Mathematics
A
QA
603
K55m
te. 01
A1
B 469256
ersity of Chicago
JOHN D. ROCKEFELLER

Metric Properties of Nets of Plane Curves
510
A DISSERTATION
SUBMITTED TO THE FACULTY
OF THE
OGDEN GRADUATE SCHOOL OF SCIENCE
IN CANDIDACY FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
(DEPARTMENT OF MATHEMATICS)
BY
H. R. KINGSTON
↓
Reprinted from
AMERICAN JOURNAL OF MATHEMATICS
Vol. XXXVIII, No. 4
October, 1916
(2)


•
LIBRARY
ARTES
a si p
ܐܩܢ
UNIVERSITY OF MICHIGAN
Kun
**S
1837
SALLARA
SCIENTIA
VERITAS OF THE
ANTHON
TUEROR
RIS-PENINGULAN AMIEHAM
CIRCUMSPICE
In
~I!
A
SAMAN
nin
ANU
117
mш m
MALIEPTREDELHIRTY
HUDBENN
HIKSH
C
Ve
2
:
The University of Chicago
FOUNDED BY JOHN D. ROCKEFELLER
Metric Properties of Nets of Plane Curves
A DISSERTATION
SUBMITTED TO THE FACULTY
OF THE
OGDEN GRADUATE SCHOOL OF SCIENCE
IN CANDIDACY FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
(DEPARTMENT OF MATHEMATICS)
BY
H. R. KINGSTON
***
Firth gener
NCAA
Vol. XXXVIII, No. 4
October, 1916
Reprinted from
AMERICAN JOURNAL OF MATHEMATICS
399
ܕ
MATHEMATICS
QA
603
•K55m
I
1
*

<
:
!
·
+
+

}
A
24 Gorne 78, Eas,
Metric Properties of Nets of Plane Curves.
By H. R. Kingston.
§ 1. Introduction.
*
In a memoir entitled "One-Parameter Families and Nets of Plane
Curves," Wilczynski has discussed the projective differential properties of
nets of plane curves, by means of a completely integrable system of three
partial differential equations of the second order. In the present paper the
foundation is laid for a study of the metric differential properties of such nets.
In order to accomplish this, it becomes necessary to consider, besides the
coefficients of the partial differential equations used by Wilczynski, the coeffi-
cients E, F and G of the square of the element of arc when this is expressed
in terms of the parameters of the net curves as coordinates. All of these
quantities taken together, which must moreover satisfy certain relations, deter-
mine a net uniquely except for a motion combined with a reflection. The
application of the general theory to the particular cases of orthogonal and
isothermal nets gives rise to interesting results.
I wish here to acknowledge my indebtedness to Professor Wilczynski, who
brought this subject to my attention, and to express to him my appreciation of
the kindly interest and patience which he has maintained throughout the
preparation of this paper.
§ 2. The Differential Equations and the Integrability Conditions.
In order to investigate metrical properties it is convenient to use Cartesian
coordinates. Let
x=x(u, v), y=y(u, v)
be the equations of the net referred to a fixed Cartesian system, which shall
hereafter be called the fundamental Cartesian system. This form of represen-
tation is contained in Wilczynski's form as a special case, if the homogeneous
* Transactions of the Am. Math. Soc., Vol. XII (1911), pp. 473-510. This paper will hereafter be
referred to as "One-Parameter Families."
299)
408
KINGSTON: Metric Properties of Nets of Plane Curves.
coordinates y¹), y(2), y), used by him, be regarded as homogeneous Cartesian
coordinates, and if we put
(8)
y=x(u, v), y(2)=y(u, v), y®=1.
(1)
In consequence of this, any other point whose homogeneous coordinates are
p(¹), p²), p‹³), will have p¹: p³) and p2): p³) as its Cartesian coordinates.
(2)
(1) (3)
(3)
The differential equations of the net have the general form *
and
Yuu=ay+by+cy, Yuo=α'Yu+b'yo+c'y, Yv=a"Yu+b″y,+c"y, (2)
y(1), y(2), y(3) must form a fundamental system of solutions of this system.
Since y=1 must satisfy (2), we find, for our special form of representation,
c=c'=c″=0.
Equations (2) will therefore assume the form
Ouu=a0u+b0。, Ouv=a'0u+b'0,,
Ouv=a'Ou+b'0v, Ovv=a″Ou+b″0%
น
and these equations must be satisfied by both 0=x(u, v) and 0=y(u, v).
The integrability conditions of (3) may be obtained from those of (2) by
merely substituting in the latter c=c'=c" =0. They are †
ba"+a₁=a'b'+au,
a'²+b'a”+a',=aa"+a'b”+αű,
}
(4)
Of course, it is evident that the equations (3) and (4) may be obtained also
from the general equations of the metric theory of surfaces by proper
specialization.
(3)
ab'+bb”+b„=a'b+b²²+bu,
a'b'+b',=a″b+b″.
§ 3. Discussion of the Triangle Determined by any Point of the Net and its
Fundamental Covariant Points.
The homogeneous coordinates of the covariant points P, and P, are
(k) =y(*) — b'y(*), σ(k)=y(*)—a′y(k), (k=1, 2, 3).
(5)
The point P, is defined geometrically as the intersection of the tangents
to two consecutive curves v-const., constructed at the points where these
curves are met by a fixed curve u const. Likewise, P, is the point of inter-
section of the tangents to two consecutive curves u=const., constructed at the
points where these curves are met by a fixed curve v const. As u and v vary,
each of these points, P, and P., describes in general a new net of plane curves
which is called a Laplacian transform of the original net. These nets are
called, respectively, the first and the minus first Laplacian transforms of the
*"One-Parameter Families," equations (4). + " "One-Parameter Families," equations (5).
KINGSTON: Metric Properties of Nets of Plane Curves.
409
original net, the reason for the terminology being that the first Laplacian
transform of the net determined by P, is the original net.*
In view of (1), equations (5) give
p¹=xu-b'x, σ(¹)=x,—α'x,
p¹²=Yu-b'y, σ (2) =y.—a'y,
ช
=—b',
Ꮕ
σ(3)——a'.
Hence, in accordance with the remark made in § 2, the Cartesian coordinates of
P, and P. are given by the equations
(2)
(3)
x=x-xu/b', x=x-x/α',
(7)
yp=y-yu/b', y。=y—y„/a',
which might also be obtained by specializing the corresponding equations in
the theory of surfaces.
We proceed to find the angle between the two curves of the net passing
through the point P,. The slopes of the lines P,P, and PP, in rectangular
Cartesian coordinates are
tan P,P,P=(xãуï—X¸уu) / (X₂X₂+Y«¥v)·
ม σ
We shall discuss later (in § 5) a more precise definition for this angle.
