MULTILINEAR FUNCTIONS
OF DIRECTION
641
MULTILINEAR FUNCTIONS
OF DIRECTION
CAMBRIDGE UNIVERSITY PRESS
C. F. CLAY, MANAGER
LONDON : FETTER LANE, E.G. 4
NEW YORK : THE MACMILLAN CO.
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CALCUTTA I MACMILLAN AND CO., LTD.
MADRAS j
TORONTO : THE MACMILLAN CO. OF
CANADA, LTD.
TOKYO : MARUZEN-KABUSHIKI-KAISHA
ALL RIGHTS RESERVED
MULT ^INEAR FUNGI IONS
OF DIRECT ION
AND THEIR USES IN DIFFERENTIAL GEOMETRY
BY
ERIC HAROLD NEVILLE
LATE FELLOW OF TRINITY COLKHfE, CAMBRIDGE
PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, READING
CAMBRIDGE
AT THE UNIVERSITY PRESS
1921
PREFACE
rjlHE distinctive feature of this work is that the functions dis-
J- cussed are primarily not functions of a single variable direction
but functions of several independent directions. Functions of a
single direction emerge when the directions originally independent
become related, and a large number of elementary theorems of
differential geometry express in different terms a few properties
of a few simple functions ; since one of the objects of the essay is
to emphasise the coordinating power of the theory, the presence of
many results with which every reader will be thoroughly familiar
calls for no apology.
In the applications to the geometry of a single surface two
functions thought to be new are described. The first, studied in
Section 4, depends on two tangential directions, reduces to normal
curvature when these directions coincide, and is called here bilinear
curvature. I became acquainted with this function in 1911 and used
it in lectures early in 1914. The second, the subject of Section 6,
depends on three directions, and reduces to the cubic function as-
sociated with the name of Laguerre ; the function is symmetrical,
and because the equations of Codazzi can be read as asserting its
symmetry I have called the general function the Codazzi function.
The theory of multilinear functions does not merely coordinate.
It affords simple proofs of the relations between the cubic functions
of Laguerre and Darboux (6'231, 6'234) and of formulae (7'242,
7*351, 7*352) for the twist of a family of surfaces, and it leads
naturally to expressions (7 '241) for the rates of change of the two
principal curvatures of a variable member of a family of surfaces at
the current point of an orthogonal trajectory of the family, expres-
sions that are interesting because their existence was deduced by
Forsyth in 1903 from an enumeration of invariants.
E. H. N.
June, 1920.
NOTE
For the sake of brevity, the space considered is real, but the
restriction operates only to the same extent as in other branches
of differential geometry. If it is removed, the intrinsic distinction
between the positive square root and the negative square root of
a given uniform function has to be replaced by a more artificial
distinction based on a dissection that is to some extent arbitrary.
And there is always a possibility that results need modification
if isotropic lines or planes are involved ; as a rule, nul vectors are
admissible as arguments but nul directions are not.
CONTENTS
PAGE
PREFACE . 5
NOTE 6
TABLE OF CONTENTS 7
PRELIMINARY PARAGRAPHS 9
O'l Vectors ; radials.
0'2 The projected product of two vectors.
0'3 Vector frames.
0'4 Cartesian axes.
0'5 Directions in a plane.
0*6 Angular differentiation.
1. LINEAR AND MULTILINEAR FUNCTIONS .... 13
1*1 Definitions of linear and multilinear functions.
1'2 Notation for multilinear functions.
1'3 The core of a multilinear function.
14 The use of reference frames.
1*5 The source of a linear function ; the projected product of two cores.
1'6 The derivative of a variable core ; the rate of change of a multi-
linear function.
1'7 The gradient of a multilinear function dependent on position in
space ; the rate of change of such a function along a curve.
1/8 The angular derivatives of a multilinear function.
2. FUNDAMENTAL NOTIONS IN THE KINEMATICAL GEO-
METRY OF SURFACES AND FAMILIES OF SURFACES . 22
2'0 The vectors to be examined.
2-l Normal curvature and geodesic torsion.
2 '2 Geodesic curvature.
2 '3 Bilinear curvature ; the relations of bilinear curvature to normal
curvature and geodesic torsion.
2'4 Swerve ; geodesic curvature as a swerve.
2 '5 The curvature and torsion of a curve on a surface.
2 6 The vectors connected with a family of surfaces.
2'7 The swing of a tangential direction about the orthogonal trajec-
tory of a family ; the twist of a family ; the spread of a family in
a tangential direction.
2 '8 The swing and the spread as spins of a frame.
3. SURFACES AND MULTILINEAR FUNCTIONS ASSOCIATED
WITH A FUNCTION OF POSITION IN SPACE ... 31
31 The surfaces denned by a scalar function of position.
3*2 Definition of the linear rate of change of a function <£ ; the gradient
and the slope of 4>.
3'3 Definition of the bilinear rate of change of the function <£.
3*4 Definition of the multilinear rates of change of $ ; the symmetry
of these functions.
3*5 Evaluation of the linear function <£Tr of an arbitrary vector r
8
PAGE
4. THE BILINEAR CURVATURE OF A SURFACE ... 35
4'1 Deductions from the bilinearity of the bilinear curvature ; Dupin's
theorem ; the mean curvature of a surface at a point.
4 -2 The bilinear curvature of a 4>-surface ; the symmetry of the bi-
linear curvature ; particular theorems implying this symmetry.
4 '3 Geodesic torsion and bilinear torsion ; angular derivatives.
4'4 The principal curvatures and directions of a surface at a point,
the amplitude of curvature ; lines of curvature.
4-5 Asymptotic directions and torsions ; Enneper's theorem ; asymp-
totic lines.
5. THE BILINEAR RATE OF CHANGE OF A FUNCTION OF
POSITION 50
5*1 The relation between the spread of the ^-family and the bilinear
rate of change of <f>.
5*2 Relations between the bilinear rate of change of <l>, the linear rate
of change of the slope of <£, and the spread of the ^-family.
5*3 The symmetrical bilinear function ^2p(dlA-/dSq}.
5*4 Concluding remarks on the formulae of this section.
6. THE CODAZZI FUNCTION 54
6*1 Definition of the Codazzi function ; its trilinearity ; the relation
of this function to the cubic functions of direction of Laguerre
and Darboux ; the form of the function in special cases.
6*2 The symmetry of the Codazzi function ; the Codazzi equations ;
relations between the function of Laguerre and the function of
Darboux.
6'3 The use of the Codazzi equations for the calculation of geodesic
curvatures.
7. THE TRILINEAR RATE OF CHANGE OF A FUNCTION OF
POSITION 60
7'1 The relation between the trilinear rate of change of 3> and the
function <£2/> (dl^/dsq}.
7 '2 The rate of change of a bilinear curvature and of the principal
curvatures along the orthogonal trajectory of a family ; the twist
of the <E>-family.
7 '3 Calculation of bilinear curvature and of twist by means of a
Cartesian frame.
8. FUNCTIONS OF DIRECTION ON A SURFACE ... 71
8'1 The fundamental difficulty ; the Darboux gradient.
8*2 The multilinear rates of change of a function <J> on a surface.
8*3 Relations between the bilinear rate of change of <£ and the geo-
desic curvatures of ^-curves and <$>-orthogonals.
8'4 The gradient of the Codazzi function.
INDEX OF DEFINITIONS 79
INDEX OF SYMBOLS . 80
MULTILINEAR FUNCTIONS OF DIRECTION
Preliminary Paragraphs
O'l. The association of a direction OR with a real number r,
which may be positive, zero, or negative, determines a vector
which will be denoted by rB\ of this vector r will be called the
amount in the direction OR. The vector rR possesses in addition to
the direction OR the reverse direction, which we shall denote con-
sistently by OR', and the amount of rR in the direction OR' is — r.
The zero vector has all directions, and its amount in every direction
is zero ; a proper vector has only two directions and two amounts.
A vector of amount unity is called a unit vector or radial. The
vector 1^ has the direction OR' as well as the direction OR, but
there is no confusion in describing the direction 0 R as the direction
of the radial.
02. There is an infinity of angles between two directions in
space, but these angles all have the same cosine. If €ES is an angle
between directions OR, OS of two vectors r, s whose amounts in
these directions are r, s, the product rs cos €RS depends only on r
and s, not on any choice which is arbitrary when the vectors are
given; this product will be called the projected product* of r and
s arid denoted by /rs. The projected product of a vector s and a
radial 1^ is the projection of s in the direction OR, and the pro-
jected product of two radials is the cosine of the angles between
their directions.
0'3. Any three vectors p1, p2, p3 which are not coplanar form a
vector frame, in which the arbitrary vector r is determined by the
three scalars f, ?;, f such that
* Many writers have not hesitated to call this the scalar product, although the
function is the negative of that for which Hamilton designed the name. There
is no universal notation ; to transfer the letter as well as the name rendered familiar
by Hamilton and to appropriate brackets of some special kind are courses equally
open to criticism, and if there is here a vacant role in the symbolism of vector
analysis it is one for which the initial of Gibbs and Grassmann may be cast with
peculiar fitness. Neither r . s nor r x s is quite secure from misunderstanding,
since Heaviside uses the one for a dyad and Gibbs the other for a vector product; I
am conservative enough to regard rs as denoting a quaternion.
10
The polar of the frame p1pap8 is the frame p^p3 such that
^ is unity or zero according as h and k are the same or
different, that is to say, such that p1 is at right angles to both p2
and p3 and £plpl is unity, p2 is at right angles to both p3 and p1
and c^p2p2 is unity, and p3 is at right angles to both p1 and p2
and c^p3p3 is unity. If
r =
then Jrpl = \,
and since the relation between the two vector frames is reciprocal,
considered as derived from the frame pxp2p3, the projected products
X, //,, v are naturally called the polar coefficients of r.
0'4. When we have occasion to use a Cartesian frame of refer-
ence, we shall not assume it to be trirectangular. We shall use
a, /3, 7 for angles between the axes of reference and A, B, T for
angles between the planes, A being an angle from the second plane
to the third round the first axis just as a is an angle from the
second axis to the third in the first plane ; A, B, F are external
angles of the spherical triangle of which or, j3, 7 are sides. Also we
shall denote by T the sine of this triangle, that is, we shall write
T = sin /3 sin 7 sin A = sin 7 sin a sin B = sin a sin 0 sin F.
Then if x, y, z are the components and I, 'in, n the projections
of any vector,
0'41 f I = x + y cos 7 4- z cos /9,
m = x cos 7 4- y 4- z cos a,
n = x cos (3 4- y cos a + z,
and on the other hand
0'42 [x = ftp-2 sin2 a + mT'1 cot F -f nT~l cot B,
y = IT~1 cot F + mT~2 sin2 /3 4- riT"1 cot A,
_z = ZT-1 cot B 4- mT~! cot A 4- wT~2 sin2 7.
The projected square of the vector, having the value Ix 4- my 4- nz,
can be expressed as a quadratic function of components alone by
means of 0'41 or of projections alone by means of 0*42 ; thus
0'43 r2 = a? 4- y*' 4- z1 4- Zyz cos a 4- ^zx cos /3 4- 2#?/ cos 7,
11
but in terms of projections alone r2 is most readily given by means
of a determinant ; eliminating x, y, z between 0'41 and
r2 == lac + my + nz
we have
= 0,
1 cos 7 cos/3 I
cos 7 1 cos a m
cos /3 cos a 1 n
I m
n r2
T-2
1
cos 7 cos /3
cos 7
1 cos a
cos/3
cos a 1
Z
m ?i
that is
0'44 ?-2 = -T-2 1 cos 7 cos/3 I
m
0
If OP is a direction perpendicular to each of two directions
OR, OS and if eRS is an angle from OR to OS round OP, the cosines
of OP are given in terms of the ratios of OR and 0$ by
0'45 (7P, mP, ??P) sin e™ = T
X
yR
'72
and the ratios of OP in terms of the cosines of OR and 0$ by
046
OP,
sn € =
The components x, y, z, and the projections I, m, w, of a vector
r in the Cartesian frame OABC are the coefficients and the polar
coefficients of r in the vector frame composed of the radials
1^, 1£, lc. But it must be observed that the polar of this vector
frame is not as a rule the Cartesian frame polar to OABG but
consists of vectors of amounts T~J sin a, T"1 sin /3, T"1 sin 7.
