(b43i
IC 28 19U
GEOMETRIC PROPERTIES COMPLETELY
CHARACTERIZING ALL THE CURVES
IN A PLANE ALONG WHICH THE CON-
STRAINED MOTION IS SUCH THAT THE
PRESSURE IS PROPORTIONAL TO THE
NORMAL COMPONENT OF THE ACTING
FORCE.
BY
SARAH ELIZABETH CRONIN
f o-ftu-b;
' DNIVSRSXXy
Submitted in Partial Fulfilment of the Requirements for
THE Degree of Doctor of Philosophy, in the Faculty
OF Pure Science, Columbia University
PRESS OF
THE NEW ERA PRINTING COMPANY
LANCASTER, PA.
1917
GEOMETRIC PROPERTIES COMPLETELY
CHARACTERIZING ALL THE CURVES
IN A PLANE ALONG WHICH THE CON-
STRAINED MOTION IS SUCH THAT THE
PRESSURE IS PROPORTIONAL TO THE
NORMAL COMPONENT OF THE ACTING
FORCE.
BY
SARAH ELIZABETH CRONIN
Submitted in Partial Fulfilment of the Requirements for
THE Degree of Doctor of Philosophy, in the Faculty
OF Pure Science, Columbia University
PRESS OF
THE NEW ERA PRINTING COMPANY
LANCASTER, PA.
1917
CONTENTS.
Introduction.
Chapter I. The Differential Equation of the System 1
Chapter II. Geometric Properties of the System , 4
Section 1. Osculating Conies 4
2. Curvature of the Conic determined by Theorem
1 5
3. Second Center of Curvature of the Conic deter-
mined by Theorem 1 5
4. Tangents to the Parabola determined by The-
orem III 6
5. Center of the Conic determined by Theorem I . . 7
6. Hyperosculating Parabolas 8
7. Inverse Curve 10
8. Circles of Curvature Hyperosculating the Conies
determined by Theorem 1 10
9. Mid-point of the Chord of the Inverse Curve
cut from the Normal to the Lineal Element 11
10. Osculating Circles having Five-point Contact . . 12
Chapter III. Conversion of the Properties stated in The-
orems I-IX 14
Section 1. Conversion of Property 1 14
II 14
III 15
IV 16
V 17
VI 18
VII 20
VIII 21
IX 23
Chapter IV. Complete Characterization 25
Section 1. Relations existing among the Arbitrary Func-
iii
*\ i'^ y^ i\ (-x ■ -
2.
3.
4.
5.
6.
7.
8.
9.
IV CONTENTS.
tions entering into the Differential Equation of
the System of Curves having Properties I-IX
if it is to be of the Special Form given in Chapter
1 25
2. Geometric Interpretations of the Relations of
Section 1 27
INTRODUCTION.
In his " Differential-Geometric Aspects of Dynamics,"*
Kasner gives a set of five geometric properties which completely
characterize a system of curves in a plane field of force {(p, yp),
along which the constrained motion of a particle is such that
the pressure exerted by the particle on the curve is a given
multiple of the normal component of the acting force. The
defining equation of the system is P = kN, when P is the pres-
sure, N the normal component of the force, and k a given con-
stant.
Denoting the tangential component of the force by T, the
equations of motion are
where
-=T P^-'-N^kN
^IT+Y" VrrP' y" '
If at the initial time to the particle is at the point (ar©, yo)
moving in the direction yo' with the speed Vq, the path of the
particle is uniquely determined. By varying the initial direc-
tion 2/0' and the initial speed Vq, we obtain 00 2 curves of the
system through the given point (xo, yo). Since the path of the
particle is determined by taking any one of its 00 ^ points as the
initial point, the system contains in all 00 ^ curves.
The differential equation of the system in intrinsic form,
obtained by eliminating v between dv/dt = T and v^jr = (A + l)N,
is _
Nr,= {n+l)T-rN, (1)
where
The equivalent equation in Cartesian coordinates is of the
third order and represents, for every value of k, a triply infinite
* Princeton Colloquium Lectures, page 91.
VI INTRODUCTION.
system denoted by Sk- The triply infinite systems of trajectories,
brachistochrones, catenaries and velocity curves are but special
cases of this more general system, being obtained from it for
particular values of k as follows:
Sq, the system of trajectories.
Si, the system of brachistochrones,
Si, the system of catenaries,
(Soo, the system of velocity curves.
In a given field of force, there are oo^ of these triply infinite
systems corresponding to the oo^ values of k. The elimination
of k from the differential equation (1) gives the differential
equation of the quadruply infinite system obtained by com-
bining all the systems Sk in the given field. The intrinsic form
of this differential equation is
where
Nts + rN T
0,
It is the purpose of the present paper to obtain a complete
characterization of this oo^ system of curves. The differential
equation of the system in Cartesian form is obtained in Chapter I.
Geometric properties of the system are derived in Chapter II.
At any point (x, y) of the plane, there are oo ^ curves of the family
having a given direction y' and a given radius of curvature r.
If the osculating conic of each of these curves is constructed, the
locus of the centers of these conies is a conic passing through
the given point in the given direction. Applying this property
to the " four special cases of physical interest," Kasner restates
it as follows: " In any plane field of force select any fixed element
of curvature, corresponding to the initial values x, y, y', r so
given, construct the unique trajectory, unique brachistochrone,
unique catenary, the unique velocity curve, and the respective
centers of the osculating conies; the four centers so formed and
the given point {x, y) will lie on a conic passing through the
INTRODUCTION. Vll
latter point in the given direction y' " The radius of curvature
of this conic is r/2.
The 00 2 curves of the family at the given point {x, y) in the
given direction y' have associated with them oo i of these conies.
The centers of these conies lie on a conic passing through the
given point, and their second centers of curvature lie on a parab-
ola passing through the given point and having its axis parallel
to the given direction. The angles (p, <pi and 6 which the conic,
the parabola and the direction of the acting force, respectively,
make with the given direction are related as follows:
tan <p = sin 20,
3 tan <pi = — sin 26.
One of the oo^ conies associated with the <»2 curves of the
family at the point {x, y) in the direction y' has four point contact
with its osculating circle at {x, y). The centers of the « ^ hyper-
osculating circles thus associated with all the curves of the family
through {x, y) lie on a cubic which has as tangents the minimal
lines at the given point.
Of the 00 1 curves of the family corresponding to a given curva-
ture element {x, y, y', r) there are two which are hyperosculated
by their osculating parabolas. The locus of the foci of these
hyperosculating parabolas as r varies in a bicircular quartic
with a node at the given point. One of the nodal tangents is
perpendicular to the direction of the acting force and the other
makes with this direction an angle which is bisected by the given
direction.
The inverse of this quartic is an hyperbola with asymptotes
parallel to the nodal tangents of the quartic. The locus of the
mid-point of the chord of this hyperbola cut from the normal to
the given direction at the given point is a circular cubic which
has a double point at the given point and its real asymptote
parallel to the direction of the acting force.