We find further
Then
(y,—y)/(x,—x), (y.—y)/(x。-x),
น
น
respectively. In virtue of (7) these slopes become yu: x and y.: x,, respect-
ively, so that
(8)
PP,=V(x,−x)2+(−y)2+= |vity /
2
ข
2
P„P¸=√(x.—x)²+(y.—y)² = |√x²+y³/a'\.
Let us introduce the following permanent abbreviations:
where &±1.
==
2
"
x²+y²=E, x,x,+YuY=F, x²+y=G, VEG—F²=H.
(6)
*"One-Parameter Families," § 4.
† In this paper all square roots are to be considered positive.
XxX¸—уuY¸=εH,
(10)
The significance of the double sign will be explained later
(in $5). Evidently all of these notations correspond to those of the classical
theory of surfaces.
(9)
v
v
The case H=0 may be excluded from consideration. For, if H=0, we
shall have 愤¸-yuy,-0. Since the left member of this equation is precisely
the Jacobian of x and y, it follows that x and y are functions of the same com-
bination of u and v, so that our net degenerates into a single curve.
51
410
KINGSTON: Metric Properties of Nets of Plane Curves.
If we put 0, for the angle P,P,P., and make use of the notations (9) and
(10), we have
tan 0,=ɛH/F, P„P,=|VĒ/b'\, P„P.=|VĞ/a'\,
(11)
and further,
P,P.=|1/a'b'
=1/a'b' | Va'E—2a'b'F+b'G.
From the first of equations (11) we see that the net will be orthogonal if, and
only if,
F=0.
(12)
The slope of the line P,P. is (y.—y,): (x,—x,) which reduces, in view of
(7), to (a'y„—b'y。) : (a'x„—b'x). The slope of the line PP, is yu: x. Hence,
we readily obtain the formula
tan P,P,P.=ɛb'H/ (a’E—b'F).
Similarly, we find
tan P„P¸P,=ɛa'H/ (b'G—a'F).
The angle P„PP, will be a right angle if G:F=a': b'. The angle P,P,P, will
be a right angle if E:F-b': a'.
Again, the angles P,P,P. and P,P.P, will be equal if E:G=b′²: a'².
Further, the triangle P,P,P. will be equilateral if
E:2F: G=b2: a'b': a'².
2
|√(a'x‚—b'x¸)²+(a'y„—b'y。)²
§ 4. The Angle formed by the two Families of the Laplacian
Transformed Nets.
Let 0, be the angle made by the two curves of the first Laplacian trans-
formed net, which meet at P.. Then, corresponding to equation (8), we shall
have
tan 0:
(ex. Jy. In, y)/ (3314
)/(a
(Ja
σ
дхо
მ
av Iv
ди
By using equations (7) and (3) this reduces to
-ɛa'a"H
tan 0,
=
𐐀х
av
Əy. Əy
Iv
+
Ju
a'a"F+(a'b" —a'—a'²) G'
Hence, the first Laplacian transformed net will be orthogonal if, and only if,
a'a"F+(a'b”—a',—a”²) G=0.
(13)
Similarly, if 0, is the angle formed by the curves of the minus first Laplacian
transformed net, which meet at P,, we have
tan 0,=
—ɛbb'H
bb'F+(ab'—b',—b'²) E *
KINGSTON: Metric Properties of Nets of Plane Curves.
411
Hence, the minus first Laplacian transformed net will be orthogonal if, and
only if,
bb'F+(ab'—b'—b'²) E=0.
(14)
§ 5. Introduction of a Local System of Cartesian Coordinates.
In order to explore in detail the properties of a net in the vicinity of one
of its points, it is desirable to introduce a system of Cartesian coordinates
having this point as origin and the bisectors of the angles between the two net
curves as axes. Let the positive direction of the tangent to a u-curve, i. e., to
a curve v=const., of the original net at the point Py be chosen as the direction
in which u increases. Similarly, let the positive direction of the tangent to a
v-curve be that direction in which v increases. Then, since a positive element
of arc is, in general, given by ds=√Edu²+2Fdudv+Gdv², the direction cosines
of the positive u-tangent, referred to the fundamental Cartesian system, are
Xu: VĒ and y„:VĒ, and the direction cosines of the positive v-tangent are
x: VG and y: VG.
ย
Further, let 0, be the angle through which the positive u-tangent must be
rotated to coincide with the positive v-tangent, this angle to be counted from
0 to +л when it is positive, and from 0 to - when it is negative. Thus 0,
is subject to the inequalities -≤0„≤я.
л
Σπ
If X and Y are the axes of the fundamental Cartesian system we have,
therefore,
0,=(X, v)—(X, u) (mod. 2л)
where (X, v) represents the angle through which the positive X-axis must be
rotated to coincide with the direction of the positive v-tangent, etc. From this
we at once obtain for the general case
sin 0,=&H/VEG.
(15)
Thus, & is positive or negative according as 0, is positive or negative, and thus
the significance of the double sign in (10) is explained. Similarly, we find
cos 0,=F/VEG,
(16)
and these formulæ are consistent with the equation (11) for tan 0,. More-
over, they may be regarded as special cases of the corresponding formulæ in
the theory of surfaces.
We shall now choose as the positive -axis of our local Cartesian system,
the bisector of the angle 0,; further, we shall choose the positive n-axis in such a
manner as to make this system congruent to the fundamental Cartesian system.
412
KINGSTON: Metric Properties of Nets of Plane Curves.
In accordance with the convention we have made regarding the angle 0,,
we find at once, by means of (16),
Ꮎ
2
COS
=+1
VEG+F
2VEG
-
(17)
If we denote by 0 the angle between the positive X-axis and the positive
E-axis, we have
0=(X, u)+0₂/2.
Hence,
cos 0=
sin";
2
X
Yu
a
VE VĒ
ปูน
sin 0= Va+ Vi· •B,
εβ,
VE
where we have made the abbreviations
VEG+F
2VEG
=α,
=ε
VEG-F
2VEG
•
εβ,
=B.
The transformation from the fundamental Cartesian system to the local
system has the form
(20)
X=¿ cos 0—n sin 0+x, Y=¿ sin 0+ŋ cos 0+y,
where cos 0 and sin ✪ are given by (18) and (19).
*
VEG-F
2 VEG
w'x₁ = [2₁(b'y¸—α'yu) —¿½ (b'x¸—α'xu) +23
$6. Transformation of Coordinates from the Covariant Triangle of Reference
PPP, to the Local Rectangular System.
w'X₂= [2₁Y,¬Z¿½X¸—≈3 (XY¸—X¸y)]/ɛH,
w'Xg= [−21Yu+ZqXu+23 (XYu—уxµ)]/εH.