0'5. For the comparison of directions in one plane actual angles
can be used, a definite direction of angular measurement being
adopted. An angle from OS to OT will be denoted by eST; this
angle is not free from ambiguity, for any restriction on the magni-
tude or sign of angles is not merely superfluous but irksome, but
cos eST and sin €ST are determinate functions of the two directions
0$, OT, and so also is the rate of change of eST with respect to any
variable on which the directions depend in a regular manner.
12
When axes of reference A'OA, B'OB are being used in the
plane, an angle eAB will be denoted by &>. To deal simply and
symmetrically with a variable direction OT, angles eAT, eTB are
both required ; the sum eAT 4- eTB must differ from o> by an integral
multiple of 2?r, and a, fi, or if necessary aT, /3r, will be used for a
pair of angles eAT, eTB subject to the convention a-ir/3 = w.
The theory of multilinear functions of direction in a plane per-
sistently associates with each direction one of the perpendicular
directions, and the direction which makes a positive right angle
with OT will be denoted by OE or by OET; for OES will be sub-
stituted OD.
0'6. A function F(T) of the direction OT in a plane regarded
as a function F(eWT) of an angle to OT from a fixed direction OTF,
requires for its study its derivative dF (eWT)ldeWT. This derivative
is itself a function of eWT, that is, of the direction OT, but since it
does not really depend on 0 W it may be called simply the angular
derivative of F(T) and will be denoted by daF(T):
0 61 daF(T) = lira [{F(8) - F(T)}/eTS].
S+T
A function of a number of independent directions in a plane has
an angular derivative with respect to each of them, and the various
angular derivatives of F(Q, R, ...) will be written daQF(Q, R, ...),
daRF(Q, R, ...), and so on.
If the directions OQ, OR, ... in a plane are made dependent on
a direction OT in that plane, the function F(Q, R, ...) becomes a
function of OT, having an angular derivative with respect to OT
given by dF_de,Ea+dF_ rfej™ +
d€Wg deWT v€WR deWT
The dependence of OQ, OR, ... on OT is a dependence of angles
€TQ, eTR, ... on OT, and since
€WQ = €WT + €TQ » €WR = eWT + €TR > ••• J
the derivatives deWqjdeWT, deWRjde]VT, ... have the values
and
0'62. The angular derivative with respect to OT of a function
F(Q, R, ...) of directions themselves dependent on OT is
(1 + daeTQ) daQF+ (1 + da €TR) daRF+ ....
13
In particular
0'63. If the directions OQ, OR, ... make constant angles with OT,
the angular derivative of a function F (Q, R, ...) with respect to OT
is the sum of the several angular derivatives daqF, daRF, —
An angular derivative in space is a function of two directions at
right angles. If ON is a direction at right angles to OT, the di-
rection of angular measurement in the plane to which ON is normal
is related to ON by the spatial convention ; a function of direction
in space becomes by the restriction of its argument to the plane
normal to ON a function of direction in that plane, with an angular
derivative whose value at OT depends no less on ON than on OT.
1. Linear and Multilinear Functions
I'll. Intrinsically, a linear function of a variable vector is a
function whose value for the sum of two vectors, and therefore also
for the sum of any finite number of vectors, is the sum of its values
for the several components ; a multilinear function is a function of
a number of independent vectors that is linear in each of them.
1'12. The value of any linear function for the argument rR is
r times the value of the same function for the argument 1^. If a
frame of reference OABC is used and the components of the variable
vector r are x, y, z, then since r is x\A + y\B +• z\c, a function F (r)
which is linear is necessarily expressible as
and conversely a function of r which is of the form scL + yM + zN
where L, M, N do not depend on r must be linear :
1*121. A linear function of the vector r is a function which is a
homogeneous linear function of the components of r in any frame.
T13. A function whose arguments are radials may be regarded
as a function simply of direction, and a function of direction is said
to be linear if the function of r and OR obtained by multiplying
its value for the direction OR by the number r is a linear function
of the vector r^. We can if we wish avoid the explicit mention of
vectors in the definition of a linear function of direction, either by
introducing implicitly the definition of the sum of two vectors or
14
by using a frame of reference. A function of direction is linear if
given any two successive steps OQ, QR, of lengths p, q, whose
resultant OR has length r, the sum of p times the value of the
function for the direction OQ and q times the value of the function
for the direction QR is r times the value of the function for the
direction OR. And a function of direction is linear if it is expressible
as a homogeneous linear function of the ratios of the direction with
reference to any frame ; with this last definition we have to notice
that a function of the direction ratios which is not given as a homo-
geneous linear function may in fact be expressible in* this form in
virtue of the quadratic identity to which the ratios are necessarily
subject.
1'14. The definitions of linear and multilinear functions of
vectors and directions are designed to restrict as little as possible
the nature of the function. In the present work the ultimate
equations are scalar, but vectors and other functions are essential
to the processes.
1'21. For an arbitrary function of the k vectors r1} r2, ... r^, the
natural notation is of the form ^(r^ ra, ... rk), but for a function
that is multilinear there is more even than brevity to be gained
by substituting the form Pr^ ... rk, or Pkrlr2 ... rfc if the degree
of the function has to be made prominent, for this form emphasises
the identities such as
P (s + 1) r.2 . . . rk = P sr2 . . . rk + P tr.2 . . . rk
involved in the definitions. It must be remembered that the order
in which the vectors are written is not irrelevant unless the function
is symmetrical in its definition: a function defined unsymmetrically
may be in fact symmetrical, in which event the order of writing the
variables does not affect the truth of any formulae but the assertion
of the symmetry is in itself significant.
1'22. The function of direction P1A1B ... 1^ is often denoted by
PAB...K- Were an attempt made to deal with functions of direction
without mention of vectors this compact alternative would be used
throughout, but since as a rule the sum of two radials and the rate
of change of a variable radial are vectors that are not radials, the
operations that are most natural commonly involve functions of
directions and functions of vectors in one equation, and the effect
15
of too persistent a substitution of PAB...K f°r P^A^B ... IJT is un-
sightly.
1*31. The advantages of detaching the symbol P from the group
PrjFa ... rk are secured by the method of Russell : P denotes the
relation of a value of the function to the set of vectors on which
the value depends, and is called a multilinear relation. The relation
P will be described as the core of the function Pr^ ... rfc and of
the corresponding function of direction PAB...K-
1'32. To prove that
1*321. The sum of any finite number of functions multilinear in
the same set of vectors is itself a multilinear function of the set
and that
1*322. The product of a multilinear function by any scalar is a
multilinear function
is easy. These propositions give further justification of the notation
we are using, and provide a basis for definitions of the addition of
cores and of the multiplication of a core by a scalar :
1*323 (2P) rira . . . r* = 2 (P r,r2 . . . r*),
1*324 (rP) rlr,...rk = r(Prlr2... rk).
1*33. In a multilinear function of degree k any h of the variable
vectors or directions may play a parametric part. The function is
then regarded as multilinear in the remaining k — h variables, with
a core which is a function of the h parameters, and we have only to
compare 1*323 with the original definition of a multilinear function
to see that this core is multilinear in the parameters ; it is a multi-
linear function which is neither scalar nor vector. Thus the bilinear
function Prs or PRS yields two linear functions which are written
as (P#s) r and (Pr#) s or (P*S)R and (P_R*)S; if the degrees of the
different functions are to be exhibited, the two linear functions of
direction subsidiary to P*ES are shewn as (P1^s)lR and (PlR*)ls-
1*41. If ^O1, %h2, %ft3 are the coefficients of rh in a vector frame
Pap2p3, the linearity of a function Pr^ ... iv^r^ in rk implies the
equality
1-411 Prlr9...rk-r =
16
and the expansion of each of the variable vectors in turn in th(
same way gives the result
T412 Pr1r2...rfc_1rfc =
S^imi X*"* • • • Xk-ink~lXX:™k P P™1?™2 . . . pmk-lpmk}
where each of the affixes mlt m2, ... mje-i, tnk stands for one of the
three symbols 1, 2, 3, and the summation extends to the 3fc possible
terms. To obtain a formula in terms of the projected products
</rhpl, c^r^p2, #Tfip9, all that is necessary is to remember that these
projected products are the coefficients %^, %h2, %V of r^ in the frame
pip^p3 polar to pap2p3, whence
1413 Pr1r,...iv.1rfc =
Particular cases of 1*412 and 1*413 are expressions for the multi-
linear function referred to a Cartesian frame, namely
1-421 P rar2 . . . rk = Sc^ic^ . . . ckm*PMM ... Mt,
where c^1, erf, ch3 are the components of rh and Mh stands for A, B,
or C according as mh stands for 1, 2, or 3, and
1-422 P 1-^2 . . . rk = Zpnip™* . . . pkm*P kmikwa . . . k?»3,
where p^, ph*, pr? are the projections of r^ and k1, k2, k3 are those
vectors normal to the planes OBC, OCA, GAB whose projections
on (L4, OB, OC are unity.
1'43. From 1'412 follow two fundamental theorems:
1*431. The value of a multilinear function is known for every set
of vectors in space if it is given for every selection from any three
vectors that are not coplanar ;
1*432. A function that is multilinear is wholly symmetrical if it
is symmetrical with respect to any three vectors or any three direc-
tions that are not coplanar.
Because of the second of these results, any two groups of theorems
which express the complete symmetry of the same multilinear
function with respect to different sets of vectors may be regarded
as equivalent : this is one of the ways in which results diverse in
form are coordinated by the theory developed here.
The two theorems of the last paragraph assume the functions
involved to be defined for all sets of vectors or directions in space.
17
If only a single plane is in question, it is sufficient in T431 for the
selection to be made from two vectors in that plane but not collinear
and in 1/432 for the symmetry to be established for two such
vectors.
1-51. That
1*511. The projected product of two vectors is a bilinear function
of these vectors,
and that
1*512. The projection of a constant vector on a variable direction
is a linear function of the direction,
follow from the elementary distributive property of the projected
product. The converse of these theorems is also true, for if a, b, c
are scalars, ax+by + cz is the projected product of the vector of
components x, y, z and the vector of projections a, b, c, and if the
former vector is the radial lr, the same sum represents the pro-
jection of the latter vector on OT:
T513. Every linear scalar function of the variable vector r can
be exhibited in one way only as the projected product of r and a
vector independent of r, and every linear scalar function of the
variable direction OT in one way only as the projection on OT of
a vector independent of OT.
It is convenient in both cases to call the vector the source of the
linear function.
1*52. The projected product of the sources of two linear functions
affords the simplest example of a scalar which depends only on
two cores, and if the cores are Q and R this projected product will
be denoted simply by QR. If Q and R themselves involve variables
QR is of course a function of these variables. Thus if from two
bilinear functions QAB, RCD are formed two linear functions
(QA*)B (R*D)C> the projected product QA% R%D is a function of the
directions OA, OD', it is in fact a bilinear function, and so can be
used to form on the same principle an infinity of other functions,
such for example as Qt£ Qt# R#D.
1'53. Any two cores of the same degree give rise to a function
corresponding to the projected product of the sources of two
linear functions, but in the absence of a direct definition of this
function in general, we must describe the function defined in the
N. 2
18
last paragraph in such a way as to indicate the line of extension.
Referred to a frame OABC, the linear function P r can be expressed
in the two forms
lpa'r 4- nt<pyr -4- npzr, xplr + ypmr + zpnr,
where lp, mp, np are the projections and a^*, yp, zp the components
of the source, and since the value of the projected product of two
linear cores Q, R is given by the two sums
not only are these sums equal but their value is independent of
the particular frame OABC. Similarly the multilinear function
Pr^Fa ... rk whatever its degree can be expanded with reference to
the frame OA EG in the two forms
and if Q, R are any two cores of the same degree k the sums
are equal and have a value independent of the frame OABC ; this
common value defines the projected product of the cores Q, R, and
is denoted by QR.
1'54. When once the projected product QR is defined for cores
of arbitrary degree, a whole group of functions is seen to be
derivable from any two or more cores, or indeed from any one core
of degree not less than two. For example, from a trilinear core P
by regarding one direction OB as parametric we derive a bilinear
core jP#jBt> and if Q is a bilinear core, the projected product P#B+Q,
better denoted by P#s\Q*-t> ig itself a linear function of OB and
gives rise by combination with any linear scalar core R to a pro-
jected product (P*§t Q*t) ^§ 5 without attempting to classify func-
tions of this kind we must recognise their nature when they
present themselves.
