniform rectilinear

motion compounded with a s. H. M. perpendicular to it is called a harmonic

curve.

The composition of s. H. M. of different periods in the same line may

be represented graphically by the super-position of harmonic curves ;

i. e. by drawing a curve whose height at any point is the sum of their

heights.

PEOP. Any s. H. M. may be resolved into two in the same line, differing

in phase by a quarter period, and one of them having any given epoch.

PEOP. s. H. M. on any number of different lines, having the same

period and phase, compound into one having that period and phase.

PEOP. Two s. H. M. on different lines, having the same period, but

differing in phase by , compound into harmonic motion in an ellipse,

(viz. an orthogonal projection of circular motion).

Its equation is

p=acos (nt- e) + /3sin(nt-e).

PEOP. Any number of s. H. M. having the same period compound into

harmonic motion in an ellipse.

Two harmonic motions in different directions and with different

periods produce a resultant which is best studied by wrapping round a

cylinder of suitable size paper on which is traced a harmonic curve.

The curve thus drawn on the cylinder may then be constructed in

wire, and when turned round the axis of the cylinder will represent

to an eye at a sufficient distance the curve of compound harmonic motion

for varying values of the difference of phase of the simple motions. The

simplest case is that in which the circumference of the cylinder is equal

88 DYNAMIC.

to the length of a wave of the harmonic curve; here the periods are

equal, and the curve traced on the cylinder is merely an ellipse.

The same result is produced by turning the cylinder round its axis

while a pencil moves with simple harmonic motion up and down a

generating line.

DEF. A motion which exactly repeats itself in the same place after a

certain interval of time is called a periodic motion.

The resultant of any number of simple harmonic motions whose

periods are commensurable is a periodic motion, its period being the least

common multiple of their periods.

Fourier's Theorem. Every rectilinear periodic motion of period P may

be resolved into a series of simple harmonic motions whose periods are

P,iP, JP, etc.

Let <f> (t) be the distance of the moving point from a fixed point on the

line at a tune t, then the periodicity of the motion is expressed by the

fact that (j> (t + P) = <p(t), whatever t is. And the theorem asserts that in

this case the quantities a, b can always be found so as to make true the

following equation, where

+ a x sin 6 + a 2 sin 26 + ...

The amplitudes and epochs of the several harmonic components may

be represented as follows. Let a vertical cylinder revolve about its axis,

while a pencil moves up and down one of its generating lines, so as to

trace out a curve on the cylinder. If the motion of the pencil is periodic,

and has a period equal to that of the cylinder or any exact multiple of it,

this curve will return into itself and be a finite curve on the cylinder.

Now let the pencil have the given periodic motion which it is required

to resolve into harmonic constituents. When the cylinder revolves once

in the period P, let the curve described be called C 1 ; when it revolves

twice in that period let the curve be called C 2 ; when it revolves m times,

let this curve be called G m . And let a circle be drawn on the cylinder

whose height is the mean height of the curve C 1 ; this will be called the

mean circle.

If a plane be drawn through the axis of the cylinder, any curve traced

on the cylinder may be orthogonally projected on that plane. It is

necessary now to define the area between this projection and the line

in which the plane is cut by the plane of the mean circle. Let AB

be this line, and let PMQNP be the projection, where PMQ is pro-

jected from the near half of the cylinder, and QNP from the further

half. Then for the near half, the area APM which is below AB must be

considered negative, and the area MQB which is above it, positive. For

SYLLABUS OF LECTURES ON MOTION.

89

the further half, QNB must be considered negative, and NPA positive.

Thus the area is

- APM + MBQ - NBQ + APN

= MPN + MNQ = MPNQ.

The same rule is to be applied when the curve cuts itself or the line

AB any number of times. Now it is found that for every closed curve

traced on a cylinder, there is one plane through the axis such that the

area of the projection on it is zero ; and that for the plane at right angles

to it the area is the greatest possible ; while for an intermediate plane

the area varies as the sine of the angle which it makes with the zero

plane. It is thus possible to draw an ellipse upon the cylinder, the area

of whose projection upon any plane whatever through the axis shall

be the same as that of a given closed curve. Let the ellipse E l have

the same projected area as the curve C v E^ half that of the curve C 2 , E m

one-mth that of the curve C m , and so on. If, while the cylinder revolves

once on its axis during the period P, the pencil be made to follow the

ellipse E v always remaining in the same vertical line, it will have a

s. H. M. with the period P. If while the cylinder revolves m times

during the period P, the pencil be made to follow the ellipse E m> it will

have a s. H. M. with the period - P. These motions are the harmonic

m

components of the given periodic motion ; and that motion may be

reproduced by compounding them all together*.

Parabolic Motion.

PROP. If rectilinear motion in which the space passed over from the

beginning is proportional to the square of the time occupied, be com-

pounded with the rectilinear motion, the resultant will be motion in a

parabola.

Its equation is p = a + fit + yt*.

[* Cf. Dynamic, p. 37, where it is said a proof of Fourier's Theorem will be given in the

Appendix.]

90

DYNAMIC.

Velocity.

DBF. If a body is in uniform rectilinear motion, and travels v centi-

metres in every second, the body is said to have at every instant a velocity

of v centimetres per second, or simply a velocity v.