(18)
If x1, x2, x3 are the homogeneous coordinates of any point referred to the
triangle P,P,P., and if 21, 22, 23 are the homogeneous Cartesian coordinates of
the same point referred to the fundamental Cartesian system, we have, by
choosing our unit point in the proper way,
wz;=X₁y(i)+X¿p(i) +230), (i=1, 2, 3),
(19)
@
where is a factor of proportionality. If we substitute in these equations
from (1) and (6), and solve the resulting equations for x1, x2, and x, we get
{εH—x (b'y,—a'y₁)+y (b'x¸—α'x₁) } ]/ɛH,
If we divide the members of these equations by 23, write X for 21/2 and
Y for 22/23, and include the common factor &H/2, in the proportionality factor
* E. J. Wilczynski, "Differential Geometry of Curves and Ruled Surfaces," p. 61.
KINGSTON: Metric Properties of Nets of Plane Curves.
413
w", we obtain as the required transformation from the triangle P,P,P, to the
fundamental Cartesian system, the equations
w″x₁=X(b'y,—a'y₁) —Y (b'x,—a'x„) +[êH−x(b'y‚—a'y„)+y(b'x¸—a′x„)],
w" x₂=Xy,-Yx,—(xy¸—уx,),
@"x3=-Xy₁+Yx₂+(xуu¬у¤…).
w"
น
น
If in these equations we substitute the expressions for X and Y given by (20)
and make use of equations (18) and (19), we obtain, as the transformation
carrying us over directly from the triangle of reference P,P,P, to the local
rectangular system, the equations
wx₁=¿[ɛb'Ha+ɛ(a'E—b'F)ẞ]+n[(a'E—b'F) a—b'Hß]+ɛH√E,
wX₂=¿[ɛHa—ɛFẞ]+n[—Fa—ɛHß],
(21)
wxg={[εEẞ]+n[Eu].
We now proceed to find the position of the covariant points P, and P. with
reference to this local rectangular system. The coordinates of P, referred to
the triangle of reference P,P,P. are x₁=X3=0, X2=1. If we substitute these
values in (21) and solve for and we find the required coordinates of P,
referred to the local rectangular system to be
¿‚=—aVĒ/b', n=ɛßVĒ/b'.
Similarly, for the point P, we find
σ
E。=-aVG/a', no=-εßVG/a'.
(22)
(23)
If we make use of equations (22) and (23) we may obtain from (21) the
following set of transformation equations, whose coefficients are expressed in
terms of p, p, E。, no, a', b' and H :
σ
wx₁ = a'b' (n,—no) §—a'b' (E,—E。 ) n +εH,
wxq=—a'n¸§+a'§on,
WX2=
wx3=b'n‚§—b'¾‚n.
d₁§²+d₂n²+2d¸§n+2d¸n+2düğ=0,
* "One-Parameter Families," equation (83).
(24)
$ 7. The Osculating Conics of the Curves of the Net.
Referred to the triangle P,P,P., the equation of the conic osculating the
curve v=const. at the point P, is
B²x²+4BB'x₂X3—2Bx₁x²+px3=0.*
When we substitute in this equation the values of x1, x2, x3 given by (24),
we get the equation of the osculating conic referred to the local rectangular
system, in the form
(25)
414
KINGSTON: Metric Properties of Nets of Plane Curves.
where
Then
d₁=a'²²² — (4a′b'BB'—2a'b'²B)n,no— (2a′b¹²B—b'²p)n²º,
— (2a'b'²B—b′²p) §%,
222
C
d₂=a¹²²² — (4a′b′BB′ — 2a′b'²B)
dz=−a¹²X² 。。+ (2a'b'²B—b'²p) {{p?p
d=eb'HBE,,
d5=-εb'HBn,⋅
4
d₁d-d² =Н²B² [p—2a’B— (2B′—b′)²],
d3d4—d2d5=Н²Ý² [—a′B§¸+b′ (2B'—b′)§,],
dзd 5-d₁d₁ =Н²Ý² [—a’Bn。+b′ (2B'—b′)n,].
If 。 and no be the coordinates of the center of this osculating conic, we
Ec
find
in which
¿。=[—a′B§。+b′ (2B'—b′)§,]/[p—2a′B— (2B'—b′)²],]
no=[—a'Bn。+b′ (2B'—b′)n,]/[p—2a'B— (2B′—b′)²]..
(26)
It follows at once from equations (26) that the conic osculating the curve
v=const. will be a parabola if
p—2a’B—(2B'—b′)²=0.
If the origin be moved to the center [supposing 4—2a'B— (2B'—b′)²‡0],
and the axes be rotated to coincide with the principal axes, equation (25) will
assume the form
๲² +øn²+ø=0,
where
Ã₁=†[V²G—2B (2B′ —b′) F' — (2a′B—q) E+√r₁],
ý=±[B²G—2B (2B'—b′) F— (2a’B—q) E—√ñ₁],
αe= -H²Ý²/[p—2a'B— (2B'—b′)²],
2
r₁ = (d₁—d₂)²+4d3
=B¹G²+4B² (2B'—b′)²EG+(2a’B—q)²E²—4B³ (2B'—b′) FG
2B² (2a’B—p) EG+4B (2B'—b') (2a’B—q)EF+4H²B² (2a’B—†).
From the first form of r₁ we see that Vr, is always real.
The conic will be an ellipse or an hyperbola according as p—2a'B
− (2B′—b′)² is positive or negative. Further, by a well-known procedure we
find the semi-axes α, and ẞ₁ to be given by
a², B²=
The eccentricity e is given by
2H²B² [p—2a'B— (2B'—b′)²]—¹
B²G-2B (2B'—b') F— (2a'B—q)E ±
Ev₁₂l.
¿²=2√r₁/ |B²G—2B (2B′ —b′) F' — (2a’B—p)E+√r₁\.
KINGSTON: Metric Properties of Nets of Plane Curves.
415
Referred to the triangle P,P,P., the equation of the conic osculating the curve
u=const. at the point P, is
4x²+4A´A″x2X3+A″²x²-2A″ x₁x₂=0.*
When the substitutions (24) are made, this equation assumes the form
d'₁ğ²+d'½n²+2d½§n+2d₁n+2d¿§=0,
(27)
where
2
d'₁ = b²²Д"²² - (4a′b′A'A” —2a'²b′X")n,。— (2a'²b′A” — a'²↓) n³,
where
#2
2
— o
d'₂ = b²² "²² - (4a′b'A'A” —2a'²b’A″) §‚§。 — (2a²²b′A″ — a'²↓) ¿ª‚
p
#12
d's —— b¹²x"² 5,n,+ (2a²²b′X″”—a¹²↓) {。。,
d'₁=—ɛa'НX"¿。,
d'=ɛa'HX″n。.