Symmetry reduces the number of distinct functions to which
a given multilinear function is related. For example, if QAB, ROD
are unsymmetrical bilinear functions, the four bilinear functions
Q*s^*r> Qs*^*r> Q*s^r*> Qs*^r* are distinct, but if the original
functions are both symmetrical, the four derived functions coincide.
1'55. If two multilinear functions Qr1r2...rA;, Rr^...^ of the
same degree are denned only for vectors in a particular plane, the
19
projected product QR can be defined as in T53, with the sole
difference that the frame of reference is two-dimensional. More-
over, if one multilinear function Qrlr2...rk is defined only for
vectors in a particular plane and another .Rr1r2...rjb of the same
degree is defined without restriction on r1? ra, ... rk, a projected pro-
duct is definable by the restriction of the arguments of Ur1ra...rfc
to the plane in which they can serve as arguments to (2r1r2...rj;
also, and no confusion can be caused by denoting this projected
product by QR ; the only point to be remembered is that if by a
change in the definitions the restriction on the arguments of
QrIr2...rfc is subsequently removed, QR will be in danger of
acquiring a second meaning inconsistent with the first.
1*61. Should the function P r^ . . . r^ involve any variables other
than the vectors rlf r2, ... r^, then if a change in these additional
variables is not necessarily accompanied by a change in the vectors
it is the core that is to be regarded as a function, and a limit of
Pi^ra...!** for variations in which r^ r2, ... r^ are constant is a
function of rlf ra, ... r^ which if multilinear can be used to define
a limit of P. It is difficult to be more precise in this assertion
without placing undue restriction on its scope ; the case which is
for us important affords the best commentary; if Pr^a...^ is a
scalar or a vector depending on a scalar variable t in such a way
that for each particular set of values of rlf r2, ... Tk there is a
derivative d(Prlr.2...rk)/dt, it follows from 1*321 and 1*322 that
this derivative is multilinear in rl} ra> ... r^, and dP/dt is defined
as the core of d (P r^ . . . rk)/dt.
1*62. It is oh the assumption that the vectors rl5 ra, ... rk not
only can be but are independent of t that d(Prlr2...rk)/dt is
multilinear and introduces dP/dt. But this derived core is of no
less service in the evaluation of d(Prlr2...rk)/dt when the vectors
vary with t, the symbols, in consequence of T323 and the defini-
tions, grouping themselves in the familiar mariner
1-621 d (P r, r2 . . . rk)/dt = (dP/dt) rx r2 . . . rk + P (drjdt) rz...rk
+ Prl(drz/dt) ... rfc+ ... +Pr1r2... (drk/dt).
This identity is sometimes of service for the calculation of
(dP/dt)rlr9...rk, but there is nothing in the formula so used to
shew why the function obtained is multilinear.
2—2
20
1*71. The multilinear functions of differential geometry are not
so much functions of directions in space as functions of directions
at a point; in other words, they are functions of direction with
cores depending on a variable point.
Let Pk rl r2 . . . rk be such a function, dependent on the position
of a point Q, and suppose Q to be confined to a curve through a
particular point 0. On this curve Pk can be regarded as a function
of the arc s measured to Q from some fixed point, and to calculate
the rate of change dPk/ds a frame of reference may be used ; then
dPk dz
=
ds dx ds dy ds dz ds '
that is,
1-712 dP*lds = PfacT + PfyT + PfzT,
where Pf, P2k, Pf are functions of position having no relation to
the curve described by Q and XT, yT, ZT are the direction ratios of
the tangent to this curve. Hence
1'713. The rate at which the core of a multilinear function depen-
dent on position changes at a point 0 ivith respect to the arc of a curve
through 0 is the same for all curves whose direction at 0 is the same,
and can be called simply the rate of change in the common
direction, and further
1*714. The rate of change of the core of a multilinear function in
a variable direction is a linear function of that direction.
If the rate of change of the core Pk in the direction OR is
dPk/dsR, the function r (dPk/dsR} rx r2 . . . r^ is linear in the vector
rB as well as in the k vectors r1} r2, ... r^ and is therefore a multi-
linear function of degree k + 1 ; its core, which depends only on
the variation of Pk in space, is called the gradient of Pk and de-
noted by P*+1. Sometimes the function Pk+l rx r2 . . . r^ rk+l is called
the gradient of the function P*rj r2 ... r%.
That linear and bilinear functions have a part to play follows
from 1*511 and 1*512, and on account of 1*714 the appearance of
functions of higher degrees is inevitable, but it is not every useful
multilinear function that is derivable from some linear or bilinear
function by the formation of successive gradients.
1*72. If r1; r2, ... r& instead of being independent variables are
definite functions of the position of the current point 0 on a curve,
21
the rate of change of the multilinear function P* i^ r2 . . . TJ. with
respect to a parameter t on the curve is given by
1-721 d (Pk r, r2 . . . rk)/dt = Pk+l rt r2 . . . rk w
+ Pk (drjdt) r.2 . . . rk + Pk r, (dr,/dt) . . . rk + . . . + Pk r, r.2 . . . (drkldt)t
where w is the velocity of 0 with respect to t. In particular, the
rate of change of a function of direction PkAB...K along a curve in
the direction OL is given by
1-722 dPkAB KldsL = ^^KL + P"^ K(d\AldsL)
the vector dlH/dsL is not as a rule a unit vector, nor is this vector
a linear function of OL unless the direction OH is independent of
the direction OL, so that in the majority of applications it is only
the first of the terms on the right of T722 that is itself a function of
direction, but if dlH/dsL can be put into the form pp + qg + rR,
where OP, OQ, OR are known directions, then
1-723
and the multilinear function of direction whose core is Pk reappears
with different sets of directions for arguments and with scalar
multipliers that do not depend on P*.
T81. If the source of the linear function QT of direction in a
plane is the vector rR, then
1*811 QT = r cos €RT
and therefore
da QT = r cos (eRT 4- \ TT) = r cos eRE,
that is,
1-812 daQT=QE:
The value of the angular derivative of a linear function in the
direction OT is the same as the value of the function itself in the
direction luhich makes a positive right angle with OT.
In other words
1*813. The angular derivative of a linear function QT of direction
in a plane is the linear function whose source is obtained by rotating
the source of QT through a positive right angle.
T82. The partial angular derivatives of a multilinear function
of independent directions in a plane are given at once by 1'812, and
22
extension to functions multilinear in interdependent directions is
made by the use of O62 and 0'63. For example, if PST is bilinear,
1*821 dasPgy = 1 DT, daTPST = P$E>
1*822 daPff = -L ET ~t~ -* TE == —
1*823 da L fE = Jr EE — -LTT ~
from 1-822,
1'824. If PST is any bilinear function of directions in a plane,
the sum of the values of the quadratic function PTT for two directions
at right angles is constant;
1'823 shews that PTE - PET also is constant, but this is merely a
second version of the same theorem, obtained by regarding PSE as
a function of OS and OT.
1'83. To look for the angular derivative in an arbitrary plane
of a scalar linear function of direction in space is to reach familiar
ground. Let RT be the linear function whose source is r, let ON
be any direction in space, and let 8 be the component of r at right
angles to ON', if OT is a direction at right angles to O^V, the pro-
jection of r on OT is the projection of s on OT, and therefore the
source of RT in the plane at right angles to ON is s ; it follows
from 1*813 that the angular derivative of RT in this plane has for
its source the vector obtained by rotating s through a positive
right angle round ON, and this we recognise as the vector product
of r and ~\.N.
2. Fundamental Notions in the Kinematical Geometry
of Surfaces and Families of Surfaces
2'0. To prepare for geometrical applications of the theory of
multilinear functions it is necessary to examine the different vectors
of the form dlH/dsK, where each of the directions OH, OK is either
constantly normal or constantly tangential to a definite surface
through 0, and the rate of change is with respect to the arc of
some curve whose direction at 0 is OK. It is assumed that by a
satisfactory convention one of the directions at right angles to the
surface is chosen to be called the normal direction, and that there
is a spatial convention by which the choice of the normal direction
determines the direction of angular measurement in the tangent
plane at 0. The normal direction is denoted by ON, and OR, OS>
23
0 T will be used for arbitrary tangential directions ; for the rest,
the notation is that described in 0'5.
2*11. The vector d\NjdsT is the velocity of the Gaussian image
of 0 as 0 moves along the surface in the direction OT\ it is at
right angles to ON, and may be described either directly as due to
the spin of the tangent plane about the conjugate tangent or by
components or projections with respect to given tangential directions.
The latter course has the advantages of involving no difficulties of
sign and of introducing two functions of prime importance : if the
velocity of the Gaussian image is resolved into a component along
the tangent and a perpendicular component, the amount of the
first of these in the direction OT' reverse to 0 T is the normal cur-
vature of the surface in the direction OT, and will be denoted by
Kn, and the amount of the second in the direction OE' with which
OT makes a positive right angle is the geodesic torsion along OT,
for which s(J will be used. Symbolically
2111 d\NjdsT — — Kn\T — s^l^,
and since the directions OT, OE are perpendicular
2112 Kn =
2113 sg =
212. Because the direction ON depends only on the position of
0, not on the direction OT, the vector dlN/dsT is a linear function
of the direction OT, and the projection of this vector in a direction
OS independent of OT is linear in both OS and OT. It follows that
2121. The normal curvature and the geodesic torsion of a surface
are quadratic functions of direction,
from which it is a corollary that
2122. Neither KH nor sg can vanish along more than two tangents
at 0 without vanishing in every direction through 0.
2'21. Analysis of the vector dls/dsT for an arbitrary relation of
the tangential direction OS to the curve described by 0 is illuminated
by the corresponding analysis of the particular vector dlT/dsT. This
latter is the vector of curvature of the curve, and being necessarily
at right angles to OT is determined by its projections in any two
directions normal to the curve. When the curve is being considered
in relation to a surface on which it lies, the directions on which the
24
vector of curvature is projected are ON, the normal to the surface,
and OE, the tangential normal to the curve : the amount of the
normal projection is the normal curvature tcn, and the formula
2-211 Kn = ^(dlT/dsT) 1N
is reconciled with 2'112 by the consideration that since £\T\N is
constant the sum £(dlT/dt) 1N + </ (dlN/dt) 1T is zero whatever the
variable t ; the amount of the tangential projection is the geodesic
curvature of the curve, and this will be denoted by /cg :
2-212 Kg = £(dlT/dsT) 1E.
Because ON and OE are at right angles, and coplanar with the
vector of curvature,
2'213 dlT/dsT = KnlN 4- KO\E.
2*22. The change in a tangential radial ls as 0 moves on the
surface is partly a motion with the current tangent plane, and
partly a motion in this plane ; the two components, of which the
first is wholly normal and the second wholly tangential, play equally
useful but dissimilar parts, and have no analytical resemblance.
2'31. The component of dls/dsT normal to the surface to which
OS and OT are tangential I propose to call, for reasons that will
become apparent, the bilinear curvature of the surface in the direc-
tions OS, OT, and to denote by KST :
2-311 KST = £(dls/dsT) 1N.
This function must be recognised in a variety of different forms
which are readily found.
The motion of 15 with the tangent plane is determined by the
spin of this plane, which if 0(7 is a direction conjugate to OT is a
spin of a definite amount p about OC :
2'312. If OC is a direction conjugate to OT and the spin of the
surface along OT is of amount p round OC, then
To avoid the use of p, which cannot be made a single- valued function
of position and direction by any satisfactory convention, all that is
necessary is to resolve the vector pc along determinate directions.
If pc is resolved into a vector along OS and a perpendicular vector,
only the second of these components affects Is, and the rate of
change of ls as far as it is due to this component has the same
25
amount in the direction ON as the component itself has in the
direction OD' with which OS makes a positive right angle round ON:
2*313. The bilinear curvature KST is the projection in the direction
with which OS makes a positive right angle of the spin of the current
tangent plane as 0 moves in the direction OT.
Another aspect is presented if the spin of the tangent plane is
related to the velocity of the Gaussian image ; the latter of these
vectors is obtained by rotating the former through a negative right
angle in the tangent plane and therefore
2'314. The bilinear curvature KST is the projection in the direction
reverse to OS of the velocity of the Gaussian image of 0 with respect
to the arc of any curve in the direction OT.