One of the oo^ curves of the family corresponding to a given
lineal element {x, y, y') has five-point contact with its osculating
fircle. The locus of the center of this five-point circle as y'
varies is a conic passing through the given point. Its tangent
at this point is perpendicular to the direction of the acting force.
Vm INTRODUCTION.
If a quadruply infinite system of curves in a plane is given at
random, there exists in general no field of force in which the
pressure along the curves of this system is proportional to the
normal component of the force. The question of the suflficiency
of these properties to insure the existence of such a field of force
is next investigated, in Chapter III.
It is found that these properties belong to other quadruply
infinite systems of curves besides the system of curves along
which P = kN, and hence are not suJ0Bcient for a complete
characterization.
Additional properties are obtained in Chapter IV which
complete the characterization.
The entire set of sixteen properties obtained is shown to be
both necessary and sufficient for a complete characterization.
The writer is indebted to Professor Kasner for helpful sug-
gestions and criticisms.
CHAPTER I.
DiFPEKENTIAL EQUATION OF THE SySTEM.
We consider the constrained motion of a particle in a plane
moving along a curve under the action of a positional force; that
is, a force whose rectangular components are functions only of
the coordinates of the point at which it acts, namely ip{x, y)
and xpixy y). Denoting by Sk the system of curves defined by
the equation P = kN for a given value of k, P being the pressure
and iV the normal component of the acting force, the problem is
to obtain geometric properties completely characterizing the
curves of all the systems Sk^
The differential equation of the system Sk is
Nr,= (n-\-l)T-rN, (1)
where
^~ k+1' ^^ ~ l + y" ^"^^
and T is the tangential component of the acting force. Differ-
entiating equation (1) with respect to S, and eliminating n, we
obtain the differential equation, in an intrinsic form, of all the
systems Sk to be
where
Nts + rN T
^ ~ l + y"
(3)
(4)
JTo obtain the equivalent of (3) in Cartesian coordinates, we
substitute for N and T the values given in (2) and (4), and for
N, T, r, Tg, Tas the values given by
1
CURVES IN A PLANE.
^ , (1 + y")y"'
2/'
_L,, .. [(1 + y")y'^ + 2y'y"y"'\ - 2(i + y")y"" \
"r "^ y'" i
(5)
1
X
Making these substitutions in (3), we obtain
, r_3 3/(^^+^) 1 3
■*" L 1 + 2/" (^ - 2/V)(^ + y'^)(l + 2/'') J ^
L (^ + 2/V)(iA-2/V) J^ •
Introducing
Pi = ^x + (^» - <p^y' - <pvy''^y
P2= <Px+ {<Pv + ^r)y' + M\
^+^ 2y'
^2-^ + y'^ ^-2/V' ^^
„ 3 3y^(^^+^)
^' ~ 1 + 2/'' (^ - y'<p){v> + 2/'^)(l + 2/'') '
QP^y' ^P^y' + Pi
^' " (^ - 2/V)(l + y") (^ + 2/'^)(l + 2/") '
{<P + y'rP){rl^-y'<p)
THE DIFFERENTIAL EQUATION OP THE SYSTEM. 3
(6) may be written
y^ = |-/ y"" + iQ^y" + Q-^)y"' + Q^y'" + Q^y"'' + Q^y"- («)
This is a differential equation of the fourth order and rep-
resents a quadruply infinite system of curves.
CHAPTER II.
Geometeic Properties of the System.
Section 1. Osculating Conies.
Consider the curves of this system through a given point
(x, y) of the plane, in a given direction y', with a given curvature.
These form a system of oo^ curves, all of which have the same
first center of curvature, but each of which corresponds to a
different second center of curvature. Each of these curves has a
definite osculating conic at the given point; that is, a conic
having fourth order contact with the curve. The coordinates
of the center of the osculating conic, if the given point is taken
as the origin of coordinates, are
^y"y"' , ^ ^y'y"y"' - ^y '"
5y"'^-dy"y^' 5y'"^ - Sy'Y
The locus of the centers of all the osculating conies correspond-
ing to the 00^ curves of the system is obtained by eliminating
y^^ and y'" between equations (8) and (9).
From (9) we have
y "y'h-k* y ~ y'h-k\_y'h-k h]'
Substituting these values in (8) and reducing, we obtain
[(3 + 3Qi y' + qzy")y"' + m^v' + Q^y")y"' + Q^y'YW
- [(3Qi + 2Q^y')y'" + (3^2 + 2Q,y')y"' + 2Q,yY]hk (10)
+ [Qzy'" + Q^y'" + Q^y"W + ^yY'h - Sy'"k = o.
This is the equation of a conic passing through the given
point; the equation of its tangent at the origin is k — y'h = 0.
Theorem I. 7f, for each of the oo ^ curves of the system corre-
sponding to a given curvature element, we construct the osculating
conic, the locus of the centers of these conies is a conic passing through
the given point in the given direction.
4
GEOMETEIC PEOPERTIES OP THE SYSTEM.
Section 2. Curvature of the Con,ic of Theorem L
Applying the formula
^kfdh^
K —
[1 + {dkldhff^
to obtain the curvature at the given point of the conic (10), we
find
Theorem II. The curvature at the given point of the conic
corresponding, by theorem I, to a given curvature element is twice
the curvature of the element.
Section 3. Second Center of Curvature of the Conic Determined by
Theorem I,
For the conic (10), we find
The coordinates of the second center of curvature are ob-
tained by the formulas
- 4(1 + y")y' , {i + y'*)\ .„
Applying these to the conic (10), we get
2(1 + y")y' 3(1+Y!)!
(1 + y'^){\ - Sy") 3 (1 + y")y'{q,y" + Q^)
2/2 - y„ 4 y//a
(11)
(11')
We now consider all curves of the system corresponding to a
given lineal element {x, y, y'). These form a system of oo^
cm-ves. In accordance with theorem I, there are ooi conies
associated with this system of oo^ curves, each conic being as-
sociated, for different values of y", with the oo ^ curves corre-
sponding to a curvature element {x, y, y', y").
6 CURVES IN A PLANE.
The locus of the second centers of curvature of these conies,
with the given point taken as origin, is obtained by eliminating
y" between equations (11).
Multiplying the first of these equations by y' and then sub-
tracting from the second, we have
J_^ 2(y2 — y'xj)
y" (1 + 2/")' *
Substituting this in the first of the equations (11), we have,
after reduction,
6^2(2/2 - y'x^y + (1 + 2/")[3(l + y'^Qx + %y']yi
+ (1 + y")[2 - Qy'' - 3Qi(l + y")y']x, = 0. ^^^^
This is the equation of a parabola passing through the given
point; the axis of the parabola is parallel to the line 2/2 = y'x^.
Theorem III. The locus of the second centers of curvature of
the 00 1 conies associated, by theorem, I, with a given lineal element
(x, y, y') is a parabola passing through the given point; the axis of
the parabola is parallel to the given direction.
Section 4' The Equation of the Tangent to the Parabola.