The coordinates of the center of this conic are given by the equat
·
Ee=[—b'A″E,+a' (2A′—a′) E。]/[4—2b′A″ — (2A′—a′)²],
nc = [—b'X″n,+a' (2A′—a') n。]/[4—2bA” — (2A′—a′)²].
Hence the conic osculating the curve u=const., will be a parabola if
4—26A" — (2A'—a′)²=0.
If the origin be moved to the center (supposing the conic not a parabola)
and the principal axes be taken as axes of coordinates, equation (27) will take
the form
2
d₁e² + d'in² + d'₂=0,
æ=}[A″²E—2A″ (2A′—a′) F— (2b′A”—¥) G+VF],
ý=† [A″²E—2A″ (2A′—a′) F— (2b′A” —4) G—Vĩí],
d's —— H² X" ²/[ 4—2b′A" — (2A'—a′)²],
12
a₁², B₁²
in which
rí=A”¹E²+4A″² (2A′—a′)²EG+(2b′A” —¥)²G²—4A″³ (2A′—a′) EF
—2A″² (2b′A” —4) EG+4A″ (2A′—a′) (2b′A″ —4) FG+4H²A″² (2b′A″ —↓),
and, as before, Vrí is always real.
The conic will be an ellipse or an hyperbola according as 4-26′A”
- (2A′—a')² is positive or negative.
The principal semi-axes a and Bi and eccentricity e are given by
2Ħ²A″² [↓—2b′A” — (2A′—a′)²]—¹
#2
1
A″²E—2A″ (2A'—a') F— (2b'A" —4) G±√r₁|
G+ V₁₂
e²=2√r'₁/ | A″²E—2A″ (2A′—a′) F — (26′A”—4)G+Vrí].
*“One-Parameter Families," equation (87).
416
KINGSTON: Metric Properties of Nets of Plane Curves.
The formulæ of this section will find their application whenever it is
desired to study the metric properties of the osculating conics of any given
net, or else if it be proposed to characterize particular nets by metric proper-
ties of their osculating conics.
§ 8. The Fundamental Theorem.
From the definitions of E, F and G we obtain, by using (3),
E=2aE +2bF, G, 2a'F+2b'G, Fa' E + (a+b') F +bG,
E„=2a′E+2b'F, G,=2a" F+2b" G, F,=a"E+(a'+b″) F+b'G.
}
uv
If we differentiate these equations again, we find that the relations E = Evu :
Fuv=Fou and Go-Gou are satisfied as a consequence of the integrability condi-
=
บน
นง
tions (4). Consequently, if we regard a, b,
b" as given functions of u
""
and v which satisfy the integrability conditions (4), equations (28) form a
completely integrable system of partial differential equations for E, F and G.
If a,
b" are analytic in the vicinity of u=u。, v=vo, equations (28)
may therefore be solved for E, F and G as convergent power series with arbi-
trary initial values. Further, if Ex, F, G, (k=1, 2, 3), be any three linearly
independent systems of analytic solutions, the most general analytic solution
may be written as a homogeneous linear combination of these three with con-
stant coefficients; thus
บน
(28)
E=ECE, F=ECF, G=ΣczGx,
where C1, C2, C3 are arbitrary constants.
Suppose now that a,
"
b" are given as functions of u and v satisfying
the integrability conditions (4), and suppose further that E, F and G are given
satisfying (28). To what extent will these nine functions determine the net?
To answer this question we must find the most general transformation on x
and y
that will leave these nine functions unaltered.
Let us, in the first place, ignore E, F and G. The most general pair of
functions which satisfies equations (3) is obtained from the particular solution
x(u, v), y(u, v), with which we started, by a transformation of the form
x=α₁x + b₁y+c₁, ÿ=α₂x+b₂y +C₂,
(29)
that is, by the most general affine transformation. Consequently, those prop-
erties of the net which are expressed by conditions independent of E, F and G
are of an affine character.
KINGSTON: Metric Properties of Nets of Plane Curves.
417
If we denote by E, F, G the fundamental quantities of the net determined
by x, y, we find
E = (a₁+a²) x²+(b²+b²) y²+2(α₁b₁+α₂b₂) Xãу µ‚
น
2
F= (a²+a²)¤¸¤¸+ (b²+b²) YuY¸‚
น
Ğ= (a₁+a²) x²+(b²+b²) y²+2(a₁b₁+ɑ₂b₂)ж„у¸‚
from which we see that E, F and G will remain unaltered by the transforma-
tion (29) if, and only if, the conditions a²+a2=1, b₁+b²=1 and a₁b₁+a2b2=0
are fulfilled. These equations give a₂=±b₁ and b₂=a₁, where a₁+b}=1,
so that the most general affine transformation (29) which leaves E, F and G
unaltered, consists of a motion and a reflection. Hence a net is determined,
except for a motion and a reflection, by the nine fundamental quantities, E, F,
G, a,
b", where the coefficients a,...., b" are subject to the integrability
conditions (4), and E, F and G must satisfy the conditions (28).
We may formulate our final theorem as follows: Let
Oμu=a0u+bo₂, Ouv=α'Ou+b′0₂, Ovv=a"Ou+b"0。,
น
(3)
be a completely integrable system of partial differential equations, so that its
coefficients satisfy the integrability conditions (4). If two linearly indepen-
dent non-constant solutions, x(u, v), y(u, v), of (3) be interpreted as the car-
tesian coordinates of a point, the equations
X=x(u, v), Y=y(u, v),
นน
UV
determine a net N. The most general net obtained in this way from (3) is an
affine transformation of N.
If the curves u=const. and v-const. of the net N be regarded as
curvilinear coordinates, the square of the element of arc assumes the form
ds²=Edu²+2Fdudv+Gdv². If the coefficients of this quadratic form are given
in any way subject to the conditions (28), the net is determined uniquely
except for its position in the plane and a reflection in any line in the plane.
Finally, it is evident that any net may be studied in this way, since the
nine quantities a, b", E, F, G are easily determined when the finite equa-
tions of the net are given.
The latter half of this theorem is, of course, an immediate consequence of
the fundamental theorem of the theory of surfaces.
$9. Orthogonal Nets.
Let us assume that the net under consideration is orthogonal; that is,
F=0, by (12). Then equations (28) become
a) E.=2aE,
b) E₁=2α'E,
c) 0=a'E+bG,
d) 0=a″E+b'G,
e) G₁=2b'G,
f) G,=2b"G.J
(30)
52
418
KINGSTON: Metric Properties of Nets of Plane Curves.