This result can be expressed in symbols in the form
2-315 KST = ~ c?(dlN/dsT) ls,
and is deducible algebraically from the definition 2*311, for since
</lsliN is always zero,
2-316. J(dls/dt) 1N + c?(dlN/dt) ls = 0,
whatever the variable t.
2*32. The relation of bilinear curvature to normal curvature is
seen immediately from 2*211 and 2'311 :
2*321. The bilinear curvature of a surface reduces to the normal
curvature when the directions on which it depends coincide.
But the part to be played by the bilinear curvature in coordinating
properties of different functions of a single direction is better ap-
preciated after a comparison of 2'315 with 2*112 and 2'113; the
identity of /cn with KTT appears again, and 9^ is seen to be KET :
2*322. If OE is the tangential direction making a positive right
angle with OT, the bilinear curvature KET is the geodesic torsion of
the surface along OT.
In virtue of 2'321 and 2*322, 2*111 may be written
2*323 d].NfdsT = — KTT\T — KET^-E>
and it follows that if P r is any linear function of a vector,
2*324 P (dlN/dsT) = — KTTPT — xETPE = — K^TP^ ,
because OT and OE are at right angles.
2*33. The apparent duplicity of 2'312 has been removed in 2'313
by means of the definite directions OS and OD; it may be removed
26
otherwise by the use of OT and OE: since the spin along OTis the
sum of Kn about OE' and 9^ about 0 T,
2*331 KST — KU cos eST — $g sin €ST,
a formula which in the form
2*332 KST = KTT sin eSE + KET sin eTS
merely expresses the linearity of the function in the direction OS.
2*41. The tangential component of dls/dsT being necessarily at
right angles to OS, its direction of measurement can be chosen and
an unambiguous scalar obtained; the amount of the tangential
component in the direction which makes a positive right angle with
OS I call the swerve of OS along OT and denote by o-Ts, or by (7s only
if the manner of the dependence on OT can be assumed:
2*411 o-Ts = c?(dIs/dsT)lD.
The swerve of OS along OT is the rate at which OS rotates about
ON as 0 moves in the direction of OT; hence if OR, OS are any
two tangential directions dependent on the position of 0,
2 412 or/ — <TTR = deES/dsT:
The swerve of OS in any direction exceeds the swerve of OR in the
same direction by the rate of change of an angle from OR to OS.
From this theorem comes a method of evaluating crs by means
of a curve in the direction of OT, for from 2'212 and 2-411 it follows
that <TTT is Kgt that is, that
2'413. If OT is the current tangent to a curve on a surface the
swerve of OT along OT is the geodesic curvature of the curve,
and therefore
2-414 as = KO + (d€TS/ds).
The swerve as is equal to /cg if €TS has any constant value, and
in particular
2-415 <rTE=Kg.
2'42. If the direction OS depends only on the position of 0, the
vector dls/dsT is a linear function of OT, arid therefore since the
swerve is the projection of this vector in a direction independent
ofOT,
2*421. The swerve along OT of a tangential direction which depends
only on tlie position of 0 is a linear function of OT.
27
Hence
2'422 <TTS sin o> = crAs sin /3 4- <TBS sin or,
and so in particular
2'423 <7-rr sin o> = o-^T sin @ 4- crBr sin a,
2-424
r jf sin ft) = <TAA sin /3 + aBA sin a,
JTTB sin ft) = O-AB sin /3 4- crBB sin a,
formulae which by 2'414 are equivalent to
2425 Kg sin &> = {kg + (cfo/c?s)} sin /3 + {^ — (d/3/ds)} sin a,
2'426 ["{#0 — (da/ds)} sin &> = ^ sin /3 4- }^ — (dco/ds)} sin a,
_{A:? + (dff/ds)} sin a> = {^ -f (dai/dl)} sin j3 + Kg sin a,
where ^ , ^ are the geodesic curvatures of the curves of reference
and s, s are arcs of these curves. The last two formulae can be used
for isolated curves, but 2*425 supposes OT to be known not merely
along a particular curve but along the reference curves also, and is
therefore available only in the discussion of the typical member of
a family of curves; in otherwords, 2'425 assumes a definite tangential
direction to be associated with every point on the surface and gives
the geodesic curvature at 0 of the particular curve which passes
through 0 and has at every one of its points the direction corre-
sponding to that point.
2'51. The definitions 2*311, 2'411 are combined in the equation
2'511 dls/dsT= KST\N + <TTS\D,
which has for particular cases 2 '2 13 and
2'512 dlE/dsT = sglN— KglT.
The three formulae 2-213, 2'512, 2111 express that
2'513. The frame OTEN has the spins 9^, — /cn, tcg;
the calculation of the vector dls/dsT by means of this moving frame
reproduces 2'511, if KST and aTs are regarded as defined by 2'331
and 2-414.
2'52. To the first writers on differential geometry, to associate
the curvatures and torsions of curves on a surface with the form of
the surface itself was the fundamental problem, and if the problem
has lost its interest with its difficulties, the solution is not the less
valuable. Supposing a curve and its tangential indicatrix both to be
free from stationary points, a choice of direction along the principal
28
normal at a single point fixes the standard direction OP along
the current principal normal everywhere, and renders determinate
the binomial direction OB and the sign* of the curvature. The
fundamental trirectal OTPR has no spin about OP, and its spins
about OB and OTare the curvature K and the torsion ? of the curve.
If a curve is on a given surface, a continuously varying angle,
determined by choice at a single point, from OP, the principal
normal of the curve, to ON, the normal to the surface, is called
the normal angle of the curve on the surface and denoted by OT.
The spin of the trirectal OTEN differs from that of the trirectal
OTPB only by the addition of a component of amount dtsjds about
OT; hence the spin of OTEN is compounded of 9 4- (d-&/ds) about
OT and K about OB, and since the latter of these components is the
sum of K cos OT about OE' and K sin OT about ON, 2513 shews that
2'521. The normal curvature, the geodesic curvature, and the
geodesic torsion, of a curve on a surface are related to the curvature
and torsion of the curve in space by the formulae
Kn = K cos OT, Kg — K sin OT, s^ = 9 + (div/ds),
where OT is the normal angle.
2'61. In dealing with a family of surfaces it is necessary to con-
template the variation of normal and tangential radials when the
current point is no longer confined to a single surface. Since a rate
of change in any oblique direction can be calculated by means of
normal and tangential rates of change, the rates of change that have
now to be discussed are normal, that is, are rates of change as the
current point describes an orthogonal trajectory of the family, and
the arc of this curve will be denoted by n. The vectors to be
examined have the forms dlsjdn and dlN/dn.
2*62. To suggest the evaluation oi'dls/dn presupposes that along
the particular orthogonal trajectory under consideration there is
associated with each position of 0 a definite direction OS tangential
to the surface through 0; the vector dls/dn is then a vector in the
plane ODN and is naturally described by its projections on OD and
ON. The vector dIN/dn is the vector of curvature of the trajectory
' The common convention that in solid geometry this sign must be positive is
mischievous beyond words. The curvature of a curve is in fact the amount of a
vector, positive if measured in one direction and negative if measured in the reverse.
29
and is not itself dependent on particular tangential directions, but
to describe it by means of scalars reference to specific directions
must be made; by a choice of tangential directions having intrinsic
relations to the surface a purely intrinsic account of dlN/dn can be
given, but not only do applications involve the projection of dlN/dn
on an arbitrary tangential direction OS, but since the directions
OS, ON are at right angles, this projection is the negative of the
projection </(dls/dn) 1N which is in any case required in connection
with dls/dn.
2'71. The tangential component ofdls/dn, which I call the swing
of OS round ON, is related to ON just as the swerve of OS along
OT is related to 0 T, and the notation of 2*41 can be adopted:
2-711 <rN8 = J(dlsldn)lD.
In fact if OS and OD are directions depending only on the
position of 0, the projection <£(dls/dsp) 1D has the same value for
all curves in the direction OP, whether this direction is tangential,
oblique, or normal, and the function aps defined by
2712 <rPs = J(dls/d8p)lD
is a linear function of OP.
The result expressed by 2*412 is true whatever the direction of
the curve involved, and in particular
2'713 a-Ns - aNR = deES/dn,
so that
2'714. If the angle bettveen two tangential directions is constant
along a trajectory the directions have the same swing about the normal.
2'72. To use 2'713 for the calculation of swings, the swing of
some one direction must be known. Anticipating acquaintance with
the principal tangents of a surface, we observe that because these
tangents are at right angles on every surface, the four principal
directions have the same swing; this swing I call the twist of the
family and denote by TV. From 2'713,
2-721 cr/ = nr
where f is an angle to OS from a principal direction of the surface;
this formula breaks down at an umbilic, and is quite useless if the
family is composed of planes or spheres, when the principal directions
30
are everywhere indeterminate, but in general the twist is the first
swing to be calculated.
Referring for a moment to a topic less elementary than will occupy
us in these pages, it may be mentioned that the vanishing of the
twist is the necessary and sufficient condition for a family not com-
posed of planes or spheres to be a Lame family, that is, to be one
of three families forming a triply orthogonal system.
2'73. We shall write
2-731 TS = J(dls/dn)ly,
and call the function TS the spread of the family along OS. Being a
linear function of OS, the spread is given with reference to any two
tangential directions OA, OB by a formula of the usual type:
2732 rssinft) = TA sin /3S + rBsmas.
274. Combining 2731 with 2711, and noting that dls/dn is
necessarily at right angles to 1>S, we have
2741 dls/dn = o-Ns!D + TS 1N.
2*81. As has been perceived in 2'62,
2-811 TS = -J(dl2f/dn)ls.
The tangential vector d\Njdn can be expressed by its projections
on any two tangential directions:
(dlN/dn) sin eST = {£(dIN/dn) 1D] 1T - {£(dlN/dn) 1E] ls,
that is,
2'812 (dlN/dn) sin eST = TE ls — rD\T.
In particular
2-813 dlN/dn = - rsls -rDlD,
which combines with 2741 to express that
2'814. With respect to the arc of the orthogonal trajectory, the
frame OSDN has spins TD, — rs, &NS.
With 2*741 and 2*813 can therefore be associated
2-815 dlD/dn = - <TNS ls + TD 1N,
but this is only another version of 2741, for &ND has the same value
as <rNs, and OS makes a negative right angle with OD.
2'82. That dlN/dn is the vector of curvature of the trajectory
31
must not be overlooked. Formulae giving the curvature in terms
of spreads are
2*821 Ac2 sin2 €ST = T/ — 2rs TT cos €ST + rr2,
which is general, and
2-822 *2 = rs2 + V,
where OD and OS are as usual perpendicular.
2'83. Comparison of 2'814 with 2*513 suggests a valuable out-
look on the functions <TNS, rs. Suppose a surface drawn to contain
the trajectory under consideration and to have OS for a tangential
direction at every point of this curve; ONS is the tangent plane
to this surface at 0, and if OD is taken for the positive normal
direction, the relation of the frame ONSD to the trajectory regarded
as a curve on this surface shews that
2*831. On any surface containing the orthogonal trajectory and
having OD for current normal along the trajectory, this curve has
geodesic torsion aNs, normal curvature — TD, and geodesic curvature
3. Surfaces and Multilinear Functions associated with
a Function of Position in Space
3'11. Referred to a frame OABC, a function <l> of position in
space becomes a function of the coordinates x, y, z of the variable
point, and in all that follows it is assumed that the functions con-
cerned are not merely absolute constants, and are regular.
If OQ denotes the value of <l> at the point Q, the aggregate of
points for which O has the particular value OQ is the class of points
satisfying the equation
3111 3> O, y, z) = cDa,
and is therefore in general a surface, the <X>-surface through Q.
Singular points are omitted, and the region considered is one
throughout which the O-surfaces compose a family of which one
and only one member passes through any point.
Conversely, any one surface is given by a set of equations of the
form
3112 x=f(u,v\ y=g(u,v), z = h(u,v),
32
and any family by a set of the same form in which the functions in-
volve in addition to u and v a parametric variable w. The eliminant
of u and v from the set of equations 3112 is a relation between
x, y, z, and w which within a sufficiently restricted domain can be
put into the form
3113 ®(x,y, z) = u).