The equation of the tangent to the parabola at the given point is
_ 2-62/-'-3Qi(l + y-V ^ .^„
If we substitute for Qi its value given in (7) and take the x
axis in the direction y', we obtain
±
slope of tangent = ~ o p • (14)
1 + 72
Calling the angle this tangent makes with the given direction
6\, and the angle the acting force makes with the given direction
di, (14) may be written
tan ^1 = — ^ sin 202- (15)
Theorem IV. At the given point, the tangent to the parabola
makes with the given direction an angle whose tangent is to the sine
of twice the angle the given direction makes with the acting force as
1 is to 3.
GEOMETRIC PROPERTIES OF THE SYSTEM.
Section 6. Center of the Conic of Theorem I.
The coordinates of the center of the conic
ax^ + bxy -\- cy^ -{- dx -{- ey =
are given by the formulas
2cd —be „ 2ae — bd
a
/3 = iJ—JZT' (16)
6^ — 4ac ' b^ — 4ac *
Applying these formulas to conic (10),
3(to" + Q2) ,.-.
" - (3Qi2 - 4Qz)y'"+ {QQ1Q2 - 4.Q,)y"+(dQ2' - 4.Q,) ' ^''^
P ~ (3Qi2 - 4^3)2/"' + {QQ1Q2 - ^Q4)y" + (3^2^ - 4Q6) * ^ ""^
The locus of the centers of the <» ^ conies associated, by the-
orem I, with the 00^ curves corresponding to a given lineal
element is obtained by eliminating y" between equations (17) and
(18). To carry out this elimination, we take
_ i2-^Qiy')y"+Q2y'
a Qiy"-\-Q2
from which
„ _ Q2W - y'ot)
y 2a- Q,W - y'cc) '
Substituting this value of y" in (17), we obtain
2[(Q2^Q3 - Q1Q2Q4 + Qm,)y" - 2(^2^4 - 2qmy'
- (3Q22 - 4Q6)]a2 _ 4[(Q2^Q3 - Q1Q2Q4 + Qi^q,)y'
- (Q2Q4 - 2QiQ5)]a^ + 2[Q2=^Q3 - Q1Q2Q4 + q^Q,,W
- 3[2Q2 + QiQ22/']a + 3Q1Q2/3 = 0.
This is a conic passing through the given point. The equa-
tion of the tangent at the given point is
, 2 + Qiy'
If we substitute for Qi its value from (7), and take the x axis
in the direction y' ,
O CUKVES IN A PLANE.
the slope of the tangent = jp . (20)
Callmg the angle this tangent makes with the given direction
Oi, and the angle the acting force makes with the given direction
62, (20) may be written,
tan di = sin 2^2- (21)
Theokem V. The locus of the centers of the <x>i conies asso-
ciated, by theorem I, with a given lineal element is a conic passing
through the given point. The tangent to this conic at the given point
makes with the given direction an angle whose tangent is equal
to the sine of twice the angle the acting force makes with the given
direction.
Section 6. Hyperosculating Parabolas.
We now determine all the curves of the system through the
point {x, y) which are hyperosculated by their osculating parab-
olas. The differential equation of all parabolas in the plane is
The hyperosculating parabola has contact of the fourth order
with its curve, hence its derivatives up to the fourth order at
the given point are the same as those of its curve. Substituting
the value of y^ from (22) in equation (8), we find
y""+ Sy"(Qiy" + ^2)2/'"+ Sy"iQ,y"'+ Q^'"+ Q,y") = 0. (23)
This is an equation of the second degree in y'". Hence two
curves of the system of 00 ^ curves corresponding to a given curv-
ature element are hyperosculated by their osculating parabolas.
The coordinates of the focus of the osculating parabola,
referred to the given point {x, y) as origin, are given by the
formulas
a =
^=-
dyV W"(y" - 1) + 2y'{Sy" - y'y'")]
2 y""-\-(^y"'-y'y"r '
^y" [(3y " - y'y"')iy"- D - 2yy^1
///2 _i_ (o,,n2 „,/„"'\2
y"" + W - y'y'")
GEOMETRIC PROPERTIES OP THE SYSTEM.
(26)
From these we have
(ay' - )8)(1 + y'')
(25)
,„ ^ _ 3(a2/^ - ^)(1 + y")[2^y' + a(l - y'^\
y 4(a2 4- (82)2
To obtain the locus of the focus of the hyperosculating parabola
as the curvature of the element changes, its direction remaining
unchanged, we eliminate y" and y'" between (23) and (25).
The result of this elimination, after substituting the values of
Qu Qi, Qs, Qi, Qb, given in (7), and reducing, is
4PiP2(a2+/3T+{[12Pi(^+2/V)-2(3P2+Piy')(^-2/V)]«
- [(6P22/'- 2Pi) (,A- 2/V) - 12Pi2/'(^+2/V)]i8 } («'+^')
+m+y")[(<pW+2v>Y- <P^W- (^+tA')(l-2/'')«/3
-^(<p4^y''-2rPY-cpxpm = 0.
This is the equation of a bicircular quartic having a node at
the given point. The equations of the tangents at the origin are
/3[(1 -/')?'+ 2y VI + «[(1 - !/'V - 2!/Vl = 0.
Each of these tangents has second order contact with the
quartic at the given point, cutting it in three coincident points,
two on one branch and one on the other.
If we take the x axis in the direction y', the equations of the
tangents reduce to
^xl^-\-a<p= 0,
0(p -\- a\l/ = 0.
Hence the first tangent is perpendicular to the direction of
acting force, and the second tangent makes with this direction
an angle which is bisected by the given direction.
Theorem VI. Corresponding to each curvature element, there
are two curves of the system that are hyperosculated by their osculating
parabolas. The hcus of the foci of these hyperosculating parabolas
as the curvature of the element varies, the direction of the element
remaining unchanged, is a bicircular quartic having a node at the
10 CURVES IN A PLANE.
given point. One of the tangents at the given point is perpendicular
to the direction of the force acting at that point, and the other makes
with thai direction an angle which is bisected by the given direction.
Section 7. The Inverse Curve.
The inverse of this quartic with respect to the origin is ob-
tained by the transformation,
« = ^2 I ■ 2» /3 =
Substituting these in (26), we obtain the conic
3(1 + y"){My"+2<pY - ^rp]e - (^^H- ^)(1 - y")^v
+ My" - 2^l^y' - <prl^W + [12Pi(v? + t/V)
- 2(3P2 + Pi2/')(^ - y'<pm - mP2y' - 2Pi)(^ - yV) ^^^^
- 12P^y'i<p + 2/V)]7/ + 4P1P2 = 0.
The discriminant of (28) is
9(1 + yy[(<p' - rl^)a - y") + ^<p^y'f >
and the conic is an hyperbola. The asymptotes of this hyper-
bola are parallel to the lines
WrPy'' + 2<p'y' - <pyp]e - {<p' + ^2)(i _ y'')^^
+ [<prh" - 2Vy' - <pW = 0.
Hence the asymptotes of the hyperbola are parallel to the
tangents to the quartic at the origin. It follows that the axes
of the hyperbola make with the given direction angles of ± 45
degrees.
The inverse of the quartic of thereom VI is an hyperbola. The
asymptotes of the hyperbola are parallel to the tangents to the quartic
ai the origin.