DBF. If a body undergo a translation whereby a point of it is carried

in any manner by any path from A to B in t seconds, the body is said to have

a mean velocity in that interval of t seconds.

t

The two quantities here defined have magnitude and direction ; they

are vectors. A velocity may be expressed in terms of other units than

centimetres per second ; in feet or miles per second, leagues per hour, etc. ;

but when expressed as a number of centimetres per second, it is said to be

given in absolute measure. In uniform rectilinear motion the mean

velocity is the same in any interval whatever, and is equal to the

instantaneous velocity at any instant ; but the latter is a property which

the body possesses at an epoch or point of tune, while the former is a

fact relating to its motion during an interval.

DBF. If any rectilinear motion of a point be compounded with a

uniform motion of unit velocity at right angles to it, the curve traced out

by the point is called the curve of positions for that rectilinear motion.

Lemma. PT is the tangent at a point P of a circle. Any angle

being proposed, it is always possible to take a point Q on the circle

so near to P that the chord of every arc pq included in PQ shall make

with the tangent PT an angle less than the proposed angle.

Let C be the centre of the circle ; make PCQ less than the proposed

angle, and draw CM perpendicular to pq. Then PCM is the angle which

the chord pq makes with PT, and it is always less than PCQ, therefore

less than the proposed angle. Q.E.D.

DBF. E, P are points on any curve, Q moves from E along the curve

towards P; if when any angle is proposed, it is always possible to take Q

so near to P that the chord of every arc pq included in PQ shall make

SYLLABUS OF LECTURES ON MOTION.

91

with a certain line TP an angle less than the proposed angle ; then the

curve is said to have TP for a tangent at the point P.

If S is a point on the other side of P and if Q moves from S towards-

P, there may be another line PT' such that an arc PQ may always be

taken in which no chord shall be inclined to PT' so much as by a

proposed angle. In this case we may speak of TP as the tangent up to

P and of PT as the tangent ore from P. When TPT' is a straight line,

the curve is said to be elementally straight or to have the property of

elemental straightness at the point P; for the more it is magnified, the

more will a portion containing P of given length in the magnified figure

approach to the straight line TPT' in shape and position. For this,,

three conditions are necessary; there must be a tangent up to P, a

tangent on from P, and these tangents must be in one straight line.

PKOP. If the curve of positions of a rectilinear motion has a tangent

at a point P, then it is possible to choose an interval ending at the

instant corresponding to the point P so that the mean velocity in that

interval (and in all intervals included in it) shall differ less than by a

given amount from a certain quantity.

Let QP be a portion of the curve of positions, PT the tangent

at P; QN, PM parallel to the (vertical) rectilinear motion considered,,

and perpendicular to the (horizontal) uniform motion with which it

is compounded; QR perpendicular to PM. Since the uniform motion

has unit velocity, the number of units of length in NM is equal to-

92

DYNAMIC.

the number of seconds in which the body has performed the vertical

73 p

motion EP. and the mean velocity in the interval NM is therefore ^^7 .

NM

Now take AB a horizontal line equal to the unit of length, and draw

AC, AD parallel to PT, PQ, meeting the vertical line through B in C, D.

Then BD represents the mean velocity in the interval NM. Similarly if

pq be any arc included in PQ (pm, qn perpendicular to NM), and if we

draw Ad parallel to the chord pq, Bd will represent the mean velocity in

the interval nm. Now it is possible by hypothesis to choose Q so near to

P that the angle QPT, which is equal to CAD, shall be less than any

proposed angle ; and that the angle which any chord pq makes with PT,

which angle is equal to CAd, shall be less than the proposed angle.

Therefore it is possible so to choose N that for every interval included in

NM the length Cd shall be less than a proposed amount ; or so that the

mean velocity shall differ from the velocity represented by BC by a

quantity less than the proposed amount. Q. E.D. The quantity Be or

MP/TM is then called the instantaneous velocity of the rectilinear motion

at the instant M.

DEF. Let Q, P be successive positions of a moving point, and let BD

represent the mean velocity during an interval included in the passage

SYLLABUS OF LECTURES ON MOTION.

93

from Q to P ; then if it is always possible to find Q so near to P that for

all intervals between Q and P the distance DC from D to a fixed point C

shall be less than a proposed length, the point at the instant of arriving

at P is said to have an instantaneous velocity B G in magnitude and

direction.

PEOP. If a moving point has a velocity, the curve described has a

tangent in the same direction; and if a length equal to the arc EQ

be measured off on a straight line as Q moves, this rectilinear motion

will have a velocity whose magnitude is equal to that of Q.

PKOP. If each of two motions has a velocity at a certain instant of

time, the motion compounded of them has a velocity which is compounded

of their velocities by the rule for addition of vectors.

Let AB and AC be the given velocities; complete the parallelogram

ABDC. Let also AB', AC' be the mean velocities during an interval

C'

which ends at the given instant; if the parallelogram AB'D'C' be com-

pleted, we know that AD' is the mean velocity of the resultant motion.