Hence geometrical properties of a <f>-surface and a <I>-family, in
so far as they do not involve the function <£ itself, are properties of
all regular surfaces and families of surfaces.
3 '21. Along a curve, defined by the expression of x, y, z as
functions of the arc s, the function <I> has a rate of change given by
3-211 _
ds dx ds dy ds dz ds'
that is, by
3-212 d^fds = <&XXT + &yyT + ®zZTt
where <&x, <3>y, <&z, the partial derivatives of <1>, are themselves
functions of position having no relation to the curve, while %T, yT>
ZT are the ratios of the direction OT of the curve. Thus
3*213. The rate of change of a regular scalar function of position
in space along any curve depends only on the direction of the curve
and is a linear function of that direction.
The linear function whose value in the direction OP is the rate
of change of <E> along any curve in that direction will be denoted
by &lPt the corresponding function of the vector p being written
4>*p; as with any other linear function,
3'214 &pp=p&p
and ^Ip is identical with <&1P.
3'22. The source of the linear function GPp is called the gradient*
of <E> at 0, and will be denoted by G :
3-221 JQIP = ®lp.
If G- is everywhere the zero vector, then <l> is an absolute constant ;
this case excepted, the region under consideration, though in special
cases it may be broken into a number of separated parts, is not
* It is not necessary to distinguish in practice between the source and the core
of a linear function.
33
sensibly contracted by the omission of the points where G is zero.
The assumption is therefore made that Gr is nowhere zero, it being
understood that the restriction implied is not on <E> but on the
domain throughout which results are asserted to hold. Within a
united region where G is nowhere zero, the two amounts of G are
separate single-signed functions of position, nowhere zero ; one of
these functions, not necessarily the one that is positive, is chosen
and called the slope of <3> ; the slope will be denoted by G.
3'23. At a point 0 where G is not zero, the directions in which
the rate of change of <E> is zero are the directions at right angles
to G. Hence
3*231. The tangent plane at 0 to the ^-surface through 0 is the
plane through 0 at right angles to the gradient of O at 0,
and the directions of the normal to the <f>-surface are the directions
of G; of these directions the one in which G has the amount G is
determinate, and is called briefly the normal direction at 0. The
normal direction, denoted always by ON, varies regularly with the
position of 0 ; hence
3*232. Every ^-surface is bifacial within a united region where
<I> is regular and the gradient of <J> is nowhere the zero vector,
and the choice of sign for the slope G determines implicitly the
direction of angular measurement in every tangent plane.
3 31. The gradient of the core <&l of the linear function O^ is
denoted by <E>2, and the bilinear function <b2P<3 is called the bilinear
rate of change of <1> in the directions OP, OQ. Differentiation of
the sum 3>xocP+ <&yyP + <&zzP with respect to a variable which is
not involved in the ratios XP, yP) zp gives
3-311 <£>2Pa = 2<&uvuPvQ, u, v = x,y, z,
where the summation covers the nine possible terms; since the
second derivatives <£>uv, <&vu are equal,
3'312. The bilinear rate of change of any regular function is
symmetrical in the two directions on which it depends.
3'32. If the direction OQ coincides with the direction OP, the
bilinear function 4>2pg becomes a function <£Vp which may be called
the quadratic rate of change of <£ in the direction OP. This function
must not be confused with the second order rate of change
N.
34
which is not the same for all curves in the direction OP : applica-
tion of 1-722 to
3-321 d^jdsp = <blP
gives
3 322 d*<$>ldsp> = <$>*pp + & (dIP/dsp) = <f>2Pp + c?G (dlP/dsP)}
and since d\pjdsP is the vector of curvature of the particular curve
along which the rate of change is being found, it is only when either
the curvature is zero or the principal normal is tangential to the
^-surface that the last term disappears from 3*322.
3'41. The conception of the bilinear rate of change, and the
fundamental theorem 3"312, are immediately extended. The core
4>2 has a gradient <£3, and so on, and the multilinear rate of
change of <l> of degree k is the function <&kPQ r, where for each
value of h in succession <&h+1 is the gradient of <&h. With a frame
of reference,
3-411 3>*p<2...r = 2<IV..«,MQ • • • wr> *, t, . . . w = a, y, z,
the coefficients being the partial derivatives when <£> is expressed as
a function of x, y, z ; hence
3'412. Every multilinear rate of change of a regular function of
position is symmetrical in the variable directions.
3*51. The rate of change of 3> along any curve on a <X>-surface
being zero,
3-511 &s = 0
if 08 is restricted as usual to denote a direction tangential to the
<£- surface at 0 ; on the other hand by the definition of the slope
3-512 &N = G.
From 3-511 and 3'512 together comes the expression for Oxr when
r is arbitrary : if r is expressed as ps + qN where OS is tangential,
then because the function <t>xr is linear
3-513 ®1r=p&8 + q&Nt
and substitution from 3*511 and 3'512 gives
3-514 &r=Gq,
that is,
3-515 &r = G£r\N\
this formula is of course obvious from the definition of G.
35
4. The Bilinear Curvature of a Surface
411. That
4111. The bilinear curvature of a surface is a bilinear function
of the two tangential directions on which it depends
is obvious equally in every expression given for KST in 2*3, and this
property alone implies such formulae as
4112 KST sin2 co = KAA sin @s sin /3r + KAB sin fis sin aT
+ KBA sm as SU1 PT + KBB sni as sm «r>
where 05, OT are independent of each other,
4114 daRKST = (1 + daReRS) KDT + (1 -f daReRT) KSE,
if 05, OT depend on a tangential direction 0-B, and in particular
4115 daKTT = KET + KTE = — da/cEE,
4116 da/cTE = KEE — KTT = da/cET,
special cases of 1/822 and T823.
412. Dupin's theorem, that
4121. At any ordinary point of a surface the sum of the normal
curvatures in two directions at right angles is a constant,
is shewn by 2*321 to be a case of 1*824, that is, to follow from the
simple fact that the normal curvature is a quadratic function.
The half of the constant sum KTT + KEE is the mean curvature of
the surface at 0, and will be denoted by B ;
4122 KTT + KEE = 2B.
The differences /cn — B and B — KH are the excess and the defect
of curvature along OT. To write
4123 KEE = 2B-tcn
is to express KEE directly as a function of OT, and the function
KEE — KTT, which appears in 4116 and in a number of other formulae,
is given by
4124 KEE ~ KTT = 2 (B - xn\
that is to say, is twice the defect of curvature.
3—2
36
An actual formula giving the mean curvature is easy to find,
for 4*112 gives
4*125 KTT sin2 o> = KAA sin2 /3 -f (KAB + KBA) sin /3 sin a + KBB sin2 a,
and substitution of a + JTT, /3 — J TT for a, ft gives
4*126 KEE sin2 « = /c^ cos2 ft — (KAB 4- KBA) cos /3 cos a + KBB cos2 a,
whence by addition we have not only a trigonometrical proof of
Dupin's theorem but the explicit result
4*127 25 sin2 CD = KAA — (KAB + KBA) cos &> + KBB.
4*21. But bilinearity alone does not account for the importance
of the function KST. Differentiation of 3*511 along a curve on a
O-surface gives
4-211 &8T + 0>- (dl8/dsT) = 0,
and substituting from 3*515 we see from 2*311 that
4*212. The bilinear rate of change of a function <3> along two
directions OS, OT tangential to the ^-surface is connected with the
bilinear curvature KST of the surface in those directions by the equation
4-Vr+ftte-O,
where G is the slope of <E>.
And this result not only enables the bilinear curvature to be calcu-
lated in specific cases, but taken with 3*312 shews that
4*213. At any ordinary point of any surface, the bilinear curva-
ture in two directions is a symmetric function of those directions.
From the combination of this result with 4*1 1 1 springs the whole
elementary theory of the curvature of a surface.
4*22. Since identically
4*221 sin /3S sin aT — sin as sin j3T = sin w sin eST,
the necessary and sufficient condition for 4*213 to follow from the
explicit formula 4'112 is the equality of the coefficients KAB) KBA :
4 222. The symmetry of the bilinear curvature for any one pair
of distinct directions at a point implies algebraically the symmetry
of this function for any other pair of directions at the same point.
With the substitution of KAB for KBA, 4*112 takes the form
4*223 KST sin2 o> = KAA sin @s sin /3r
+ KAB (sin /3S sin aT + sin as sin /3T) + KBB sin as sin aT>
37
giving
4*224* /cn sin2 o> = KAA sin2 ft + 2/cAS sin /3 sin a + tfBJ5 sin2 a,
4*225 90 sin2 a> = — #^ sin /3 cos /3 — A;^ sin (a — j3)
+ KBB sin a cos a,
and 4127 becomes
4*226 2B sin2 &> = /e^ — 2tcAB cos &> + /CB^.
It need hardly be said that all relations between bilinear curvatures
of one surface in different pairs of directions at one point are de-
ducible from 4*223 by pure trigonometry, or that this method of
deduction has nothing to recommend it.
4'23. A simple case of 4'213 is the assertion that if /e8T, as de-
scribed in 2*3, is zero for a particular pair of directions, then so also
is KTS. To say that KST is zero is to assert that as 0 moves in the
direction OT the tangent plane at 0 rotates about 0$; in other
words
4'231. A pair of directions for which the bilinear curvature is
zero is a pair of conjugate directions.
Hence 4*213 includes the familiar theorem that
4*232. If OS is conjugate to OT then OT is conjugate to OS,
and the application of 4*222 to this result takes the form that
4*233. If a surface is known to have a single pair of mutually
conjugate distinct tangents at a point, the symmetry of the bilinear
curvature at that point can be inferred.
4*24. By means of 2*331 the symmetry of the bilinear curvature
can be expressed as a relation between the normal curvatures and
geodesic torsions in two directions without explicit mention of the
bilinear function; comparing the two formulae
4'241 KST = /CTT cos €ST — KET sin eST,
^TS ~~ ^ss cos €$T \ &DS ^1^1 GST
we have
4*242 ( KTT — KSS ) cos eST — (KET + KDS) sin €ST>
* This formula shews that a geometrical theory without the bilinear curvature is
as incomplete as an analytical theory without the function for which M is used by
Scheffers, Forsyth, writers in the Encyk. d. Math. Wiss., and others, D' by Bianchi,
and D'M(EG - F2) by Gauss and Darboux.
38
or to use a notation convenient with reference curves
4*243 (Kn - Kn) cos w = (93 + 90) sin &>,
a result given in other symbols and used again and again by Darboux.
4*31. The relation between geodesic torsions in perpendicular
directions is simpler in form than the relation between normal
curvatures asserted in Dupin's theorem, but belongs in fact to a
more advanced stage, depending as it does not on the bilinearity
alone but on the symmetry of the bilinear curvature. To write
down this relation from 4'225 or 4' 243 is of course simple enough,
but an appeal to first principles shews more clearly on what the
result depends. The linearity of KST in the direction OT implies
4'oJ.J. KST' == — ^ST'
and in virtue of the symmetry of the function this equation gives
4'312 KST + KT'S ~ 0 J
hence in particular
4'313 KET + KT'E = 0,
and since OT' is the direction making a positive right angle with
OE the function KT>E is the geodesic torsion along OE :
4*314. The sum of the geodesic torsions in two directions at right
angles is zero.
This result, like 4*232, is a special case of 4*213 and implies the
more general theorem in which it is included; thus 4*314 and
4'232 in spite of their diversity of form are theorems implying
each other, that is, are equivalent theorems, on account of the
bilinearity of the bilinear curvature.
4*32. Brevity is often achieved by the use of the function
2 (KDT + KSE)> which is the symmetrical bilinear function of OS and
OT that reduces to the geodesic torsion KET when OS coincides
with OT', it is natural to write
4*321 ?sr = | (KDT + KSE)
and to call this function the bilinear torsion, but it must be
recognised that the function has none of the fundamental impor-
tance of the bilinear curvature. Identically,
4*322 9TT = KET = <>g ,
4*323 9^ = | ( KEE — KTT) — B — Kn,
39
and 4"314 can be expressed in the form
4-324 *EE = -*g-
Being bilinear and symmetrical, the function <$ST has its value
given in terms of directions of reference OA, OB by
4-325 9sr sin2 w = SAA sin /3S sin jBT
+ ?AB (sin fis sin a.T + sin as sin /3T) + SBB sin as sin aT.