Section 8. Determination of the Conies, Obtained in Accordance
vrith Theorem I, which are Hyperosculated by Their Circles of
Curvature at the Origin.
The differential equation of all the circles in a plane is
y'" = r^7- (29)
GEOMETRIC PROPERTIES OF THE SYSTEM. 11
If the conic is hyperosculated by its circle of curvature at the
origin, the first three derivatives are the same for both curves
at that point.
For the conic (10), we find
fh=y'- i=2,". g=6(to"+w. (30)
Substituting these values in (29), we obtain
-[Qiy"+Q2](l-hy") = 2yy'. (31)
This is linear in y". Hence in the system of oo^ conies cor-
responding to a lineal element, by theorem I, there is one which
is hyperosculated by its circle of curvature at the origin.
The coordinates of the center of this hyperosculating circle
are
The locus of the center of this hyperosculating circle as y'
varies, obtained by eliminating y" and y' between (30) and (31),
is
{xf^y<p - 2<pyyP)X^ + {<py<p + 2<p^^P - 4^^<p - rP,rP)X^Y
- {<pyyp - 2xl^y<p - xl^^rP + <Px<p)XY' + {<p,yp - 2^x^)P (33)
This is a cubic having a conjugate point at the given point:
the tangents at the given point are the two minimal lines.
Theorem VII. Of the oo ^ conies eorresponding, in aceordance
with theorem I, to a given lineal element, there is one which is hyper-
osculated by its osculating circle at the given point. The locus of
the center of this hyperosculating circle, as y' varies, is a cubic with
a conjugate point at the given point. The tangents at this point
are the minimal lines.
Section 9. Coordinates of the Mid-point of the Chord of the
Hyperbola {28) Cut from the Normal to the Corresponding
Lineal Element.
The equation of the normal is
^=-^^ (34)
12 CURVES IN A PLANE.
Regarding (28) and (34) as simultaneous equations, we obtain
3(1 + yy{<p + 2/V) (^ - y'<p)e
The discriminant of this quadratic,
4(1 + y'W\^ - 2/V)Pi[Pi(^ - 2/V) + 12P2(^ + yV)],
is a polynomial of even degree in y', with the real linear factor
^ ~ y'<P' Hence corresponding to each point of the plane
there is a range of values of y' for which the discriminant is
> 0. Throughout this range of values of y', (34) intersects its
corresponding hyperbola (28) in two real and distinct points.
From (35) and (34) we find the coordinates of the mid-point
of the chord to be
_ 2Pyy' ^ 2Px
3(^ + t/V)(i + y'')' ^{<P^y'^){i + y")' ^ ^
With each lineal element through a given point, we may as-
sociate the mid-point of the chord of the hyperbola corresponding
to the element cut from the normal to the element. The locus
of this mid-point, as the lineal element revolves about the given
point, obtained by eliminating y' between equations (36), is
3(<p7-^X)(Z2+ p)_2[^,y2 _ (^^ _ ^,)XY- VyX^] = 0. (37)
This is a circular cubic with a double point at the given point.
The real asymptote to this cubic is parallel to the direction of
the force acting at the given point.
Theorem VIII. The inverse of the quartic of theorem VII is
an hyperbola. The locus of the mid-point of the chord of this
hyperbola cut from the normal to the lineal element, as the direction
of the element varies, is a circular cvhic with a double point at the
given point. The real asymptote of the cubic is parallel to the
direction of the force acting at the given point.
Section 10. Curves of the System which Have Five-Point Contact
with Their Osculating Circles.
The differential equation of all circles in the plane is
!'"' = f$7- (38)
GEOMETRIC PROPERTIES OP THE SYSTEM. 13
DiJBFerentiating this, we have
y '- (1+7^2+ r+p. (38)
If a curve has five-point contact with its osculating circle,
the first four derivatives at the point of contact are the same for
both curves. Substituting the values of y'" and y^ from (38)
and (380 in the differential equation of the system (8), we have
[r|^2+^4]2/"+Q5=0. (39)
Hence corresponding to each lineal element there is one curve
of the system which has five-point contact with its circle of
curvature. If we eliminate y' and y" between
X^-^r^, Y^'-±^ (39')
y y
and (39), we get the locus of the centers of these five-point circles,
as the direction of the element varies, to be
^xP - {<Py + ^.)XY + 4'yX' +m+ <pX) = 0. (40)
This is a conic passing through the given point. Its tangent
at this point is perpendicular to the direction of the force acting
at the point.
Theorem IX. Of the oo^ curves of the system corresponding to
each lineal element, there is one which has five-point contact with
its osculating circle at the given point. The locu^ of the centers of
these circles, as the direction of the element varies, is a conic passing
through the given point. Its tangent at this point is perpendicular
to the direction of the force acting at the point.
CHAPTER III.
Conversion of the Properties Stated in Theorems I to IX.
The quadruply infinite system of curves defined by (8) pos-
sesses the properties stated in the theorems I to IX. We now
consider the converse question, if a quadruply infinite system of
curves possesses these properties, does there exist a field of force
such that the pressure on any curve of the family is some constant
multiple of the normal component of the acting force?
Section 1. Conversion of Property I.
The differential equation of any oo* family of curves in the
plane is y^ = G{x, y, y', y", y'"), when G is an arbitrary function
of its arguments.
If this family possesses property I, the centers of the osculating
conies of the oo ^ curves of the family defined by any curvature
element {x, y, y', y") lie upon a conic through the point {x, y)
in the direction y'.
The most general equation of such a conic, referred to the
point {x, y) as origin, is
Ayo? + A^a^ + Az^ + {ay' - ^) = 0, (41)
where A\, A2, Az are arbitrary functions of x, y, y', y".
The coordinates of the center of the osculating conic are
^y"y"' o ^y'y"y"' - %'"
^ K'l.f'f^ Q-..".iiIV ' P
by'"^ - Sy'Y"^' '^ ~ 5y"'^ - Sy"y^ '
Substituting these in (41), we have after reduction
y"^ = B^y"" -\- B^y'" + Bz, (42)
where the B's are arbitrary functions of x, y, y', y" . Hence if a
system of oo^ curves of the plane has property I, its defining
equation must be of type (42).
Section 2. Conversion of Property II.
We now find those systems of type (42) which have property
II.
14
CONVERSION OF PROPERTIES. 15
The locus of the centers of the o?^ osculating conies of the
system (42) corresponding to a curvature element {x, y, y', y")
is obtained by substituting
Making this substitution, we obtain
[Iby'" - ^B,y"' - 3B2y'y - B^y^W
(43)
+ m^y" + 2Bzy')hk - Bzk^ - Sy'yh + dy"k = 0.
The curvature K of this conic is
2[3B^y'' - 5]y"
If the system (42) has property II, then
2[SBiy" - 5]y" 2y"
from which, Bi = 2ly".
The differential equation of all quadruply infinite systems
having properties I and II is then
y"^ = ^rV'" ■\- B^y" + Bz, (44)
when Bz and ^3 are arbitrary functions of x, y, y', y".
Section 3. Conversion of Property III.