Now the interval may be so chosen that for it and all shorter ones

included in it BB' and CC' are each less than half of any proposed

length; and therefore DD', which is their vector- sum, less than the

proposed length. Consequently AD is the velocity of the resultant

motion at the given instant. Q. E. D.

It is to be noticed that in accordance with our definitions a motion

may have one velocity up to a certain instant and another velocity on

from that instant ; or, as we may say, an arrival and a departure velocity.

Such motions are for mathematical convenience supposed to take place

in the theory of collisions ; but it is believed that they do not occur in

nature, and that the arrival and departure velocities are always identical.

If a point has an arrival and a departure velocity at a given instant

and if they are identical, its motion is said to be elementally uniform;

for if a small portion of the path containing the position of the point

94 DYNAMIC.

at that instant be magnified to a definite length, and the times of traversing

different parts of it be preserved in their proportions, then the smaller

the portion taken, the nearer will the path approach to a straight line

and the motion to uniform motion along it.

PKOP. The velocity in the s. H. M.

p=acos (nte)

is p na sin (nt e),

(when the position-vector of a point is called p, its velocity is denoted

The s. H. M. has a velocity, because its curve of positions has a tangent,

being produced by unrolling an ellipse from a cylinder. Now uniform

circular motion being compounded of two simple harmonic motions, its

velocity is compounded of their velocities by the law of addition of vectors.

Thus the velocity of P is compounded of the velocities of H and N; but

these velocities are respectively perpendicular to the lines CP, CM, and

MP, the vector CP being equal to CM +MP. The velocities are there-

fore proportional to the lengths of these lines, and as the velocity of P is

n . CP along the tangent, the velocities of M and N are n . MP and n . CM

along MC and CN respectively. But a length n. MP =n.AC sin PCM

along M C is equal to - na sin (nt - e). Q. E. D.

PROP. The velocity in the elliptic harmonic motion,

p = acos(nt-e)+f}sin(nt-e),

is p -na sin (nt - e) + nfi cos (nt - e)

=n jacos ( nt-e + y) -h/Ssin ( nt-e + - JV ,

and is therefore proportional to the conjugate diameter.

SYLLABUS OF LECTURES ON MOTION. 95

PEOP. The velocity in the parabolic motion

is

Let t v 2 , t 3 , t be successive values of t, these quantities being there-

fore in ascending order of magnitude ; p v p 2 , p 3 , p the corresponding values

of p. Then the mean velocity in the interval from t z to t s is

Since t 2 and t 3 are intermediate between ^ and t, this vector differs

from /3 + 2yt less than /3 + 27^ does ; that is, less than 2y (t - a ). Now it

is possible so to choose ^ that this shall be shorter than any proposed

length yx ; that is, it is possible to choose an interval ending at t, so that

the mean velocity for every interval included in it differs from /3 + 2yt by

less than a proposed amount. The same thing may be shewn for intervals

beginning at t. Therefore the motion is elementally uniform and has

/3 + 2yt for its velocity. Q. E. D.

PROP. In the motion whose equation is

p=at n

(n a positive integer), the velocity is

p=nat n ~\

With the notation of the previous proposition, the mean velocity in

the interval from t 2 to t 3 is

Since ( 2 and t 3 are intermediate between ^ and t, this quantity differs

from not" 1 less than nat^ 1 does ; that is, less than na(t n ~ l - tj 71 " 1 ), which

by proper choice of ^ can be made less than an assigned quantity. Whence

as before.

(B.)

I. Divisions of the Subject. Dynamic is the doctrine of the cir-

cumstances under which motion takes place. It is preceded by the

description of motion, or Kinematic. The simplest motions are transla-

tions, with no change of aspect or shape ; then motion of rigids, with no

change of shape ; then strains or changes of shape. In Dynamic we

may consider circumstances of actual motion, or circumstances of

possible rest ; these are sometimes called Kinetic and Static ; and these

again may be of particles, rigids, or elastic bodies. In the latter the

theory of fluids is specially distinguished.

II. Translations and their composition. The change of position or

total effect produced by a translation is represented by a line of given

length drawn in a given direction a vector. It is determined by assigning

three quantities. Quantities can only be described B

in words or numbers approximately; the right

way of specifying a quantity (of length, time, or

weight) is by drawing a line which represents it

on a certain scale. A vector may then be

specified by 3 Lines marked off on the same

scale ; or by 2 lines in descriptive geometry.

The composition of vectors arises out of

relative motion. If any carriage or vehicle have

the motion represented by one vector o, and a thing carried have relative

to it the motion represented by another /3, the actual motion of the

thing carried is represented by a third vector y, which is called the

resultant or sum of the other two. This composi-

tion is represented by the sign + , thus a+(i=y, or

AB + BC=AC ; this equation gives the rule of addi-

tion. If two motions be successively undergone by

a body, the resultant change of position is their

sum. In all cases a + /3 = /3+a. Another rule: if

AD = DB,OA + OB = 20D. If AD :DB = q: p,(p + q)OD=p.OA + q.OB.

For triangle, p + q + r. OG=pOA + qOB + rOC, if etc. Hence properties

ABSTRACT OF DYNAMIC.

97

of centre of triangle and tetrahedron. Expression of vector in terms of

3 vectors : ix + jy + kz. Special representation on plane : complex

numbers. Projection of lines on lines and planes.