But unlike the coefficients KAA, KAB, KBB, the coefficients SAA, SAB,
<f£B are not numerically independent, for the sum 9rr + <$EE is not
merely constant but is zero :
4'326 SAA — ^AB cos a) + ?£B = 0.
4'33. The angular derivatives of the normal curvature and
geodesic torsion are given in 4'] 15 and 4*116. Since KTE as well
as KET is 9^ the first of these formulae becomes
4-331 daKn = ^g,
a familiar result; 4*116 is equivalent to
4-332 da 90 = 2 (£-*„),
which is therefore more elementary than 4*331 since it is proved
without reference to the symmetry of KST. There is a temptation
to replace 4'331 by
4-333 da(icn-B) = 2sg
and to treat as correlative the geodesic torsion and the excess of
curvature, but the suggested analogy must not be pressed too far.
Written in the forms
4*331, 4-332 are seen to be corollaries of the more general theorem
that
4*334. //" OS is inclined to OT at a constant angle, then
an immediate deduction from 4*1 14.
4'41. A function of direction that is not a mere constant must
have at least one direction of maximum value and one of minimum.
If OT is a direction along which the value of /cn is a minimum,
then the value along OT' is the same minimum, while it follows
from Dupin's theorem that along OE and OE' the value is a
maximum. Hence unless /cn has the same value in every direction
40
from 0, there certainly are two distinct tangents along which the
value of /cn is stationary. On the other hand, 4'331 implies that a
tangent along which /cn is stationary is a tangent along which 9^ is
zero, and since sg is a quadratic function there cannot be more
than two of these tangents unless sg is zero in every direction.
4*411. At an ordinary point of a surface, either the normal
curvature is the same in all directions and the geodesic torsion is
zero in every direction, or there is one tangent along which the
normal curvature has its least value and one along which the normal
curvature has its greatest value, these tangents are at right angles
and are the only tangents along which the geodesic torsion is zero,
and the normal curvature in a variable direction increases or de-
creases steadily as the direction rotates from one of these tangents
to the other.
A point at which the normal curvature has the same value in all
directions is an umbilic ; the constant value is of course equal to
the mean curvature B at the point.
4-42. From 4'332 and 4'333 it follows that the sum (KU - B)~ + 9/
does not vary with OT but is a function only of the position of 0
on the surface, in general positive but zero if and only if 0 is
umbilical. Spheres and planes are composed wholly of umbilics,
but from a surface that is neither plane nor spherical the umbilics
can be removed, for it can be proved that they are isolated points
or compose isolated curves. Throughout a region which is nowhere
umbilical, the two square roots of (/rn — B)z + sy* are separate single-
valued functions of position ; one of these, selected and called the
amplitude -of curvature, will be denoted by A :
4-421 (*n-fl)' + 9/ = 4a.
From 4 421 the extreme values of Kn at a point 0, corresponding
to the directions along which 9^ is zero, are B — A and B + A ;
these are the principal curvatures of the surface at 0, and I write
4-422 ^ = B-A, *t = JB + A.
The principal tangents, that is, the tangents along which the
normal curvatures are »T, xz, are individually determinate when
the sign of A has been chosen. To secure complete freedom from
ambiguity, definite directions along these tangents must be chosen
also; the choice along one principal tangent at one point is arbi-
41
trary,and determines the standard direction along the corresponding
tangent at all neighbouring points ; the standard direction along
the other principal tangent is then fixed by the convention that
4"423. One of the angles from the first principal direction to the
second is a positive right angle.
The principal directions at 0 will be denoted by 0(7t, OCZ, but
as affixes t, z will be substituted for (7T, Cz.
4'43. The equation
4-431 KET = Q
which characterises the directions of curvature implies of course
4-432 KE'T=Q,
and is therefore equivalent to the combination of
4-433 KST = 0
with the condition that OS and OT are at right angles :
4'434. At any ordinary point that is not an umbilic, the principal
tangents are the only two conjugate tangents at right angles.
4'44. From the definitions and the convention of 4*423,
4 4:41 KU = Xl) KIZ = U, Kzz = HZ)
4-442 6l5 = i7r.
Substitution in 4'223 gives for any two directions
4*443 KST = #t cos fs cos fy + #s sin £s sin fy ,
where f denotes an angle to the variable direction from the first
principal direction; the forms corresponding to 4*224 and 4'225
which are special cases of 4*443 are
4*444 Kn = #T cos2 f + *5 sin2 f,
4-445 93 = (#g — ^T) cos f sin f ,
the formulae of Euler and Bonnet, of which the first was transformed
by Euler himself into the shape
4-446 Kn = B-A cos 2f
and the second is
4-447 93 = A sin 2?,
A, B having the meanings assigned in 4*11 and 4*42. Corollaries
of 4*444 are
4-448 • Kn-» = 24sin8, •v-*n = 24 cos* ,
42
which with 4 '447 give
4--44.Q c 2 _ (v u- \( u- v\
•x Tf±«/ S^ -- \-"z Kn)\Kn *t/»
a relation which is otherwise evident from 4'421.
4'45. To relate 4'443 to the fundamental property of a direction
of curvature is a most instructive exercise, if OC is a direction of
curvature and # is the corresponding principal curvature tccc, the
spin along OC is a vector of amount » in the direction with which
OC makes a positive right angle, and therefore the projection of
this vector on the direction OE' with which OT makes a positive
right angle is # cos 6CT:
4'451. If OC is a direction of curvature and * is the correspond-
ing principal curvature, the bilinear curvature KCT has the value
y* cos ^cj1'
Thus
* iO^j K\T ^~ i COS Cy, f£~-T =~ 5 Sin CT")
because KST is linear in OS and the principal directions are at right
angles,
4'453 KST = KIT cos fs + KZT sin %s,
and substitution from 4 '452 reproduces 4'443.
4'46. Nor is the proof of 4'443 just given the only use, or the
chief use, of 4'451 ; it is from 4'451 that come formulae for deter-
mining the principal curvatures and tangents in terms of magni-
tudes related to arbitrary tangential directions of reference.
Because KAT and KTB are linear in OT,
4'461 \~KAT sm w ~ KAA gin P + KAB sm a>
_KTB sin co = KAB sin 0 + KBB sin a,
for any tangential direction, while from 4'451
4'462 KAC = * cos ac, K(]B = # cos /3C,
for a principal direction OC. Hence
4'463. A direction of curvature in which the normal curvature
is x is characterised by the pair of equations
~KAA sin ft + KAB sin a = * sin co cos a,
_KAB sin ft + KBB sin a = ^ sin co cos ft.
Elimination of » reproduces the equation obtained more simply by
43
equating to zero the geodesic torsion as given by 4*225. On the
other hand, since identically
sin co cos a = sin ft + cos co sin a, sin co cos ft = sin a + cos co sin ft,
the equations of 4*463 can be written as
4'464 \~(KAA ~ x) sin 0 + (KAB ~ * cos co) sin a = 0,
(./• _ -v /TkO /-.i i eiTi /Q i ^ «/• •!/ j em n — (\
>*-AB ** wVQ / ""-1 r-' i^ V BB / "•*•*•*• M, — w.
Elimination of the ratio sin /9 : sin a yields an equation which x
must satisfy, and since this equation is quadratic, it has no roots
except #T and *z\
4*465. The principal curvatures of a surface are the roots of the
equation
(* - KAA) (» ~ *BB) = 0 cos co - KAB}\
The equation of 4*465 expands to
4*466 #2 sin2 co — * (KAA — 2tcAB cos co + KBB)
-I- (if if -if 2^ — 0
» \KAA KBB KAB ) — V,
and therefore 2B, which is the sum of the principal curvatures,
and the product of these curvatures, which is the specific or
absolute curvature of the surface at 0, and is denoted always by
K, are given by
4*467 2B sin2 co = KAA — %KAB cos co + KBB)
which has been obtained already in 4*226, and
4*468 K sin2 co = KAA KBB — KAB* ;
the amplitude of curvature is determined numerically from the
identity
4*469 &-A*=K.
4*47. The fluctuations of the geodesic torsion 9^ are seen most
readily from Bonnet's formula 4*447 ;
4*471. The extreme values of 9^ are —A and A, and these are
assumed in the directions midway between consecutive principal
directions.
The discussion of STT as the function denned by identifying OS
with OT in 4*325 is parallel to the discussion of *:rras the function
given by identifying OS with OT in 4*223, and therefore the extreme
values of <$TT have for their sum (?AA — 2s\4B cos co + 9BB) cosec2 co and
44
for their product (9^ SBB — SAB} cosec2 a>. Thus 4'326 is reproduced,
and the amplitude of curvature is seen to be given by the equation
4-472
4'48. If sin a and sin /9 are both known, the direction OT is
determinate, but the ratio of the sines alone does not distinguish
OT from OT'. Thus in general when n has a definite one of its
possible values, either equation in 4P464 defines the corresponding
principal tangent but not the corresponding principal direction.
These formulae however render precise a detail left vague in 4'42.
If
4*481 sin a/p = sin (3/q = sin co/r,
where p, q, r are functions of position on the surface, then
4'482 r- = p2 + 2pq cos w + cf,
and throughout a region where p and q do not vanish simultaneously,
r is a single-signed function determined everywhere by 4 482 if its
sign is known. Hence the choice of sign of a single radical deter-
mines the principal direction corresponding to the principal curva-
ture * throughout the whole of a region provided that no points
are included where simultaneously
4'483 KAA—M> KAB = xcosa>, Kr,K — y'
But
KAB = KAA COS o>
implies that OA is a direction of curvature,
KAB = *BB cos o>
implies that OB is a direction of curvature, and since by hypothesis
OA, OB lie along distinct tangents, KAA and KBB are the extreme
values of the normal curvature, and the additional equality
KAA — KBB
implies that 0 is umbilical: having excluded umbilical points for
the purpose of separating the principal curvatures, we have actually
obtained a region in which the various principal directions also are
separated.
4'49. On any surface, a curve whose tangent at every point is a
principal tangent of the surface there, or in other words whose
geodesic torsion is everywhere zero, is called a line of curvature of
the surface. Throughout a united region containing no singular or
45
umbilical points, the two principal directions at 0 are definite
directions depending regularly on the position of 0. It follows from
the theory of differential equations that over such a region there
are two distinct families of lines of curvature and that through each
point passes one and only one member of each family.
If F is a regular function of position of any kind on the surface,
the values at 0 of the rates of change of F in the two positive
directions along the two lines of curvature through 0 depend only
on the position of 0 and define by their relations to 0 two functions
of position which will be denoted by dF/dsl, dF/dsz. These functions
are not partial derivatives; if in order to use sl and sz as actual
coordinates we go so far as to define the position of 0 by its distances
from two selected trajectories measured along lines of curvature, it
is still impossible to secure that every curve along which s^ has a
constant value is itself a line of curvature or has sl for its arc; thus
even in this case the partial derivative dF/dsl is not the rate of
change in the direction to which it does correspond and bears no
intrinsic relation to dF/ds^ It follows that although there are rates
of change cPF/dsf, d'2F/dszdsl derivable from dF/dsl and rates of
change dPF/dsidSs, d2Fjds£ derivable from dF/dsz, there is no reason
to anticipate equality ofd?F/d8idst to d^F/ds^ds^, in point of fact it
is easy when Fis scalar to evaluate the difference between d?Fjdsldst
and d2F/dszdsl and to recognise the rare cases in which this difference
vanishes.
4'51. A direction of curvature is a direction in which the geodesic
torsion is zero. If there are directions in which the normal curvature
is zero, these directions, which are called asymptotic, have properties
not less interesting than have the directions of curvature.
Since the normal curvature at 0 varies continuously between its
extreme values #r, #r., the existence of asymptotic directions depends
on the relation between the signs of these two curvatures, that is,
depends on the sign of the product K. If K is strictly positive, there
are no asymptotic directions and 0 is said to be an elliptic point on
tjje surface. If K is zero, one if not both of the principal curvatures
vanishes, and 0 is said to be parabolic. For both of the principal
curvatures to vanish, that is, for a point to be umbilical as well as
parabolic, is altogether exceptional on any surface but a plane. At
an ordinary parabolic point, one only of the principal curvatures
46
vanishes, and the asymptotic directions are the corresponding direc-
tions of curvature. A developable is a surface composed wholly of
parabolic points, but on a surface that is not developable the para-
bolic points in general, if there are any, compose a curve or a number
of distinct curves separating regions throughout which K is positive
from regions throughout which K is negative. In discussing
asymptotic directions attention is confined in the first place to a
united region composed wholly of hyperbolic points, that is, of points
where K is strictly negative.