The conic corresponding, in accordance with property I, to a
curvature element of the system (44), may be obtained from (43)
by replacing Bi by 2jy". This gives
W + 3B,y'y + Bzy")h? - {ZB^y"' + 2Bzy')}ik
+ Bzk + Zy'yh - Zy"'k = 0.
For this conic, we have
d^k ^„ „ d^k „ „ dh
^z=-^B,y, ^,= 2y, ^=y.
Substituting these in the formulas (11) for the second center
of curvature,
16 CURVES IN A PLANE.
(45)
4(1 + y'')/ 6(1 + yyB2
^ (1 + y'^jl - Sy") 6(1 + y")y'B2
2/2 2y" 8y"^
If system (44) has property III, the coordinates of the second
center of curvature obtained in (45) must satisfy the equation
of a parabola passing through the given point with its axis
parallel to the given direction.
The most general equation of such a parabola is
(2/2 - y'x^y + Ciy2 4- C2X2 = 0, (46)
when X2, yz are the coordinates of any point on the parabola
referred to the given point (x, y) as origin, and C\, C2 are arbi-
trary functions of x, y, and y'.
Substituting the values from (45) in (46), we obtain
Solving this for B2, making a slight change in notation, we
obtain
B2 = Hiy"-{-H2,
where Hi and H2 are arbitrary functions of x, y, 3^.
All the quadruply infinite systems having properties I, II,
III are then defined by the differential equation
2/"^ = ^ y'"^ + ili^y" + H2)y'" + Bz, (47)
where Hi and H2 are arbitrary functions of x, y, y' and B% is an
arbitrary function of x, y, y' , y" .
Section 4- Conversion of Property IV.
In order to convert Theorem IV, we state it in a form which
does not assume the existence of a force, as follows: At each point
{x, y) of the plane, there exists a certain direction of slope co(x, y)
svxih that the sine of twice the angle this direction makes with the
given element is equal to three times the tangent of the angle the given
element makes with the parabola corresponding to it by theorem III.
CONVERSION OF PROPERTIES. 17
The parabola corresponding, in accordance with property III,
to the 00 2 curves of system (47) passing through a given point
{x, y) in a given direction y', is found by eliminating y" between
4(l + y^V 6(1 + y-^)'^
^2 = 2y^' '^^ — ^^'^y + ^2;»
(1 + y") ,, . „, 6(i + y-W(giy" + g2)
2/2 = 22/" ^^ ~ ^^ ^ W" *
Carrying out this elimination, we obtain
6^2(2/2 - yW) + (1 + 2/'=')[3(l + y")Hr + 82/']z/2
+ (1 + 2/")[2 - W' - 3Fi(l + y")y']x2 = 0.
The slope of the tangent to this parabola at the given point is
_ 2-Qy"-SHra-^y")y'
3(1 + y")Hi + 8y' '
If the system (47) has property IV, we have
2(0, - y')
of 'J 2 - 6y^^ - 3gi(l + y- y i
' ^L^-^ S{l-\-y")Hi-\-Sy' J _
[2 - 6y-' - Bffid + y")y']y' f co-y- T'
^ 3(1 + 2/'=^)^!+ 8/ ■^Ll + coy'J
Solving this for iTi, we find
1 + ic"^ 2y'
Hi
(CO - 2/')(l + W) 1 + 2/"'
where w is an arbitrary function of x and 2/-
The differential equation
defines all the quadruply infinite systems having properties I,
II, III, IV.
Section 5. Conversion of Property V.
Property V may be stated in a form that does not require the
existence of a field of force by replacing the direction of the
acting force by the function o){x, y) defined in Section 4.
The most general conic fulfilling the conditions of theorem V is
Gia^ + GiaP + Gs8'' + fi - ma = 0, (49)
18 CURVES IN A PLANE.
where the G's are arbitrary functions of x, y, and y', and m is
determined from the relation
m
m
^Ll + co/J
(1 + co2)(l + y")y' + 2(co - y'){\ + cot/')
to be
(1 + y")0. + d") - 2(co - y'){l + coy')2/'*
The center of the conic corresponding, according to property
I, to the system (48) is
Z{H,y"+my"
a = —
where
S{Hiy" + H,)Y^ -
453'
3(22/" + H^y'y + H,y')y"
SiH^y" + ^2)Y'
-4^3
1 + 0)2
22/'
(co-2/')(l + <^y') 1 + 2/''*
By property V, these satisfy (49). Substituting, we have
Liy'" + L2y"' + L,y"
+ 2F2(co - 2/0(1 + co2/')[(3Fiy" + H^Yy" - 4^3] = 0,
when the L's are functions of Gu G2, G3, H2, x, y, and y' and there-
fore arbitrary functions of x, y, and y'.
Solving this for Bz,
Bz= Miy'"-\-M,y'"+Mzy",
when the M's are arbitrary functions of x, y, and y'.
Hence all systems possessing properties I, II, III, IV and V
have the form
2^ -y"^' +iL(l + co2/0(co-2/') l+2/'^J^+^^P (50)
+ Miy"'+M2y"'+Mzy".
Section 6. Conversion of Property VI.
We state theorem VI without assuming the existence of a
force by replacing the direction of the force acting at the given
point by the function (a{x, y) defined in Section 4.
CONVERSION OF PROPERTIES. 19
We determine the slopes mi and mi of the tangents to the
quartic of theorem VI from the conditions
1 (a— y' y' — tni
TOl = —
CO
1 + C02/' 1 + 2/'wi2 *
(51)
(52)
Solving the second of these equations for m2, we find
2y^+coy^'-co
We now find the locus of the foci of the hyperosculating
parabolas of the curves of (50) which correspond to a given lineal
element (ar, y, y'). It is obtained by substituting the values of
y", y"', y" from (22) and (25) in (50). The result is
m^{c? + ^f + {2[M2(1 + 2/") + 6ilf ii/']/3
+ 2[3F2(1 - y'^) - Miy\\ + y'^)\a\ {o? + ^)
+ [121/'=' + m^y'iX + y") + M i(l + y'^X^ (53)
+ [12t/'(l-2/'^)-2Mi2/'(l+2/'^)2+3Fi(l+2/'=')(l-3y"')]a/3
+ [3(1 - y'^) + iWi2/''(l + y'''? - ^H^y'a - y")W = 0,
where
l + co^^ 2y[_
^' (CO - 2/0(1 -I- coi/O l + 2/'='*
This is a bicircular quartic. Hence if an qo* system has
properties I to V, it also has the property that the foci of the
hyperosculating parabolas of the curves corresponding to a
lineal element lie upon a bicircular quartic.
In accordance with theorem VI, the slopes of the tangents to
(53) at the given point must equal mi and m2 of (51) and (52).
Forming the product of these slopes, we have
3(1 - y") + M^y'W + y^ - SH^y'd - y")
12y" + QHiy'a + y'') + Mi(l + y'Y
_ _ 2y' + <^y'^ 1 - CO
co(l + 2co2/'-i/'^)-
This solved for Mi gives
M,= ^ 32/'(l + co^)
(54)
1 + 2/'' (co-2/')(l + co2/0(l + 2/'')-
The differential equation of all the systems having properties
HI
(55)
20 CURVES IN A PLANE.
I, II, III, IV, V, VI is of the form
when Hi, If 2, Mi are arbitrary functions of x, y, and y', and w
is an arbitrary function of x and y.