III. Eepresentation of Motion. Uniform Motion. The motion of a

particle is specified when its' path is laid down and when we are told at

what point of this path it was at each instant

of time. This may be done by a curve whose

abscissae represent the time and ordinates the

distance traversed. In uniform motion equal

distances are traversed in equal times : hence

the distance is always proportional to the time.

This is represented by a straight line. Also by

the equation s = vt. Uniform rectilinear motion is represented by the

vector equation p = a. + (3t ; uniform circular by equation p = a cos nt +

/3 sin nt + y. Period, phase, amplitude, epoch. Two uniform rectilineal

motions compound into uniform rectilinear ; so two circular of same

period and direction.

IV. Harmonic Motion. Definition, amplitude,

period, epoch, phase, equation. Combination of

2 S.H.M. of one period in one line; tidal clock, etc.

Curve of positions; construction by ellipse on cylinder.

Properties of projection (orthogonal). Parallel lines

remain parallel and are all altered in a certain ratio.

Properties of ellipse so derived ; centre, conjugate

diameters, tangents at extremities of these axes.

General projection of uniform circular motion is

elliptic H. M., equation p = a cos nt + /3 sin nt where

a, /3 are conjugate semi-diameters. Any S. H. M. of

same period may be reduced to two sets, each set having same phase, but

phases differing ; each set may be reduced to one S. H. M. ; hence resultant

is always E. H. M. Case of two S. H. M. variation of resultant with varying

difference of phase. Revolving ellipse on cylinder.

Combination of different periods. Note and

octave. Curves of position. Different directions ;

figure of 8. Method by curves on cylinder. Bird's

wing. Periodic motion generally. Statement of

Fourier's Theorem.

V. Parabolic Motion. Definition of parabola.

Proof that abscissa oc (ordinate) 2 . Path of motion

p = a + (it + yt 2 . Combination of Parabolic Motions

with same or different axes.

C.

98 DYNAMIC.

VI. Direction of Motion (Tangents).

Proof that in circle or ellipse TA : TB

= AM : MB, and CA Z = CM . GT. In

parabola AT=AN. Harmonic situation

of 4 points. Tangents at P, P' intersect

on diameter bisecting PP'. In parabola

SP=ST-SG. Solution of problem to

draw parabola from P to Q having given tangent at P and direction of

axis. All this is on assumption that tangency is unaltered by projection,

or tangent regarded as line two of whose intersections coincide.

VII. Velocity. Of uniform rectilinear and circular motion. Velocity

is directed quantity. Assumption that velocity may be compounded as

vectors : true for two uniform rectilinear or circular motions. Velocity of

S. H. M. Kepresentation by curve of positions ; tangent to the curve of

sines. Problem the same as that of tangents. Velocity of parabolic

motion, of elliptic H. M.

VIII. Hodograph. Acceleration. In Harmonic and Parabolic

Motions.

IX. Fluxions. Exact definition of tangent and velocity. Proof of

law of composition of velocity. Tangent of circle. Tangency retained

in projection. Eate of variation of a quantity. Fluxion of at n . Change

of independent variable. Fluxion of/(P, Q), etc.

X. The Inverse Method. Space traversed = area of curve of velocities.

Wallis's integration of of.

XI. Curvature. Pliicker's apparatus for plane. Stoppage of motion

of point or line gives cusp or inflexion. Generally angular velocity -f linear

velocity is curvature-mean and total curvature expression p". Circle of

curvature ; construction for ellipse and parabola. Resolved parts of

acceleration: fluxion of p(=vp') is p=v z p" + vp'. Tortuous curve ; point

line and plane. Equation of helix. No acceleration perpendicular to

osculating plane. Total curvature of tortuous curve. Tortuosity.

XII. Logarithmic Motion. Quantity equally multiplied in equal

times increases at rate proportional to itself. Density of air. Intensity

of light in water, s =ps .p called intrinsic rate. If such a quantity is

science of Elasticity.

These divisions may be represented by the following scheme :

Science of Motion.

Kinematic, -v /- Particles (Translations),

Dynamic, or | J Static, L of -1 Kigid Bodies (Twists),

Energetic, viz. J ( Kinetic, J ( Elastic Bodies (Strains).

Translations.

DBF. If two bodies A and B are in motion, the motion of B is said

to be compounded of the motion of B relative to A, and the motion of A.

PKOP. Translations represented by the sides of a parallelogram

compound together into a translation represented by the diagonal.

DBF. A vector is a quantity having magnitude and direction. A

translation is a particular kind of vector, and the composition of

translations is equivalent to their addition as vectors ; it satisfies the law

DBF. Uniform rectilinear motion is that in which equal spaces are

traversed in equal times.

Its equation is

p = a + pt.

PROP. Two uniform rectilinear motions compound into a uniform

rectilinear motion.

Harmonic Motion.

DBF. Uniform motion in a circle is that in which equal arcs are

traversed in equal times.

DBF. If a point P move uniformly in a circle, and a perpendicular

PM be always drawn from it to a fixed diameter A A' of the circle, the foot

M of the perpendicular will oscillate to and fro in the diameter; this

motion of the point M is called a Simple Harmonic Motion.