If we differentiate E, with respect to v, and E, with respect to u, and equate
r
the resulting values, we obtain the equation
α=αu.
Then the first equation of (4) gives
From (31) we conclude that
b'₁=b",
whence, by the fourth of equations (4), we have
a'b'=a″b,
as before. Equation (34) tells us that
b'=Yu,
ba"-a'b'.
a=&u,
where
is an arbitrary function of u and v.
If we differentiate G, and G., as expressed by (30), with regard to v and
u, respectively, we find
(34)
a₂+2aa'
b₂+2bb'
a+2a'2
b₂+2bb"
a'=4%,
***
of (36), we readily find
b"=&,,
where
is an arbitrary function of u and v.
If we differentiate c) and d) of equations (30) with respect to u and v,
making use of a), b), e) and f), and equate the resulting values of the ratio
-G:E, we obtain the equalities
If we make use of equations (33) and (35), these equalities become
Puv+20uPv
b u + 2 b ¥ v
Pvv +243
b₂+264.
a"+24₁a" a"," +24,a" Φο a"
+av+24u¥o ¥u
¥uu+242
b
นน
From the equalities
aű+2aa"
a"+2a'a" a' a"
b'₁+2b'2 b'+2b′b" b b'
=
=
-
(31)
ט
(Puv+24uPv)/(b„+2b¥„)=4,/b and (P…+2p3)/(b+2b¥,)=4,/b
(32)
(33)
(35)
(36)
log blog 4,+2(p−4)+c₁(v) and log b=log 4,+2(p−4)+c₂(u),
where c₁(v) is an arbitrary function of v alone and c₂ (u) is an arbitrary func-
tion of u alone. From these two values of log b we get c₁(v)=c₂(u)=K₁,
where K₁ is an arbitrary constant. Hence, we have
1
b=k₁₂e²(-4).
KINGSTON: Metric Properties of Nets of Plane Curves.
419
In the same way, by using those equalities of (36) which involve a", we
find
a” —k₂¥„e−²(ø−4).
Here k, and k₂ are arbitrary constants.
But we have found that a"b must be equal to p., so that kik₂ must be
equal to unity. Hence, we have
b=kq,e²(*-4), a"
1
2
(37)
where k is an arbitrary constant which does not vanish if we assume that
neither family of the net is composed of straight lines.*
We shall suppose
here that k0, and shall later consider the case where k vanishes.
We have now made use of all our conditions except the second and third
of (4), which must still be satisfied. If we substitute in the second equation
of (4) the values we have found in (33), (35) and (37), we get
น
2
(4²+¥uu—Pu¥u) e²4 —k (P? +P„v—P,¥₁) e²=0.
−2(0—4)',
1
k
(38)
We have then found the following result:
Similarly, if we substitute in the third of equations (4), we obtain precisely
the same result.
Let & and
and let
be any two functions of u and v satisfying the condition (38),
a=&u, a'=&„, a″
¥₂e~2(0-4)
b=ko,e²(-4), b'=4u, b"=4,•
(39)
Then any orthogonal net may be regarded as a solution of a system of partial
differential equations of the form (3) with the coefficients (39). Of course,
not all nets which satisfy such a system of form (3) are orthogonal; they are
all affine to an orthogonal net. However, we may find E and G by quadratures
from (30). We get
E=ce²/ (adu+a'dv) = ce², G=c'e²/(b'du+b''dv) = c'e²¥,
(40)
where c and c' are constants satisfying the condition c+kc'=0, imposed by c)
and d) of (30). If we adjoin the conditions that E and G shall have these
values and that F shall be equal to zero, the corresponding orthogonal net is
determined uniquely except for its position in the plane and for a reflection.
But this result may be simplified a little. The constant k, appearing here, has
*"One-Parameter Families," § 4, p. 490.
420
KINGSTON: Metric Properties of Nets of Plane Curves.
no geometrical significance for we may, by a linear transformation of the inde-
pendent variables, make this constant assume any value we please. To show
this, let us make the transformation
ū=au, v=ßv,
นน
where a and ẞ are arbitrary constants. Then P₂=Ÿã•α, Puu=Puu a², etc., and
(38) becomes
and let
a² (4₁+Yuu—Pu¥u) e² —ẞ²k (9²+P‰ï—Põ¥%) e²=0.
บน
2
We may now put a²: 62 equal to any constant we please and thus make our new
k have any arbitrary value. In particular, put the ratio a²: ẞ² equal to —k,
and our new k becomes equal to -1. The constants c and c' of (40) then
become equal, the coefficients take the form of (39) with −1 in the place of k
and with u and v barred,* and our theorem becomes as follows:
Let & and ↓ be any two functions of u and v satisfying the condition
Moreover, let
(¥%+Yuu−Pu¥u) e² + (P%+P„v¬P¸¥%) e²=0,
vv
2
a=&u, a' =&v,
b=−0₂e²(x-4),
a"=-4e-26–4),
b'=4u, b"=4..
a=u, a'=0, a” = To↓ „e²(4),
b=0, b'=4u, b" =4,
นน
}
E=ce², F=0, G=ce24,
where c is an arbitrary constant. Then there exists an orthogonal net, uniquely
determined by these quantities except for a motion and a reflection, and any
orthogonal net, neither of whose families is composed of straight lines, may be
obtained in this way.
Let us now consider the case when k=0 in (39). If we proceed in a
manner similar to that followed above, we readily obtain
ū=U(u), v=V (v),
(41)
* "One-Parameter Families," equations (17).
where is any function of u alone, and
any function of u and v, such that
¥₁+Yuu—Pu¥u=0, and where I is an arbitrary constant.
2
นน
If we now make the transformation
(42)
KINGSTON: Metric Properties of Nets of Plane Curves.
421
where U and V are defined by the equations
U”—ò̟„U'=0, V=v,
we readily obtain the following result:
Let be any function of the form
¥= log (Vou+V₁),
where V。 and V₁ are arbitrary functions of v alone, and let
0
1
Moreover, let
a=0, a'=0, a" =—₁e²,
b'=4u, b"=4..
b=0,
E=c, F=0, G=ce24,
where c is an arbitrary constant. Then there exists an orthogonal net, of
which the family of curves v=const. are straight lines,* uniquely determined
by these quantities except for a motion and a reflection.
If in (39) we had taken k equal to infinity, that is, if we had taken a"
equal to zero, we should have obtained a similar theorem regarding the
existence of an orthogonal net of which the curves u=const. are straight
lines.