4*52. Between consecutive directions of curvature at a hyperbolic
point, there is one and only one direction in which KH, changing in
sign from the sign of #t to the sign of #5, is zero; thus there are
four distinct asymptotic directions, and since the reverse of an
asymptotic direction is itself asymptotic these are the four directions
along two asymptotic tangents.
With a direction of curvature is associated the corresponding
normal curvature, which is a principal curvature of the surface. In
the case of an asymptotic direction it is the geodesic torsion that
survives, and this magnitude is called the asymptotic torsion
associated with the direction.
The fundamental relation of an asymptotic direction 01 to an
arbitrary direction OT corresponds to 4'451. The spin of the surface
as 0 moves in the asymptotic direction 01 has no component at
right angles to 01 but is simply a spin of amount ?/7 about 01, if
?/7 is the asymptotic torsion along 07; the projection of this spin
in the direction OE' is therefore <?7/ sin e/r:
4'521. // 01 is an asymptotic direction and ?7/ is the correspond-
ing asymptotic torsion, the bilinear curvature KIT has the value
9/j sin eIT.
If OJ, OK are two asymptotic directions at 0, the bilinear
curvature KJK is shewn by 4'521 to be expressible both as <tjj sin eJK
and as 9^ sin eKJ\ it follows without reference to the principal
directions that
4*522. The two asymptotic torsions at a hyperbolic point of a
surface are equal in magnitude and of opposite sign,
and it follows also that if the existence of two distinct asymptotic
tangents is known 4'522 implies the complete symmetry of the
47
bilinear curvature. To find the actual values of the asymptotic
torsions we have only to compare 4*521 with 4*451 : if 0 is an angle
from 01 to the first principal direction, then
4*523 <?// sin 0 = Kn = »t cos 0,
and since 0 + \TT is an angle from 01 to the second principal
direction,
4*524 ?// cos 0 = KK = — xr, sin #;
the combination of 4*523 with 4*524 gives
4-525 *// = -*i*«,
that is,
4*526. The square of the asymptotic torsions is the negative of
the specific curvature,
a theorem usually ascribed to Enneper but in fact announced by
Beltrami four years earlier than by Enneper.
4*53. For the determination of asymptotic directions from arbi-
trary directions of reference 4*521 is again useful. Comparing
4*531 tcAI = — ?7/ sin or/, KIB — ?7/ sin /37,
which are implied by 4*521, with the general formulae 4*461, namely,
4-532
KAT sin w = KAA sin j3 + KAB sin a,
KTR sin ft> = KAB sin ft -f KBB sin a,
we find that
4*533. An asymptotic direction with asymptotic torsion ?/7 is
characterised by the pair of equations
~KAA sin ft + (KAB + ?/7 sin &>) sin a = 0,
~(KAB ~ Sn sin &)) sin ft + KBB sin a = 0.
To eliminate ?7/ is to obtain the equation expressing that the
normal curvature is zero; the elimination of sin ft : sin a gives
4-534 ?//2 sin2 a> = KAB* - KAA KBB,
a formula which 4*468 shews to be equivalent to 4'525.
4*54. Throughout a region where K is strictly negative, the
asymptotic tangents are distinguished by the asymptotic torsions,
which are separate functions of position. One of these square roots
of — K is chosen and called the first asymptotic torsion: it will be
denoted by — ?a; the second asymptotic torsion is ?a. Since the
four quadrants into which the tangent plane at 0 is divided by
48
the principal tangents Cl'OCl) CZ'OCZ are distinct, and the four
asymptotic directions lie one in each of these quadrants, the
asymptotic directions also are distinct. One of the directions of
the first asymptotic tangent, chosen arbitrarily at one point and
in consequence determinate elsewhere, is called the first asymptotic
direction and denoted by OJ, and an angle from this direction to
the first principal direction will be denoted by Ju. Thus
4'541 9a sin J v = — xl cos J v , 9a cos \v = yz sin \ v,
implying
4-542 #, cos2 1 v + »z sin2 \ v = 0,
an equation which is of course deducible immediately from Euler's
formula 4'444. The direction making an angle \v with the first
principal direction is one of the directions of the second asymptotic
tangent, and is denoted by OK and called the second asymptotic
direction. The angle v is an angle from one asymptotic tangent
to the other, and is given with as little ambiguity as possible by
the equation
4-543 B - A cos v = 0,
a corollary of 4*446.
4'55. In the use of the asymptotic directions OJ, OK as direc-
tions of reference, there is an embarrassing choice, for the bilinear
curvature KJK and the asymptotic torsion ?rt are connected by the
relation
4-551 K JK = - 9rt sin v.
For any pair of tangential directions,
4'552 KST sin2 v = rcJK (sin eSK sin eJT -f sin ejs sin €TK),
and for a single direction
4'553 Kn sin2 v = 2/cjx sin eTK sin eJT ,
4-554 93 sin2 v = KJK sin (eTK - €JT)',
the last formula can be replaced by
4-555 90 sin f = - 9« sin (e^ — eJT).
The principal curvatures are given by
4*556 *i = — 5a tan | v, #c = 9a cot J v,
and therefore
4*557 B = 9^ cot v, A=<sa cosec v, K = - 9a2.
49
4*56. From asymptotic tangents are defined asymptotic lines;
these on a united anticlastic region without singular or parabolic
points compose two families, every point lying on one member of each
family. The relation of an asymptotic line to a surface is in a sense
more intimate than that of a line of curvature. If an asymptotic line
has curvature tc and normal angle OT, the normal curvature, which
is zero, is K cos w, and three cases are distinguishable : if tc is not
zero, then cos -GT must be zero; if a point where K is zero is a limit
of points where K is not zero, continuity requires cos i& to be zero
there also; if fc is zero everywhere on the line, the line is straight,
and while as a curve in space it has no determinate principal normal
at any point, to assign it in its capacity as asymptotic line a definite
normal by the convention* that cos •cr is zero leads inevitably to
consistent interpretations of general theorems. Thus cos iz is zero
at every point of any asymptotic line, and continuous variation of
-TO- being on this account out of the question, there is far more gain
than loss in a further convention to fix absolutely the value of OT,
which is taken to be \TT\
4'561. The normal angle of an asymptotic line on a surface is
everywhere a right angle.
In other words,
4'562. At every point of an asymptotic line on a surface the
principal normal to the line is its tangential normal and the bi-
normal is the normal to the surface;
further, because & is constant,
4'563. The torsion of an asymptotic line is its geodesic torsion,
that is, is the asymptotic torsion of the surface in the direction of
the line, and because -sr is a positive right angle,
4'564"|*- The curvature of an asymptotic line is its geodesic curva-
ture,
in sign as well as in amount. In consequence of 4*563, theorems
concerning asymptotic torsions may be read narrowly as theorems
* A straight line on a surface is geodesic as well as asymptotic, and as a geodesic
has for principal normal the normal to the surface.
t This is one of the theorems to whose simplicity the convention that curvature
itself must be positive is fatal. The vanishing of cos •& is consistent with a value
- \ir for w, and if the direction OP is predetermined by the sign of /c, two cases have
to be admitted; either w itself, or a symbol for sin -or, must then be retained if the
cases are to be treated together.
N. 4
50
concerning the torsions of asymptotic lines, and in particular 4*522
and 4'526 imply that
4*565. The torsions at a hyperbolic point 0 of the two asymptotic
lines through 0 are equal in magnitude and opposite in sign and
their product is the specific curvature of the surface at 0.
It is to be remarked that for an asymptotic line to be straight the
specific curvature of the surface need not be zero: as an asymptotic
line on a given surface containing it, a straight line has a definite
torsion which is the rate at which the tangent plane, which in
general varies from point to point, rotates about the line; on a
ruled surface the rotation disappears if the same plane is the tangent
plane at every point of the line, and this is precisely the degenerate
case in which K is zero along the whole line.
5. The Bilinear Rate of Change of a Function of Position
5*11. The equation
5111 &s = 0
is of course true only if OS is tangential to the <l>-surface, but the
derived equation
5112 3>*SP + <£! (dl8/d8p) = 0
involves no restriction on the direction OP, and leads not only to
5113 <£V + GKST = 0,
the relation used to establish 4*213, but also in virtue of 3*515 and
2-731 to
5114 &
where rs is the spread of the <J>-family along OS, or the negative
of the geodesic curvature of the "^-orthogonal regarded as a curve
on a surface to which OS is tangential.
5'21. By means of 5*113 and 5'114 the bilinear rate of change
<I>2P(3 can be transformed whenever the direction OP is tangential
to the <l>-surface : if the vector 1$ is the sum qT + SN where UT is
tangential, then <&*SQ is q&*ST + s<&'2SN and therefore
5-211 3>* + G
51
If OP is the normal direction ON the change is of another kind ;
from the definition of the slope,
5-212 3>V = #,
and therefore along any curve in a direction OQ
5-213 d&N/dsQ=GlQ;
but by 1-722 and 3'515
5-214 d&N/dsQ = &NQ + O1 (dlN/d8Q) = <&NQ + G£(dlN/dsQ) 1N,
and since a rate of change of a radial is necessarily at right angles
to the radial itself, the last term vanishes and there remains the
formula
5-215 d&N/dsQ = 3>*NQ,
which taken with 5!213 shews that
5"216. Whatever the direction OQ,
*•>«=• GV
5"22. The general relation of 5'216 is a synthesis of the par-
ticular relations
5-221 <Dv^=GV,
5-222 &NS=Gls.
The first of these can be written in the form
5-223 O2^ = d^jdn^
and suggests a reference to 3'32. The second can be compared
with 5-114, and since the bilinear rate of change is symmetrical
gives
5-224 G1
whence*
5-225 T =
5*226. The spread of the ^-family in any tangential direction is
the negative of the rate of change of the logarithmic slope of <X> in
that direction.
5'23. If application is to be made of 5'113, 5*114 and 5*216 to
<£>2PQ when the directions OP, OQ are both oblique, the radials 1P,
* Allowance must be made for the possibility that G is negative, and for this
reason the logarithmic slope is defined as ^ log G2.
4—2
52
\Q must both be resolved into normal and tangential components.
Assuming
5-231 lp = l>s + rjv» IQ = qT+sN)
the bilinearity of the function implies
5-232 &
and according to the purpose in view the useful transformation
will be
5'233 <£2pQ - rsG\ = -G (psrs 4- <?rrr + pqtcST)
or
5-234 4>2P -psGls - qr@lT - r«G V =
5-31. From 3-515
5-311 <£> (dlg/dsp] = G£(dIs/dsP) 1N,
and since c^l^l^ is zero,
5-312 £(dls/dsp)
hence 5'112 is equivalent to
5-313 3>* =
for an arbitrary direction OP and a tangential direction OS. In
contrast to this result, G</(dlN/dsP) 1N is necessarily zero, but 4>2^P
is zero only in special cases : the tangency of OS is essential to the
truth of 5'313. Multiplication of 5'313 by a scalar shews that as
a linear function of a tangential vector s the bilinear function
<f>2Ps is obtained by multiplying by G the projected product of
d\NjdsP and s. In particular, since a rate of change of the radial
IN is necessarily tangential,
5-314 02P (dlN/dsQ) = G£(d\N/dsP) (dlN/dsQ)
whatever the directions OP, OQ.
5 32. The function 3>2P(dIN/dsQ) will reappear at a later stage ;
5'314 shews that the function is in fact symmetrical in OP and
OQ and indicates the geometrical magnitudes with which it is
connected, which depend on the relations of OP and OQ to the
^-surface. If OS, OT are tangential directions, d\Nldss, dl^/dsf
are the corresponding Gaussian velocities; dlN/dn. is the vector of
curvature of the orthogonal trajectory. Since neither KST nor TT
53
is defined except for tangential directions, the notation described
in 1"55 is applicable and it is possible to write
5-321 t£(dlff/d8$) (d1N/dsT) =
5-322 £(dlN/dss) (dlN/dn) =
5-323 J'(dlN/dn)2 = T^;
the last of these functions is the square of the numerical curvature
of the trajectory. To discover analytical expressions in which the
same projected products are involved, let q(p) denote temporarily
the vector, dependent upon OP, which is such that for an arbi-
trary direction of OR the value of ^Vs ig the projection of q(p) on
OR, and let this vector be resolved into a normal and a tangential
component. The projection of q(p) on ON, which by the definition
of q(p) is <&ZPN, is the projection of the normal component of q(p)
on ON, and therefore the normal component of q(p) is ^p^l^.