Section 7. Conversion of Property VII.
The most general cubic having the properties required by
theorem VII is
TiX^ + TiX^Y + TzXY' + T,r+X^+Y^ = 0, (56)
when the 7"s are arbitrary functions of x and y.
The conies of theorem I corresponding to system (55) may
be obtained from (10) by replacing Qi, Q2, Q3, Qi, Qh by Hi, H2,
Ml, Mi, Mz respectively. Hence dkjdh, (Phjdh^, d^kjdh? of the
conic may be obtained from (30) by replacing Qi and Q2 by Hi
and Hi respectively. We find
dh , d?k ^ ,, d?k ^,tx » ■ ttv ..
Substituting these in the differential equation of all the circles,
y 1 + 2,'*'
we have, after substituting for ^1 its value in (53),
r l + co^ ^y' 1 „ , „ _ 2yy^
lio>-y'){\ + <.y')-l + y''\y + ^^ " " 1 + y'- ^^7)
In accordance with property VII, if we eliminate y' and y"
between (57) and equations (32), we get a cubic of the form
(56).
Substituting in (57) the values y'= - (Z/F), 2/" = (Z'+ P)/l^,
and denoting by Hi the function of X and Y which Hi becomes
on replacing y' by — (X/Y), we obtain
g,'[ ^^"^|j^>J-"^ J + X^+P=0. (58)
CONVEESION OP PROPEKTIES. 21
In order that (58) may have the form (56), we must have
^2' r Y(^y+^)(Y-o>X) -\ ^ y^^3_^ T2X'Y+ TzXY'+ T,Y\
_ iTiX^+ T2X^Y+ TzXY'+ rP)(l + oy")
>^^ " F(a)y4-X)(F-coZ)
(a,+ f)(l-cof)
^2' =
Hence,
Tj _ K,y"+K,Y"+KzY' + K, .^^.
^' - (0,-2/0(1 + 0,/) ' ^^^^
when the K's are arbitrary functions of x and y.
The differential equation defining all 00 ^ systems having the
first seven properties is
2/^ = 4 y"" + [Hiy" + my'" + M^y"' + M,y"' + ilf 32/", (60)
where
_ 1 + 0,2 22/'
^' ~ (0, - 2/0(1 + 0,2/') ~ fTF'
__ K,y"-^K,y''+K^y' + K, .^^^
H2 - («_ 2,0(1 +0,2/0 ' ^^^^
^ 3 32/'(l + o,2)
1 + 2/" (o,-2/)(l + 0,2/0(1 + 2/'*)'
Xi, X2, K3, Ki are arbitrary functions of x and 2/ and M2, Mz
are arbitrary functions of x, y, and 2/',
Section 8. Conversion of Property VIII .
We find the quartic corresponding, in accordance with
theorem VI, to (60) by substituting in (53) the particular values
of Hi, H2, and Mi given in (61). The inverse of this quartic is
the hyperbola
22 CURVES IN A PLANE.
[3(1 - y") + Miy'W + y^ - 3H^y\l - y'*)]^
+ [Wil - y") - 2Miy'(l + yy
+ SH,{1 + y'Yl - Sy")]^V
+ [W + 6H^y'(l + y") + Mi(l + y'')]^^ (62)
+ 2[M2il + y") + m,y']r}
+ 2[3Zr2(l - y") - M,y'{l + 2/'^]^ + ^M, = 0,
where Hi, H2, and il/i have the values given in (61).
We find the coordinates of the mid-point of the chord of this
hyperbola cut from the line v = — (1/2/0^ to be
_ 2M,y' 2H,y"
^~ 3 '^l + y'''
(63)
2M2 2H2y'
Y= -
1 + y'"
In accordance with theorem VIII (which we restate by re-
placing the direction of the force acting at the given point by
CO (a:, y)) these coordinates must satisfy a cubic of the form
(Z2 + Y^)io}X - Y) + AX^+ BXY + CP = 0. (64)
Substituting the values of X and Y from (63) in (64), we obtain
[^ + ff^] (1 + /')(1 + «/) + ^^" - By'+ C = 0.
Solving this for if 2 and introducing Ni = — ^A, N^ = }5,
Nz= — f C, we obtain
^ ^Niy^+N^y'+Nz SH^y'
(1 + 2/'=')(l + coy') a + y")'
where « and the N*s are arbitrary functions of x and y, and Hi
has the value given in (61).
The differential equation of all 00 * systems possessing the first
eight properties is
f'-Y'^""
CONVERSION OF PROPERTIES. 23
"^lL("-2/')(H-coy') l-\-y"V'
^ (CO -2/0(1+ cot/') \y
+
r
L (1 + 2/'')(l + CO2/0
(co- 2/0(1 + W2/0 J
when the iiC's, the N's, and co are arbitrary functions of x and y,
and ilf 3 is an arbitrary function of x, y, and 2/'.
Section 9. Conversion of Property IX.
We obtain the curve of system (65), corresponding to a
given lineal element, which has five point contact with its os-
culating circle, from (39) by replacing Q2, Q4, and Q5 by
Kxy" + K<,y" ^ K^y' + K,
H2 =
if2 =
(<0 - 2/0(1 + C02/0
(l + 2/'')(l + co2/0 1 + y"
and ikfa respectively. The result is
(1 + 2,")(1 + .,/) ^ +il/3-0.
Hence
_ il/3(l + y'^W + 1) .„,
y - N,y"+N,y' + N, ' ^^'
The center of the corresponding osculating circle, obtained
by substituting this value of y" in (390, is
y_ y'[Niy"+N2y' + Nz]
Mzil + C02/O
(67)
_N^y;^_±N2y^±Nz
Mzil + a;2/0 •
24 CUKVES IN A PLANE.
In accordance with property IX (restated without assuming
the existence of a force), these coordinates satisfy a conic of
the form
EiZ2 - IkXY + Rzr + (X + wF) = 0, (68)
when Ri, Rz, R3 and w are arbitrary functions of x and y.
Substituting, we obtain
[Ni Y" + N^y' + N,][R^y" + R,y' + Rz]
+ iy' -«)(! + o}y')M, = 0.
Solving for Mz,
^ _ [Njy'' 4- Njy' + NzWR^y" + Riy' + Rz]
(CO -2/0(1 + "2/')
when the iJ's are arbitrary functions of x and y.
Hence the systems of type
IV _ A n,7A__l±jl 2y^l u
y y"y "^L(«- 2/0(1 +"2/0 i + y"r
T Kiy"^-K,y"■\-Kzy' + K, ■^ ,„
■^L (CO - /)(1 + C02,') J 2^
4. r_J 3y-(l + co^) 1
^ L 1 + 2/'' (1 + 2/'0(co - 2/0(1 + C02/O J ^
■^L(l + 2/")(l + co2/0
(69)
3(ii:i y^* + ^22/^' + Ji^ay^' + T^^v' 1 ,
+ 2/'0(«- 2/0(1 + C02/O J^'
+
(1
l^^y'^ + i\^22/' + iV^3][/?i2/'' + ^^22/' + M
\y".