Again, let us suppose that the arbitrary constant k in (42) vanishes. We
then find the following result:
A system of partial differential equations of the form (3), in which all the
coefficients vanish, when considered along with the equations
E=c, F=0, G=c',
where c and c' are arbitrary constants, determines uniquely, except for a
motion and a reflection, an orthogonal net both of whose families of curves are
composed of straight lines. Moreover, any such net can be represented in this
Day.
This system of partial differential equations with vanishing coefficients,
S readily integrated, and gives as one fundamental set of solutions, the
equations
x=u, y=v,
which represent the two families of straight lines parallel to the axes of
coordinates.
* "One-Parameter Families," p. 489.
422
KINGSTON: Metric Properties of Nets of Plane Curves.
Let us consider again equation (38), and let us suppose first that k‡0.
Then if
¥å+Yuu-Pu¥u=0,
(43)
P²+Pov¬P¸¥。=0
(44)
and conversely. By comparing these equations with (13) and (14), and
remembering that in the present case F is equal to zero, we see that the equa-
tions (43) and (44) are respectively the conditions that the minus first and
the first Laplacian transformed nets are orthogonal.
Again, if in (37) the constant k should be zero, the family of curves
v=const. is composed of straight lines, from which it follows that the minus
first Laplacian transformed net degenerates. Further, for this case we found
that a' vanishes, from which it follows by (13) that the first Laplacian trans-
formed net is orthogonal unless it should be degenerate.
A similar conclusion is true if in (37) the constant k should become infinite.
We have then found the
it follows that
THEOREM: If a given net is orthogonal, and if either of its Laplacian
transformed nets is orthogonal or is degenerate on account of the fact that one
of the families of the original net is composed of straight lines, then the other
Laplacian transformed nets will also be orthogonal or degenerate.
Let us suppose we have the conditions
2
น
2
¥₁+Yuu—Pu¥u=0, P²+P₁₂-Pv¥.=0,
นน
fulfilled. Then, by integration, we find the equations
¥₂=√。ex-4, &o=U。e*-*, &₂¥u=U。。,
Po¥u=UoVo,
น
where U, and V, are, respectively, functions of u alone and of v alone. By a
0
0
transformation of variables we may reduce these equations to
¥„=(Vo/Uo)e*-*, •.=(Uo/Vo) e¥-*,
if we assume that neither o, nor 4, is identically zero.
น
du
=S%xdv+U₁, += Su。
น
log U。-log Vo+V₁−U₁+Su。
1
χ
χ
which gives rise to the following integral equation for x,
du
Pu¥u=1,
Putting (Uo/Vo) e¥-*=X;
(45)
+ V₁,
19
— Soxdv=log x-
* "One-Parameter Families,” p. 490.
(46)
KINGSTON: Metric Properties of Nets of Plane Curves.
423
Hence, we may state the following result:
1
If U。, U₁ are arbitrary functions of u, Vo, V₁ arbitrary functions of v, if
u。, v。 are arbitrary constants, and if x is any function of u and v which satis-
fies the integral equation (46), then the equations (41) and (45) together with
E=ce20, F-0,
G=ce²,
will determine (except for a motion and reflection) an orthogonal net whose
Laplacian transforms are also orthogonal. The integral equation (46) may
be replaced by the partial differential equation
J² log X Əx ax-¹
ах
+
Juəv
Ju
obtained from it by differentiation.
From (30) we obtain readily
§ 10. Isothermal Nets.
Let us now suppose that the net is not only orthogonal but also isothermal,
that is, such that
F=0, E:G=U:V,
where U is a function of u alone, and V a function of v alone. Then
Ə² log E/G
Əuəv
Ə² log E
Judv
Hence,
Similarly,
Əv
Ə² log G
ƏuƏv
=0.
=2a,,
=
=0,
=26',.
Uniting the results of equations (47), (48) and (49) we find
Ə² log E/G
მმ.
=2(a„—b')=0.
(47)
(48)
(49)
a=b'”.
(50)
a=b".
(51)
But since the isothermal net is a special case of the orthogonal net, the results
of § 9 must hold here, and we obtain from equations (33), (50) and (51)
a=&u, a'=&v, b'=Pu+s, b"=❀o+t,
where s is an arbitrary function of u alone, t an arbitrary function of v alone,
-
424
KINGSTON: Metric Properties of Nets of Plane Curves.
and is a function of u and v. Hence the function of $9 is replaced here
by +fsdu+Stdv. Putting this value of 4 in (37) we get
a”
and (38) reduces to
(Puu+$u$+8x+s²) e²/adu—k (Poo-P¸t) e−2/idv=0.
(52)
These expressions may be simplified a little by replacing fsdu by s and
Stdv by i, after which the bars may be dropped. Thus we see that the coeffi-
cients of the system of partial differential equations of an isothermal net may
be written in the form
a=&u, a'=&,, a″= 12/12 (44
(Pu+Su) e²(8+t),
k
b=kò̟,e−2(s+t), b'=Qu+su, b″=qv+tv,.
where & is a function of u and v, s a function of u, t a function of v, and k is
an arbitrary constant (which we assume here to be different from zero), and
where these quantities satisfy the relation
1
Ž (Px+8) e²/ (adu+ido), b=ką,e−2/(sdu+idv)
k
2t —0.
(Puu+&usu+suu+s²) e²³ —k (Pvv—P₂t₂) e²=0.
(54)
In order to make (53) more symmetrical, let us put +s=7, and then
omit the bar. We shall find
1
a=Qu—Su, a'=¶„‚ a″ == $₁e²(0+1),
b=ko,e-2(s+t), b'=qu, b"=&v + tv,.
and the condition (54) becomes
Then (56) becomes
where
Let us now transform the independent variables by putting
ū=U(u), õ=V (v).
2
(Pux+Pu³u) e²³—k (P。.—Pvt.) e−2t=0.
[U'²Quu+Qu(n+U'§ã) U′] e² —k [V¹Òïï+9% (Š—V't.) V′]e¬²t =0,
n=U"/U', (=V"/V'.
If U and V are so determined that
n+U'sã=0, S-V't=0, i. e., U'=ae', V'=ße*,
where a and ẞ are arbitrary constants, (57) reduces to
a²Þùu—kß²ïï=0.
(53)
นน
(55)
(56)
(57)
KINGSTON: Metric Properties of Nets of Plane Curves.