Its equation is

p = acos(nt-e).

DEF. The radius of the circle is called the amplitude of the s. H. M.

DBF. The time which P takes to go once round the circle is called

the period of the s. H. M.

DBF. The circular measure of the arc described by P from the era of

reckoning till it came to the positive end of the diameter A A' is called the

epoch of the s. H. M.

SYLLABUS OF LECTURES ON MOTION. 87

DEF. The portion of the whole period which has elapsed since

the point M last passed through its middle position in the positive

direction is called the phase of the s. H. M.

PEOP. Two s. H. M. of the same period compound into a s. H. M. of

that period.

The construction here made use of for compounding two s. H. M.

is exemplified in the Tidal Clock of Sir W. Thomson. The clock has

two hands whose lengths are proportional to the solar and lunar tides

respectively, while their periods of revolution are made equal to the

periods of these tides. A jointed parallelogram is constructed, having the

hands of the clock for two sides; the height of that extremity of the

parallelogram which is furthest from the centre will then be proportional

to the height of the compound tide. For this purpose a series of

horizontal strings at equal distances are stretched across the face of

the clock, and the height is read off by running the eye along these to a

vertical scale of feet in the middle.

DEF. The curve described by a point which has a uniform rectilinear

motion compounded with a s. H. M. perpendicular to it is called a harmonic

curve.

The composition of s. H. M. of different periods in the same line may

be represented graphically by the super-position of harmonic curves ;

i. e. by drawing a curve whose height at any point is the sum of their

heights.

PEOP. Any s. H. M. may be resolved into two in the same line, differing

in phase by a quarter period, and one of them having any given epoch.

PEOP. s. H. M. on any number of different lines, having the same

period and phase, compound into one having that period and phase.

PEOP. Two s. H. M. on different lines, having the same period, but

differing in phase by , compound into harmonic motion in an ellipse,

(viz. an orthogonal projection of circular motion).

Its equation is

p=acos (nt- e) + /3sin(nt-e).

PEOP. Any number of s. H. M. having the same period compound into

harmonic motion in an ellipse.

Two harmonic motions in different directions and with different

periods produce a resultant which is best studied by wrapping round a

cylinder of suitable size paper on which is traced a harmonic curve.

The curve thus drawn on the cylinder may then be constructed in

wire, and when turned round the axis of the cylinder will represent

to an eye at a sufficient distance the curve of compound harmonic motion

for varying values of the difference of phase of the simple motions. The

simplest case is that in which the circumference of the cylinder is equal

88 DYNAMIC.

to the length of a wave of the harmonic curve; here the periods are

equal, and the curve traced on the cylinder is merely an ellipse.

The same result is produced by turning the cylinder round its axis

while a pencil moves with simple harmonic motion up and down a

generating line.

DEF. A motion which exactly repeats itself in the same place after a

certain interval of time is called a periodic motion.

The resultant of any number of simple harmonic motions whose

periods are commensurable is a periodic motion, its period being the least

common multiple of their periods.

Fourier's Theorem. Every rectilinear periodic motion of period P may

be resolved into a series of simple harmonic motions whose periods are

P,iP, JP, etc.

Let <f> (t) be the distance of the moving point from a fixed point on the

line at a tune t, then the periodicity of the motion is expressed by the

fact that (j> (t + P) = <p(t), whatever t is. And the theorem asserts that in

this case the quantities a, b can always be found so as to make true the

following equation, where

+ a x sin 6 + a 2 sin 26 + ...

The amplitudes and epochs of the several harmonic components may

be represented as follows. Let a vertical cylinder revolve about its axis,

while a pencil moves up and down one of its generating lines, so as to

trace out a curve on the cylinder. If the motion of the pencil is periodic,

and has a period equal to that of the cylinder or any exact multiple of it,

this curve will return into itself and be a finite curve on the cylinder.

Now let the pencil have the given periodic motion which it is required

to resolve into harmonic constituents. When the cylinder revolves once

in the period P, let the curve described be called C 1 ; when it revolves

twice in that period let the curve be called C 2 ; when it revolves m times,

let this curve be called G m . And let a circle be drawn on the cylinder

whose height is the mean height of the curve C 1 ; this will be called the

mean circle.

If a plane be drawn through the axis of the cylinder, any curve traced

on the cylinder may be orthogonally projected on that plane. It is

necessary now to define the area between this projection and the line

in which the plane is cut by the plane of the mean circle. Let AB

be this line, and let PMQNP be the projection, where PMQ is pro-

jected from the near half of the cylinder, and QNP from the further

half. Then for the near half, the area APM which is below AB must be

considered negative, and the area MQB which is above it, positive. For

SYLLABUS OF LECTURES ON MOTION.

89

the further half, QNB must be considered negative, and NPA positive.

Thus the area is

- APM + MBQ - NBQ + APN

= MPN + MNQ = MPNQ.