(CO- 2/0(1 + C02/O
where co, iVi, iVz, ^3, ^1, -K^2, -K^s, -K^4, i^i, jR2, and Rz are arbitrary
functions of x and 2/, are completely characterized by properties
I, II, III, IV, V, VI, VII, VIII, IX.
CHAPTER IV.
Complete Chaeacterization.
Types (69) and (6) involve the derivatives y', y", y'", y^
in exactly the same way; but (69) is more general than (6) with
respect to the arbitrary functions of x and y appearing in the
coefficients.
Section 1. Relations Existing among the Arbitrary Functions
Entering into the Differential Equation of the System of Curves
Having Properties I-IX if it is to be of the Special Form given
in Chapter I.
We now determine the relations that must exist among these
arbitrary functions in order that (69) may be of the special
type (6).
By direct comparison of (69) and (6), we have
<a
Y
(70)
Ni
_<Py ^^^<P'-^v ^ >Px
<P <P <P
(71)
Ri
^i T, ^X-h (Pv Jf <PX
<P <P <P
(72)
2(Py\l/ — \l/y<P ^^ (py(p " xf/ x<P " ^y^ + '^'(pX^
ipyip — 2\py<p + (Pj,(P — yffgxf/ ^^ (p^xp — 2\l/x(p
^2 > ^^4- ^2
(73)
itroc
lucing (f> = log <f) and substituting xp — aj0, we have
Ni
= <t)y, Ni = <f>x — W0y — Uy, Nz = — 03x — Oi4>x,
(74)
Ri-
= Ul^y + COj,, JB2 = 01/ + (^4>X + i>ix, RZ = <f>X,
(75)
Ki = W0I, — COy, iL2 = (1 — W )0J/ + W0a; — COs — C0CO«,
- - - (76)
Kz= {\ — (»)^)<f>x — oi(t>v ~ 2a)y — cowx, i^4= —2(ax—oi(f>x'
From the first and third equations of (74), we have
25
26 CUEVES IN A PLANE.
<i>y=Nr, <!>:,= --^-5. (77)
CO
Substituting these values in the second equation of (74), it
becomes
wWi + U)N2 + Nz+0}^+ UUy = 0. (78)
From (77) we have
(78) and (79) are necessary conditions. To show that they are
also suflBcient: if (79) holds, we can find a function (f> which
satisfies both of the equations (77). Then on account of (78),
equations (74) hold. Finally (70) and <f> = log <f> determine a
pair of functions <f> and ^ in terms of which we can express Ni,
Nz, Nz and u in the forms given in (70) and (71).
From (75) and (77), we have
Ri = oiNi + oiy, R2=Ni-N3, Rs= - ^^+^" , (gQ)
CO
and from (76) and (77),
Kl = CoiVi — C0„, ^2 = (1 — CO^)Ni — N3— 2cOx — COCOy,
__ (1 — CO^) .^ ... 0)x „ rr HT ^^^^
K3= - -Nz - coiVi 2uy, Ki= Nz- cox.
CO CO
If relations (80) and (81) hold, we can express Ri, R2, Rz,
Kl, K2, Kz, and K^ in terms of (p and \J/ in the forms given in
(72) and (73).
We now derive relations equivalent to (80) and (81) which admit
of geometric interpretations.
From (75) we have 0x = Rz, <f)y = {Ri — cS)yfo).
Substituting these in the second equation of (75), it becomes
ui^Rz — <aR2 -{- Ri — o}y-{- coco* = 0. (83)
From (80),
Ri = coiVTi + CO,, Rz= - ^^^^^t-^. (83')
CO
The equations (83) and (83') are equivalent to the equations (80).
From (76) we have
_ K i + o}y - Ki + 2co»
COMPLETE CHABACTERIZATION. 27
Substituting these values in the second and third equations
of the set (81), we obtain
(1 - o)^)Ki - o)Ki - UK4 + uy- 2co2aj„ - Scoco^ = 0, (84)
uKi + ojKs + (1 - o}^)K4, + 2co, - oi'u}^ + 3ajco„ = 0. (85)
Multiplying (85) by w and adding to (84), we obtain
Ki - coiiTz + oi^Kz - o)'K, + (coj, - (acc^Kl + w2) = 0. (86)
Multiplying (84) by co and subtracting from (85), we obtain
u^Ki + 03^K2 + uKz + Ki + 2{o3^ + «a)„)(l + <a^) = 0. (87)
From (82) we have
Ki = uNi - o3y, Ki= N3- w,. (88)
Relations (86), (87), and (88) are equivalent to relations (82).
Section 2. Geometric Interpretations of the Relations of Section 1.
We now obtain geometric interpretation of relations (78),
(79), (83), (86), (87), (88).
Interpretation of {78). — The radius of curvature of the bi-
circular quartic
(^2 + ^2)2 _^ ^a<x + 6/3) (q:2 + ^2) + ca2 + ^^^ + e/32 =
at the origin is
{l-\-^"yi\d+2e^')
^= W+W) • ^^^^
If in the bicircular quartic (26) we take y' — rp/cp, the slopes
of the tangents at the origin are j8' = \l//(p and /3' = — (p/\l/.
We now apply formula (91) to the branch of this quartic for
which j3' = \l/(<p. We obtain, after replacing the direction of
the force acting at the given point by the (a(x, y) considered in
Section 4, Chapter III,
1 (1 + aj2)3/2
R^lr^. 1 T— n- (92)
£0 CO'' I
Substituting in (92) ^xlf = — Nz, {ipx — ^v)l^ = -^2,
<pj<p — Ni from (71), we obtain the radius of curvature of the
quartic corresponding to (69), by theorem VI, for 2/' = \A/v and
28 CURVES IN A PLANE.
/3' = ^jip. This gives
1 (1 + oiy^
Consider now the lines of force defined by,
y' = u{x, y).
The radius of curvature of the curve of this family passing
through the given point is
From (93), (94), and (98) we obtain
4E = pi. (94')
Theorem X. The slopes of the two branches of the bicircular
quartic corresponding, by theorem VI, to the lineal element (x, y, w)
at the given point, are co and — (l/w). The radius of curvature of
the branch of slope o) is ^ of the radius of curvature of the line of
force passing through the given point.
To each point of the plane there corresponds, by theorem
VIII, a definite cubic passing through the given point. From
(64) and (64') we obtain the cubic of 69 to be
3(^2 + P)(a,Z - 7) - 2[iViZ2 - N^XY + N^Y^] = 0. (95)
The line Y = — 1/coX meets this curve in the point N,
whose coordinates are
2 (Ni + (cN2 + coW3)co 2 (Ni + uN^ + C0W3)
^3 (l + w2)2 ' ■'^ 3 (l + «2)2
The distance
3 (1 + «2)3/2 • Ky^)
From (96), (94), and (78), we have
iON = -. (97)
Pi
This relation is equivalent to the relation (94').