425
If we put a²-kẞ2, this becomes
Püü+Pöö=0,
and this is the fundamental condition which must satisfy. The coefficients
(55) now assume the very simple form*
น
ã=&u,
ā=
b=—&v,
We then obtain from (30), by quadratures,
ā'=‡ã‚ ā"
b' =&u
b" =&v ·
E=ce², G=ce²º,
where c is an arbitrary constant. Thus we have found the following
THEOREM: If is any function of u and v satisfying the condition
(58)
Puu+Pvv=0,
then any net which satisfies the differential equation (3) with the coefficients
a" =—&u ›
Фи
I
a=&u
b=—,,
(59)
a'=v,
b'=&u ›
b'=ɖu, b"=&v ›
}
whence
นา
If we have besides
F=0, G=ce20
will be affine to an isothermal net.
E=ce²,
(60)
where c is any constant, the net will be isothermal, and moreover the parame-
ters will be isometric parameters.
To complete the proof of this theorem it suffices to remark that among the
integral nets of a system of form (3) with the coefficients (59) there always
exists an isothermal net; namely, one for which the E, F and G have the values
(60). All other integral nets of the system are, of course, affine to this par-
ticular one.
It is of interest to note here that since the function satisfies equation
(58), which is Laplace's equation, this function must be either the real or the
imaginary part of some monogenic function of a complex variable.
Again, if we put the coefficients (59) in the first and last of equations (3),
we get
@uu=Qu°u—4v0v, Ovv=Quu+8₂0%,
น
บบ
Ouu+Ovv=0,
บ
which says that each of the cartesian Coordinates of the generating point P,
of our net, is the real or the imaginary part of some analytic function of a com-
* "One-Parameter Families," §3, equation 17.
53
426
KINGSTON: Metric Properties of Nets of Plane Curves.
plex variable, a very familiar theorem. The whole theorem might, of course,
be proved on this basis.
Further, the Laplace-Darboux invariants, H and K, are given by the
equations
H=C'+A'B'-A, K=C'+A'B'—B',.*
If in these expressions we substitute the coefficients (59), and remember that
C'=0, we find
H=&uPo—Puv, _K=&uPv—Puv,
and, therefore, the invariants H and K are equal. A very elegant geometrical
interpretation of the meaning of the equality of these two invariants has been
given by Wilczynski in a memoir entitled "Flächen mit unbestimmten Direc-
trixkurven." This interpretation is embodied in the following
THEOREM: "Let us suppose that P is an arbitrary point of a net of plane
curves with equal Laplace-Darboux invariants, and let P1 and P-1 be the corre-
sponding points of the first and of the minus first Laplacian transformed nets.
Then there exists a curve of the first net which touches the line PP at P₁, and
a curve of the second net which touches the line PP at P-1. Let M₁ and N-1
be the conics which osculate these two curves at P₁ and P-1, respectively.
Then either the conic of the pencil determined by M₁ and N_1,
which passes
through P, intersects the segment P₁P_, harmonically, or this conic degenerates
into a pair of straight lines, of which one constituent part is identical with
P₁P-1, or else the conics M, and N_₁ are coincident."
1
1
1
-1
1
-1
We now observe that this theorem is applicable to all isothermal nets,
since any isothermal net may be represented by a system of form (3) with
coefficients of form (59), and hence with equal Laplace-Darboux invariants.
We can easily prove a more specific theorem regarding isothermal nets
than we have yet developed. Under conditions (59) the equations (28) become
E„=28„E—28„F,
F₁=q,(E—G)+29,F, F.=-u (E—G) +2,F,
Gu=24,F+24uG,
F。
E¸=24„E+24F,
น
G₁=-24uF+29.G.
(61)
น
Suppose the net is orthogonal. Then, by the third and fourth of equations
(61), either E=G or qu=4,=0. In the latter case the coefficients (59) all
vanish and the net consists of two families of straight lines. This case we
exclude and therefore E must be equal to G. We then have the result:
*"One-Parameter Families," equations (29).
KINGSTON: Metric Properties of Nets of Plane Curves.
427
Any integral net of a system of form (3) with the coefficients (59) is iso-
thermal if it is orthogonal, and the parameters u and v are isometric parameters.
Again, suppose that E=G, F-0 for some particular regular point u=u。,
v=v。, of the plane. We may express this hypothesis by saying that the net
is isothermal at that particular point. Then E, F and G may be expanded by
Taylor's theorem as power series whose coefficients have as factors the suc-
cessive derivatives of E, F, and G at the point (uo, vo). When equations (61)
are differentiated it is found that the successive derivatives of F may all be
put in the form, p (Eo-Go)+qFo, where E., F. and Go denote the values at
(uo, v。). Also, the successive derivatives of E-G have this same form. Con-
sequently, we find F=0, E-G at all points of the plane. We then have the
result:
If an integral net of a system of form (3), with the coefficients (59), is
isothermal at a single regular point, it is isothermal over the entire plane, and
the parameters are isometric.
§ 11. Effect of the Laplace Transformation on an Isothermal Net.
1
Let E₁, F₁ and G₁ be the fundamental quantities of the first order for the
first Laplacian transformed net. Then
(a'u—a'b')²
a'
4
with corresponding expressions for F, and G₁. If we make use of (7) and (3)
we find
1
a₁-a'b'
a'4
2
2
2
E₁₂ = (de) + (ve)",
E₁ =
F₁=
1
G₁ = [a'a”²E—2a′a″ (a',—a'b" +a'²)F+(a',—a′b" +a'²)²G].
a'4
•
V
G,
[—a'a"F+(a',—a′b” +a'²)G],
Let us suppose that the given net is isothermal with coefficients given by (59).
Then F=0, E=G, and we find
E₁ = (?-?₁₁)". G, F₁ = ??₁P.. 4. G, G₁ = ?₁₂+P:P:
(Puv—PuPv)²
Puv
PuPv
Påv+Pipi. G.
=
1
Φ
Φ
01
The first Laplacian transformed net will be orthogonal also, if F₁=0, that
is, if either ..-PuP。=0, or P=0. The first case is not permissible, otherwise
Pvv
E₁ would vanish and the Laplacian net would degenerate, the curves u=const.
uv
428
KINGSTON: Metric Properties of Nets of Plane Curves.
reducing to points, and the curves v-const. to one and the same curve.
second condition gives, by (58), Puu-0. Hence we can only have
4=au+ßv+Suv+y,
where a, B, Y,
j,
င် are constants. Thus, we find in this case
E₁ [S—(a+Sv) (B+Su)]²
1
G₁
=
Ə² log E₁/G₁
Əuəv
(a+dv)¹(B+du)²
If we take the second logarithmic derivative of E/G₁ with respect to u and v,
we get
-288
2
2
[d—(a+dv) (B+du) ]² *
Hence
Ə² log E₁/G₁
JuƏv
will vanish if, and only if, d=0. It follows that the ratio
E₁: G₁ is of the form, a function of u alone divided by a function of v alone, if,
and only if, 8=0, in which case E₁=G₁. Therefore, the first Laplacian trans-
form will be isothermal together with the original net, if, and only if,
p=au+Bv+y.