The same rule is to be applied when the curve cuts itself or the line

AB any number of times. Now it is found that for every closed curve

traced on a cylinder, there is one plane through the axis such that the

area of the projection on it is zero ; and that for the plane at right angles

to it the area is the greatest possible ; while for an intermediate plane

the area varies as the sine of the angle which it makes with the zero

plane. It is thus possible to draw an ellipse upon the cylinder, the area

of whose projection upon any plane whatever through the axis shall

be the same as that of a given closed curve. Let the ellipse E l have

the same projected area as the curve C v E^ half that of the curve C 2 , E m

one-mth that of the curve C m , and so on. If, while the cylinder revolves

once on its axis during the period P, the pencil be made to follow the

ellipse E v always remaining in the same vertical line, it will have a

s. H. M. with the period P. If while the cylinder revolves m times

during the period P, the pencil be made to follow the ellipse E m> it will

have a s. H. M. with the period - P. These motions are the harmonic

m

components of the given periodic motion ; and that motion may be

reproduced by compounding them all together*.

Parabolic Motion.

PROP. If rectilinear motion in which the space passed over from the

beginning is proportional to the square of the time occupied, be com-

pounded with the rectilinear motion, the resultant will be motion in a

parabola.

Its equation is p = a + fit + yt*.

[* Cf. Dynamic, p. 37, where it is said a proof of Fourier's Theorem will be given in the

Appendix.]

90

DYNAMIC.

Velocity.

DBF. If a body is in uniform rectilinear motion, and travels v centi-

metres in every second, the body is said to have at every instant a velocity

of v centimetres per second, or simply a velocity v.

DBF. If a body undergo a translation whereby a point of it is carried

in any manner by any path from A to B in t seconds, the body is said to have

a mean velocity in that interval of t seconds.

t

The two quantities here defined have magnitude and direction ; they

are vectors. A velocity may be expressed in terms of other units than

centimetres per second ; in feet or miles per second, leagues per hour, etc. ;

but when expressed as a number of centimetres per second, it is said to be

given in absolute measure. In uniform rectilinear motion the mean

velocity is the same in any interval whatever, and is equal to the

instantaneous velocity at any instant ; but the latter is a property which

the body possesses at an epoch or point of tune, while the former is a

fact relating to its motion during an interval.

DBF. If any rectilinear motion of a point be compounded with a

uniform motion of unit velocity at right angles to it, the curve traced out

by the point is called the curve of positions for that rectilinear motion.

Lemma. PT is the tangent at a point P of a circle. Any angle

being proposed, it is always possible to take a point Q on the circle

so near to P that the chord of every arc pq included in PQ shall make

with the tangent PT an angle less than the proposed angle.

Let C be the centre of the circle ; make PCQ less than the proposed

angle, and draw CM perpendicular to pq. Then PCM is the angle which

the chord pq makes with PT, and it is always less than PCQ, therefore

less than the proposed angle. Q.E.D.

DBF. E, P are points on any curve, Q moves from E along the curve

towards P; if when any angle is proposed, it is always possible to take Q

so near to P that the chord of every arc pq included in PQ shall make

SYLLABUS OF LECTURES ON MOTION.

91

with a certain line TP an angle less than the proposed angle ; then the

curve is said to have TP for a tangent at the point P.

If S is a point on the other side of P and if Q moves from S towards-

P, there may be another line PT' such that an arc PQ may always be

taken in which no chord shall be inclined to PT' so much as by a

proposed angle. In this case we may speak of TP as the tangent up to

P and of PT as the tangent ore from P. When TPT' is a straight line,

the curve is said to be elementally straight or to have the property of

elemental straightness at the point P; for the more it is magnified, the

more will a portion containing P of given length in the magnified figure

approach to the straight line TPT' in shape and position. For this,,

three conditions are necessary; there must be a tangent up to P, a

tangent on from P, and these tangents must be in one straight line.

PKOP. If the curve of positions of a rectilinear motion has a tangent

at a point P, then it is possible to choose an interval ending at the

instant corresponding to the point P so that the mean velocity in that

interval (and in all intervals included in it) shall differ less than by a

given amount from a certain quantity.

Let QP be a portion of the curve of positions, PT the tangent

at P; QN, PM parallel to the (vertical) rectilinear motion considered,,

and perpendicular to the (horizontal) uniform motion with which it

is compounded; QR perpendicular to PM. Since the uniform motion

has unit velocity, the number of units of length in NM is equal to-

92

DYNAMIC.

the number of seconds in which the body has performed the vertical

73 p

motion EP. and the mean velocity in the interval NM is therefore ^^7 .

NM

Now take AB a horizontal line equal to the unit of length, and draw

AC, AD parallel to PT, PQ, meeting the vertical line through B in C, D.

Then BD represents the mean velocity in the interval NM. Similarly if

pq be any arc included in PQ (pm, qn perpendicular to NM), and if we

draw Ad parallel to the chord pq, Bd will represent the mean velocity in

the interval nm. Now it is possible by hypothesis to choose Q so near to

P that the angle QPT, which is equal to CAD, shall be less than any

proposed angle ; and that the angle which any chord pq makes with PT,

which angle is equal to CAd, shall be less than the proposed angle.

Therefore it is possible so to choose N that for every interval included in

NM the length Cd shall be less than a proposed amount ; or so that the

mean velocity shall differ from the velocity represented by BC by a

quantity less than the proposed amount. Q. E.D. The quantity Be or

MP/TM is then called the instantaneous velocity of the rectilinear motion

at the instant M.