Theorem XP. The cubic associated, by theorem VIII, with any
point intersects the line through perpendicular to its asymptote
at a distance equal to ^ of the curvature of the line of force passing
through 0.
COMPLETE CHARACTERIZATION. 29
Interpretation of (79). — The intercepts of the cubic (95) are
O CO
From these we have
Ni=iu)' OA, N3= - iOB, (98)
Substituting these values in (79), it becomes
^(a,-0^)-^(-j + 3 '^ ' =0. (99)
If through we draw OF in the direction u{x, y)
and let A' and B' denote the points at which this
line meets the perpendicular to the axes through
the points A and B respectively, (99) may be
written
S d 2i cocOt» — OirijOti
ai t-^^') - a-y f-^^') + 3 -^V^ = "• "««)
This is an intrinsic property, since it is true for any choice of
rectangular axes.
Theorem XI. When the point is moved, the cubic associated
ivith it by theorem VIII changes in the following manner. Take
any two fixed perpendicular directions for the x direction and the
y direction; through draw lines in these directions meeting the
cubic again in the points A and B. Construct the tangent to the
line of force at 0. At A draw a line parallel to the Y axis meeting
this tangent in the point A' and at B draw a line parallel to the x
axis meeting the tangent in B'. Then the distance AA' and BB'
and the slope a of the line of force satisfy relation (100).
Interpretation of (83). — The equation of the conic correspond-
ing to (69) by theorem IX may be obtained from (40) by
substituting
R3.
Making these substitutions, the result is
RiX' - R2XY + EsF^ + (coF + Z) = 0. (101)
The line Y = coX meets this conic in the point M whose coor-
dinates are
— /il.
— it2,
<Px
<p
<P
<P
30 CURVES IN A PLANE.
y_ 1 + co^ „_ (l + «>
■A — — rt ni or>t / — —
ill- coi^+ w2^3* ill - coilz + w'iJi*
The distance
<'^ = B^r^-i;«;- (102)
Now consider the orthogonal trajectories of the lines of force
defined by
y'=--r—\' (103)
^ co(a:, y)
The radius of curvature of the one of this family passing
through the given point is
pi = . (104)
From (102), (104) and the first relation of (83), we have
OM = p2. (1040
Theorem XII. Of the oo^ curves of the system (6) which pass
through a given point in a direction normal to the direction of the
force acting at the point, there is one which has five point contact
with its osculating circle. The radius of curvature of this element
is equal to the radium of curvature of the orthogonal trajectory of the
lines of force which passes through the given point.
Again, consider the radius of curvature of the quartic of
theorem VI corresponding to the lineal element [x, y, — (1/co)]
for the branch of slope — (1/w). By formula (91) we obtain
(1 + w2)3/2
R = -r V, Tn • (105)
— w* w H I
<p <p <p J
Substitute Rz = <Pxl<p, R2 = i<Pu + ^x)l<p, Ri = ^vf<p in (105),
to get the radius of curvature of the quartic corresponding to
(69) for y' = - 1/w and /3' = - 1/w. This gives
(1 + 0)2)3/2
^ " 4(ili - 0iR2 + o^'Ri) ' ^^^^^
From (104), (106) and the first relation of (83), we find
4J? = P2.
This relation is equivalent to the relation (104').
COMPLETE CHAEACTEEIZATION. 31
Theorem XIP. The slopes of the two branches of the bicircular
quartic corresponding, by theorem VI, to the lineal element
[x,y, — (1/co)] are co and — 1/co. The radius of curvature of the
branch of slope — (1/w) is equal to ^ of the radius of curvature of
the orthogonal trajectory of the lines of force which passes through
the given point.
Interpretation of (83')- — The intercepts of the conic (101) are
ill il3
Then
_ 1 _ ^
Substituting these values and the values of iVi and N3 from
(98) in (830, we have
30,2-0^ +2 r^+a)„) = 0, 3-OB + 2(^-a),)=0. (107)
These are intrinsic relations, since they hold for any choice of
rectangular axes.
Theorem XIII. The intercepts of the conic associated, by
theorem IX, with any point {x, y), and the intercepts of the cubic
associated, by theorem VIII, with the same point, satisfy the
relation (107).
Interpretation of (87). — The cubic corresponding to (69) by
theorem VII may be obtained from (33) by substituting
\l/y<P — 2^y^
<P'
-Ki- ^ ,
The cubic is
KiX'-K2X^Y+K3XY'-K,Y'- {X'+ F2)(l + w^) = 0. (108)
This cubic is intersected by 7 = — (l/w)X at the point Mi
32 CURVES IN A PLANE.
whose coordinates are
^ ^ i(l + C02)2C0
Y= -
(1 + c^r
The distance
From (109), (94), and (87), we obtain the relation
20M = pi. (110)
Theorem XIV. Of the oo ^ conies of theorem I which pass through
a given point in the direction of the line of force through that point,
there is one which is hyperosculated by its circle of curvature; the
radius of curvature of this conic is ^ of the radium of curvature of
the line of force passing through the given point.
Interpretation of (86). — The cubic (108) is intersected by
Y = wX at the point M^ whose coordinates are
y =
The distance
(111), (104) and (86), give the relation
OM2 = P2. (112)
Theorem XV. Of the 00 ^ conies of theorem I which pass
through a given point in a direction normal to the direction of the
line of force through that point, the radium of curvature of the one
hyperosculated by its circle of curvature is equal to the radiums of
curvature of the orthogonal trajectory of the lines of force which
passes through the given point.
Interpretation of (88). — The intercepts of the cubic (108) are
1+0)2 Q I 2)
0^, = ^!^^, 0B2=-'^^-^,
^1
-coK2-{-co'Kz- o>^K^
a;(l + co2)2
Kr
-oiK2 + o>'Kz-oi^Ki
(1 + 0)2)5/2
COMPLETE CHAKACTERIZATION. 33
Then
rr _ L+^' K -- i±^'
Substituting these values and the values of Ni and Nz from (98)
and (88), we have
^•^^-^W'-'^^]^'-
(113)
This is an intrinsic relation.
Theorem XVI. The intercepts of the cubic associated, by
theorem VI, with any point (x, y), and the intercepts of the cubic
associated by theorem VIII with the same point, satisfy the relation
(113).
Any quadruply infinite system of curves in a plane along
which the pressure is proportional to the normal component of
the acting force possesses properties I, II, III, IV, V, VI, VII,
VIII, IX, X, XI, XII, XIII, XIV, XV, XVI; and, conversely,
if a quadruply infinite system possesses these properties (restated
by replacing the direction of the acting force by the function
o){x, y) defined in section 4, chapter III, there is a field of force in
which this system represents the family of curves along which
the pressure is proportional to the normal component of the
acting force.
VITA
Sarah Elizabeth Cronin, Bachelor of Science, University of
Iowa, 1903; Master of Science, University of Iowa, 1905; In-
structor in Mathematics, Iowa State College, 1905-1907; In-
structor in Mathematics, Iowa State University, 1907-1913;
Graduate student at Columbia University, 1913-1915.
84
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