(62)
Again, if we let E-1, F-1, G-1 be the fundamental quantities of the first
order for the minus first Laplacian transform, and proceed as before, we find
that the minus first Laplacian transform will also be isothermal together with
the original net, if, and only if, the condition (62) is satisfied.
When & is of the form (62), the coefficients (59) become
a=α, a'=ß,
b=-ß, b'=a,
where a and B are arbitrary constants.
statement:
The isothermality of a net and of one of its Laplacian transforms implies
that of all of the others. Every net of this kind may be regarded as an integral
net of a system of partial differential equations with constant coefficients of the
form (63).
uv
The
bv
This system of partial differential equations with constant coefficients is
readily integrated. The system, written out explicitly, is
?
a" ——a,
b" =ß,
}
(63)
We may now make the following
Xuu=xx₁-ẞx,, Xμv=Bx₂+αx, x¸v=-αx₁+Bx。.
น
น
Xvv
(64)
It may be satisfied by putting x=eau+b², where a and b are constants to be de-
termined. We readily find the non-constant solutions
X=e(B−ia)(iu+v) y=e(B+ia)(−iu+v)
KINGSTON: Metric Properties of Nets of Plane Curves.
429
The corresponding net is imaginary for real values of a and B. However,
these solutions may be put into the form
whence
we obtain
x=eau+Bv [cos (ẞu—av) +i sin (Bu-av)],
(ßu—av)],
y=eau+ßv[cos (ẞu—av) —i sin (Bu—av)],
x+y
2
If we make the imaginary affine transformation
—¿au+Bv cos (ßu—av), X—Y _ ¿au+ßv sin (Bu—av).
2
{=(x+y)/2, n=(x—y)/2i,
=eau+Bv cos (Bu-av), n=eau+ßv sin (ẞu-av),
(65)
and these equations represent a net which satisfies (64) and which is real if a
and ẞ are real. If we now put V¿²+n²=r, n/§=tan 0, we obtain
2
r=eau+Bv, 0=ẞu—av+2nл,
(66)
as the polar coordinates of the generating point of the net. For the curves
u=const. and v=const. in (66) we find
r
นน
do
dr
a
B
β
and r
do
B
dr a
do
respectively. But r
is the tangent of the angle at which the radius vector
dr
cuts the curve. Hence, our net is composed of two families of equiangular
spirals, which intersect their radii vectores at the angles whose tangents are
a/ß and 3/a, respectively. Hence the two families are actually orthogonal
to each other and therefore (by a theorem of $ 10) form an isothermal net.
Hence, we have the result:
uv
"
The isothermal net composed of two orthogonal families of equiangular
spirals has the property that all of its Laplacian transformed nets are also
isothermal. Moreover, it can be readily shown that this is the only real iso-
thermal net having this property.
In order to study the Laplacian transforms of the nets considered in the
foregoing paragraph, we set up the Laplacian transforms for the general net
(3). If we proceed in a way similar to that followed by Wilczynski,* using
equations (7), we find for the first Laplacian transform the system of partial
differential equations,
σuu=ɑ₂u+b₁₂, Ouo=α₁ou+bio,, σ =ᨤ„+b″σ,,
น
น
vv
น
* "One-Parameter Families," § 4.
430
KINGSTON: Metric Properties of Nets of Plane Curves.
;
I
where
a1=
a₁ =
απ
b₁ =
where
1
a₂-ab [b'a' + a'a + a'd'
a'—a'b'
a"
1
`q;a'b' [-a'b' — b'a² + a',
12
a'—a'b'
1
"
a — a 'b' [ — b'ï'a' +œ¦‚ — b'a″
a′
a'
K
a_1=a-
1
b'—a'b'
-b'a'a-
αγαπ
a'' +α'a']
+a'an],
a'
07/0
a'b' —au b₁ = a'b'—d', b₁ =b" — 2a; +a
=
a"
a'
a' a"
Similarly, we find for the minus first Laplacian transformed net, the
equations
_b'a'a" b'a'+' 20"],
a"
a'
+
a_1=
บ
a'
b " a "d
a"
Puu=a_1Pu+b_1Pv, Pwv=a_19x+b_1v, Pov=a_1Pu+b"1Pv,
a'b' —b'.
26'u
+
b'
b'
-b"as- +2a'a'. — ª,"a¹³],
ana'²
"
uu
αΰα,
a"
b_1=
1
bub',
_— —
b's= b; — a'b⋅ [ — b'a' — a'b¹² + b', - b₂b² + bb; ]·
นท b'
b"₁ =
- b₁-a'b² [a'b'+b;b" + b₂b' -
1
b'—a'b'
-1
b
a'b'-b',
b
a"₁=
bubu
bub′2
-
—ab'+b'—a'bb'+abb' — ab' — b₂b²+2bb b₂b]
น
นน
a'b'b,
b
12
-a'b'b" -a'b' + b² - 26"]
บท
b'
When the coefficients (63) are substituted in these two sets of equations,
it is found that the coefficients of each transformed net are exactly the same
as the corresponding coefficients of the original net. Hence the Laplacian trans-
formed nets are affine to the original net. We may state our result as follows:
The only real isothermal nets both of whose Laplacian transformed nets
are isothermal, are those which are composed of two families of equiangular
spirals, and then both Laplacian transforms are affine to the original net.

VITA.
Harold Reynolds Kingston was born on June 26, 1886, at Picton, Ontario,
Canada. His secondary education was received at Picton High School. He
later attended Queen's University, Kingston, Ontario, from which he graduated
in 1908 with the degree of M. A. Since then he has attended the University
of Chicago during the following quarters: Summers of 1908, 1909, 1910, 1912,
1913, 1914, the entire year 1912-13, and the spring of 1914. In 1913 he was
appointed lecturer in mathematics at the University of Manitoba, Winnipeg,
Car
Canada. While at Chicago he studied under Professors Bliss, Bolza, Dickson,
Laves, Lunn, MacMillan, Moulton, Slaught and Wilczynski, to whom he wishes
to express his gratitude for their kindly interest at all times, and especially to
Professor Wilczynski under whose guidance the doctor's thesis was written.
1
bra
1
1


UNIVERSITY OF MICHIGAN
AUTHOR
MATHEMATIO
QA
603
455m
Metric properties of Nets of one
TITLE
Curves
Be the
F
BOOK CARD
Mingston
SIGNATURE
---
I
ISS'D RET'D
- - - - -
3 9015 05116 2165
¿
FAC
#
K.