DEF. Let Q, P be successive positions of a moving point, and let BD

represent the mean velocity during an interval included in the passage

SYLLABUS OF LECTURES ON MOTION.

93

from Q to P ; then if it is always possible to find Q so near to P that for

all intervals between Q and P the distance DC from D to a fixed point C

shall be less than a proposed length, the point at the instant of arriving

at P is said to have an instantaneous velocity B G in magnitude and

direction.

PEOP. If a moving point has a velocity, the curve described has a

tangent in the same direction; and if a length equal to the arc EQ

be measured off on a straight line as Q moves, this rectilinear motion

will have a velocity whose magnitude is equal to that of Q.

PKOP. If each of two motions has a velocity at a certain instant of

time, the motion compounded of them has a velocity which is compounded

of their velocities by the rule for addition of vectors.

Let AB and AC be the given velocities; complete the parallelogram

ABDC. Let also AB', AC' be the mean velocities during an interval

C'

which ends at the given instant; if the parallelogram AB'D'C' be com-

pleted, we know that AD' is the mean velocity of the resultant motion.

Now the interval may be so chosen that for it and all shorter ones

included in it BB' and CC' are each less than half of any proposed

length; and therefore DD', which is their vector- sum, less than the

proposed length. Consequently AD is the velocity of the resultant

motion at the given instant. Q. E. D.

It is to be noticed that in accordance with our definitions a motion

may have one velocity up to a certain instant and another velocity on

from that instant ; or, as we may say, an arrival and a departure velocity.

Such motions are for mathematical convenience supposed to take place

in the theory of collisions ; but it is believed that they do not occur in

nature, and that the arrival and departure velocities are always identical.

If a point has an arrival and a departure velocity at a given instant

and if they are identical, its motion is said to be elementally uniform;

for if a small portion of the path containing the position of the point

94 DYNAMIC.

at that instant be magnified to a definite length, and the times of traversing

different parts of it be preserved in their proportions, then the smaller

the portion taken, the nearer will the path approach to a straight line

and the motion to uniform motion along it.

PKOP. The velocity in the s. H. M.

p=acos (nte)

is p na sin (nt e),

(when the position-vector of a point is called p, its velocity is denoted

The s. H. M. has a velocity, because its curve of positions has a tangent,

being produced by unrolling an ellipse from a cylinder. Now uniform

circular motion being compounded of two simple harmonic motions, its

velocity is compounded of their velocities by the law of addition of vectors.

Thus the velocity of P is compounded of the velocities of H and N; but

these velocities are respectively perpendicular to the lines CP, CM, and

MP, the vector CP being equal to CM +MP. The velocities are there-

fore proportional to the lengths of these lines, and as the velocity of P is

n . CP along the tangent, the velocities of M and N are n . MP and n . CM

along MC and CN respectively. But a length n. MP =n.AC sin PCM

along M C is equal to - na sin (nt - e). Q. E. D.

PROP. The velocity in the elliptic harmonic motion,

p = acos(nt-e)+f}sin(nt-e),

is p -na sin (nt - e) + nfi cos (nt - e)

=n jacos ( nt-e + y) -h/Ssin ( nt-e + - JV ,

and is therefore proportional to the conjugate diameter.

SYLLABUS OF LECTURES ON MOTION. 95

PEOP. The velocity in the parabolic motion

is

Let t v 2 , t 3 , t be successive values of t, these quantities being there-

fore in ascending order of magnitude ; p v p 2 , p 3 , p the corresponding values

of p. Then the mean velocity in the interval from t z to t s is

Since t 2 and t 3 are intermediate between ^ and t, this vector differs

from /3 + 2yt less than /3 + 27^ does ; that is, less than 2y (t - a ). Now it

is possible so to choose ^ that this shall be shorter than any proposed

length yx ; that is, it is possible to choose an interval ending at t, so that

the mean velocity for every interval included in it differs from /3 + 2yt by

less than a proposed amount. The same thing may be shewn for intervals

beginning at t. Therefore the motion is elementally uniform and has

/3 + 2yt for its velocity. Q. E. D.

PROP. In the motion whose equation is

p=at n

(n a positive integer), the velocity is

p=nat n ~\

With the notation of the previous proposition, the mean velocity in

the interval from t 2 to t 3 is

Since ( 2 and t 3 are intermediate between ^ and t, this quantity differs

from not" 1 less than nat^ 1 does ; that is, less than na(t n ~ l - tj 71 " 1 ), which

by proper choice of ^ can be made less than an assigned quantity. Whence

as before.

(B.)

I. Divisions of the Subject. Dynamic is the doctrine of the cir-

cumstances under which motion takes place. It is preceded by the

description of motion, or Kinematic. The simplest motions are transla-

tions, with no change of aspect or shape ; then motion of rigids, with no

change of shape ; then strains or changes of shape. In Dynamic we

may consider circumstances of actual motion, or circumstances of

possible rest ; these are sometimes called Kinetic and Static ; and these

again may be of particles, rigids, or elastic bodies. In the latter the

theory of fluids is specially distinguished.

II. Translations and their composition. The change of position or

total effect produced by a translation is represented by a line of given

length drawn in a given direction a vector. It is determined by assigning

three quantities. Quantities can only be described B

in words or numbers approximately; the right

way of specifying a quantity (of length, time, or

weight) is by drawing a line which represents it

on a certain scale. A vector may then be

specified by 3 Lines marked off on the same

scale ; or by 2 lines in descriptive geometry.

The composition of vectors arises out of

relative motion. If any carriage or vehicle have

the motion represented by one vector o, and a thing carried have relative

to it the motion represented by another /3, the actual motion of the

thing carried is represented by a third vector y, which is called the

resultant or sum of the other two. This composi-

tion is represented by the sign + , thus a+(i=y, or

AB + BC=AC ; this equation gives the rule of addi-

tion. If two motions be successively undergone by

a body, the resultant change of position is their

sum. In all cases a + /3 = /3+a. Another rule: if

AD = DB,OA + OB = 20D. If AD :DB = q: p,(p + q)OD=p.OA + q.OB.

For triangle, p + q + r. OG=pOA + qOB + rOC, if etc. Hence properties

ABSTRACT OF DYNAMIC.

97

of centre of triangle and tetrahedron. Expression of vector in terms of

3 vectors : ix + jy + kz. Special representation on plane : complex

numbers. Projection of lines on lines and planes.

III. Eepresentation of Motion. Uniform Motion. The motion of a

particle is specified when its' path is laid down and when we are told at

what point of this path it was at each instant

of time. This may be done by a curve whose

abscissae represent the time and ordinates the

distance traversed. In uniform motion equal

distances are traversed in equal times : hence

the distance is always proportional to the time.

This is represented by a straight line. Also by

the equation s = vt. Uniform rectilinear motion is represented by the

vector equation p = a. + (3t ; uniform circular by equation p = a cos nt +

/3 sin nt + y. Period, phase, amplitude, epoch. Two uniform rectilineal

motions compound into uniform rectilinear ; so two circular of same

period and direction.

IV. Harmonic Motion. Definition, amplitude,

period, epoch, phase, equation. Combination of

2 S.H.M. of one period in one line; tidal clock, etc.

Curve of positions; construction by ellipse on cylinder.

Properties of projection (orthogonal). Parallel lines

remain parallel and are all altered in a certain ratio.

Properties of ellipse so derived ; centre, conjugate

diameters, tangents at extremities of these axes.

General projection of uniform circular motion is

elliptic H. M., equation p = a cos nt + /3 sin nt where

a, /3 are conjugate semi-diameters. Any S. H. M. of

same period may be reduced to two sets, each set having same phase, but

phases differing ; each set may be reduced to one S. H. M. ; hence resultant

is always E. H. M. Case of two S. H. M. variation of resultant with varying

difference of phase. Revolving ellipse on cylinder.

Combination of different periods. Note and

octave. Curves of position. Different directions ;

figure of 8. Method by curves on cylinder. Bird's

wing. Periodic motion generally. Statement of

Fourier's Theorem.

V. Parabolic Motion. Definition of parabola.

Proof that abscissa oc (ordinate) 2 . Path of motion

p = a + (it + yt 2 . Combination of Parabolic Motions

with same or different axes.

C.

98 DYNAMIC.

VI. Direction of Motion (Tangents).

Proof that in circle or ellipse TA : TB

= AM : MB, and CA Z = CM . GT. In

parabola AT=AN. Harmonic situation

of 4 points. Tangents at P, P' intersect

on diameter bisecting PP'. In parabola

SP=ST-SG. Solution of problem to

draw parabola from P to Q having given tangent at P and direction of

axis. All this is on assumption that tangency is unaltered by projection,

or tangent regarded as line two of whose intersections coincide.

VII. Velocity. Of uniform rectilinear and circular motion. Velocity

is directed quantity. Assumption that velocity may be compounded as

vectors : true for two uniform rectilinear or circular motions. Velocity of

S. H. M. Kepresentation by curve of positions ; tangent to the curve of

sines. Problem the same as that of tangents. Velocity of parabolic

motion, of elliptic H. M.

VIII. Hodograph. Acceleration. In Harmonic and Parabolic

Motions.

IX. Fluxions. Exact definition of tangent and velocity. Proof of

law of composition of velocity. Tangent of circle. Tangency retained

in projection. Eate of variation of a quantity. Fluxion of at n . Change

of independent variable. Fluxion of/(P, Q), etc.

X. The Inverse Method. Space traversed = area of curve of velocities.

Wallis's integration of of.

XI. Curvature. Pliicker's apparatus for plane. Stoppage of motion

of point or line gives cusp or inflexion. Generally angular velocity -f linear

velocity is curvature-mean and total curvature expression p". Circle of

curvature ; construction for ellipse and parabola. Resolved parts of

acceleration: fluxion of p(=vp') is p=v z p" + vp'. Tortuous curve ; point

line and plane. Equation of helix. No acceleration perpendicular to

osculating plane. Total curvature of tortuous curve. Tortuosity.

XII. Logarithmic Motion. Quantity equally multiplied in equal

times increases at rate proportional to itself. Density of air. Intensity

of light in water, s =ps .p called intrinsic rate. If such a quantity is