And from 5'313 the projection of the tangential component of q(p)
in any tangential direction is the same as the projection of
Gdly/dsp in that direction, whence since GdlN/dsP is itself tan-
gential the tangential component of q(p) is nothing but Gd\NjdsP.
Thus,
5-324 q(p> = 0 (dlN/dsP) + &P2fIN.
But if OP, OQ are any two directions the projected product
c^q(p)q(Q) is the bilinear scalar function of OP and OQ denoted by
^P* 02Q* , and this is calculable with the greatest ease by means
of any frame of reference. Hence from 5 '3 24 and the corresponding
formula giving q(Q)
5-325 <n*<*V = G*J(dlyld8P) (dlN/dsQ) + &PN&QN.
The three distinct theorems comprehended in 5'325 can be ex-
pressed in a variety of forms ; among the results are
5-326 &s* 3>2r* = G* (KS* KT* 4- rs TT),
5-327 <n* <S>V» = G*,cs* r* + &8 &N,
5-328 ^V = (02^y-(^)2.
5-41. In 5'226 and 4'213 we have two distinct and independent
deductions from the symmetry of the bilinear rate of change of a
scalar function of position. It is important to observe that there
can be no deductions independent of these two, a result implied by
1*432 : if OS, OT are distinct tangential directions and O^V is
54
normal, the complete symmetry of ^2PQ is deducible from and
therefore involves no consequences independent of the set of
equalities
5-411 &ST = <D2rs, &NS = &SN, ' <&NT = &Tjr,
of which the first is equivalent to 4'213, and the second and third
express for different directions the single theorem 5'226.
5'42. To suppose that such formulae as 5'113, 5*114 and 5'216
assist in the calculation of multilinear rates of change is completely
to misvalue these formulae. Whatever the system of coordinates,
the multilinear rates of change are among the functions most
easily found, and in application to particular surfaces and functions
it is rather for the sake of the other magnitudes involved that
results of this kind are desirable.
6. The Codazzi Function
6*11. The bilinear curvature KRS is not a function from which a
gradient can be formed, for as a rule if the position of 0 is changed
the directions OR, OS cannot remain unaltered. But there is an
elegant function which plays as far as possible the part of a
gradient, and it is with this function that the present chapter is
concerned.
From the equation
6111 &ES+G,cES = 0
it follows that if OR, OS are specified functions of the position of
0 on a curve with direction OT on a ^-surface, then
6112 (d&xs/ds,,) + GITKES + G (dKRSjdsT) = 0 ;
also by 1'722 and 2'511
6113
<t>2^ (dls/dsT)
where OC, OD make positive right angles with OR, OS, and
therefore
6114
- & (°'TR>CCS
55
on substitution from 5 '2 16 and 4'212. Thus 6112 gives
6115 &]&.,, + G*RK8T + 018KRT + GITKRS + G\RST = 0,
where
6116 \RST = (d/cRS/dsT) - O-TRKCS — <rTs KRD.
The function \RST denned by 6116, which I propose to call the
Codazzi function, belongs like the bilinear curvature to the geo-
metry of a single surface, for this definition contains no reference
to the function <1>. But 6115 is of value as shewing at once that
the value of \RST depends only on the three directions OR, 08, OT,
not on the variation of OR, OS along any particular curve in the
direction OT, and moreover that
6117. The Codazzi function is linear in each of the three direc-
tions on which it depends.
As a formula for the calculation of the Codazzi function 6115
may be modified to
6118
*>RST = ~ (&N&RST ~ &VM&ST ~ &N8&XT ~ ^NT^ES^K^N^
612. The Codazzi function \R$T takes special forms if two of
the directions on which it depends coincide or are perpendicular.
Whatever the angle between OS and OT,
6121 \SST = (dtcss/dsT) — 2<rTs/cDS
= (dKss/dsT)-2o-Ts<;ss,
6122 \STS = (dfCsT/dss) ~ &S KDT ~ °~s KSE
= (dtcST/dss) - 2<r/9sr - (d€8T/d88) KSE)
6123
6124 \DTS = (dfcDT/dss) + O-SDKST - O-STKDE
613. More familiar functions are among those of a single direc-
tion OT which appear as degenerate forms of the Codazzi function
and can be regarded as defined by means of a single curve in the
direction OT\ that the function depends only on the direction and
not on any particular curve is in no case self-evident. In the most
elementary notation,
6131 \TTT = (dtcn/ds) — 2^?^,
6132 \TET = (d?g/ds) + 2fcg («n - B).
56
Thus Xrrr> the simplest of cubic functions, is the function associ-
ated with the name of Laguerre who first shewed it to depend on
direction alone, and Xr^r is the cubic function of Darboux. As
actually given by 6*116,
6133 ^EET — (d/cEE/ds) — 2fcgKET>,
but on substitution from 4*123, this becomes
6134 \EET = 2 (dB/ds) - {(dH:n/ds) - 2*^},
that is
O -LOO h'EET ~~ *•*•*-* T ~~ ^TTT'
On account of the multilinearity of the Codazzi function, \TTE,
\ETE bear to the direction OE the relations of \EET, — \TET to OT\
hence
while \ETE is the negative of the Darboux function of the direction
OE; \EEE is of course the Laguerre function of this last direction.
6'14. Naturally it is when the three directions involved are all
principal or all asymptotic that the Codazzi function is most simply
expressed. If Kgl, K^ are the geodesic curvatures of the lines of
curvature,
6141 <7TT = a? = Kgl, azl = azz = K^,
and therefore
6142 Xnt = d^ljdsl , XI8l = — 2 A tcgl , \x.l = d
6-1 4.3 "\ — rlv //7<? "X —
-L^rO A'HZ — U""i/Gtoj5, ^IZZ
where A denotes as before \ (#5 — 3/t), the amplitude of curvature.
The corresponding functions for the asymptotic directions OJ, OK
are simplified by the relation
6144 KJK = ?a sin v ;
if KgJ, KgK are the geodesic curvatures, which are the actual curva-
tures, of the asymptotic lines,
6145
^jjj = - ZsaKgj> ^JKJ = (dsa/dsj) sin v , X^^ = 2?a \KgJ + (dv/dsj)},
6*146
: - 2?a {KgK - (dv/dsx)}, \JKK = (dsa/dsK) sin u, \KKK = 2?a«^.
6'21. We are now in a position to appreciate the fundamental
property of the Codazzi function, which is apparent from 6*115 :
57
6*211. The Codazzi function is a symmetrical trilinear function
of the tangential directions on which it depends.
That the function XBST denned by 6'1 16 is linear in each of the
directions OR, OS, OT can be proved without difficulty from the
most elementary considerations*; indeed, it is by its linearity in
OT that \RST ni>st attracts attention in the geometry of a single
surface. The symmetry of \B8T in the two directions OR, 08 is
manifest from the symmetry of KRS in the same directions, but the
discovery that ^RST depends on OT in the same way as on OR and
OS is both unexpected and fertile.
6*22. Because \RST is trilinear,
6*221 A, sin3 w
sin j3B sin /38 sin j3T + \BAA sin aR sin 0S sin $T
sin (3R sin cts sin @T + \AAB sin f*R sin f}8 sin aT
sm PR sin as sin a.T + \BAB sin OLR sin f$8 sin aT
4- \BBA sin OR sin ofe sin (BT 4- X^^^ sin OR sin a^ sin aT>
and the complete symmetry of the function is implied by the tri-
linearity if the equalities
are known for any one pair of distinct directions. On account of
the symmetry of KAB, there is no distinction between \BAA and \ABA
or between \ABB and \BAB, and therefore the equations necessary
to imply 6*211 are two only, namely
"'223 ^AAB = ^ABA) ^ABB = ^BBAi
which on reference to 6*121 and 6*122 are readily identified with
the equations associated with the name of Codazzi :
6*224. The Codazzi equations for any pair of families of curves
of reference express the symmetry of the Codazzi function for the
directions of reference and imply the complete symmetry of this
function,
and it is for this reason that I have proposed to attach Codazzi's
name to the function itself.
From 1*43,
6*225. Any two pairs of Codazzi equations are equivalent,
and this result adds interest to a comparison of different forms
which the equations assume.
* See 8-1 below.
58
6*23. An important interpretation of the Codazzi equations comes
from 6 '136, which can now be read as a relation between \TET, the
Darboux function of OT, and \EEE, the Laguerre function of OE :
6*231. The sum of the Darboux function of any direction OT and
the Laguerre function of the perpendicular direction OE is a linear
function, equal to twice the rate of change in the latter direction of
the mean curvature of the surface.
Since the two equations
D AOA >^TTE = ^TET> "*TEE == "'EET
differ only in the direction which is denoted by OT, 6*231 implies
them both*, and is equivalent to any pair of Codazzi equations.
Angular differentiation gives relations of another kind between
the functions of Laguerre and Darboux. If the variable directions
are independent,
u Zoo da j< \.ftgji = A-^S-JJ.
Hence because the Codazzi- function is symmetrical,
O ^jOiJ aa A. fTE ~~ TEE ~~ ^"fTT ^~ T ~~~ *J
it is easy to express these results in words.
6 31. The Codazzi equations derived from 6*142, 6143, and 6*145,
6*146 are
6*311 ZA/cp = - dxjdsz, '2AfcffZ = - dxz/dsl}
and
6*312 r 2?a {/Cgj + (dv/dsj)} = (d^ajdsK} sin v,
[_ 2?0 {tcgK - (dv/dsK)} = - (dia/dsj) sin v,
and these are inevitably regarded as formulae for the calculation
of the geodesic curvatures which they involve. The same view may
be taken of the Codazzi equations in general, for although as a rule
each equation involves two geodesic curvatures, the pair of equations
"'313 ^AAB ~ ^ABA > ^ABB ~ ^BBA
is linear in the pair of geodesic curvatures /cg, /cg, and has for its
discriminant SAB*— SAA*>BB> which has been seen in 4*472 to be
equal to A2 sin2 co and therefore vanishes only at an umbilic.
*- Formulae equivalent to 6-232 were discovered in 1911 and announced to the
Fifth International Congress of Mathematics (Cambridge, 1912; Proceedings, vol. 2,
p. 34) ; I have not hitherto published a proof.
59
6"32. Since any two pairs of Codazzi equations are equivalent,
the geodesic curvatures in any one pair of families of reference
curves can be calculated from those in any other pair ; this is in
accordance with 2'425 and 2'426, but if in illustration we deduce
Kgj from 6'311 we shall see the economy effected by the enlarging
of our ideas. Because the swerve aTs is linear in the direction OT,
6'321 KgJ + (dv/dsj) = arjK = o-^K cos J v - azK sin \ v
= {xgi + 2 (dv/dsj] cos Ju - {xgz - % (dv/dsz)} sin Ju,
and therefore from 6'311
/» orirt c* A (d*z . . A dv . 1
6-322 ZA K+ - = sm u +^ cos
t . dv .
— -^-7— cos *t> — A -r- sm
dsz
dv
as anticipated. It would be rash to assume that every useful formula
for a geodesic curvature is given by some Codazzi equation ; in fact
an example can be given to the contrary. Identically,
C.QOQ « o a
6 323 -y — 2?a , cosec v = — - — 5— - tan2 1 v,
dsl dsl ds\
a.ooA dsa 0 dv a ol
6 324 -^ — h 2?rt -y- cosec v = - -*—L cot2 \ v ;
dsz dsz dsz
utilising the relations
6-325 9a2 cot4 Ju = ?a3 cot3 Ju/?a tan \ v = - ^53/^t,
6-326 ?a2 tan4 J u = 9a3 tana J u/9a cot J u = - »T3/^ ,
it is easy to deduce from 6'312 the expression
6'327 KgJ={d log (- V/tfO/^i) sin | v cos2 J v
+ (<i log (- »T3/:y5)/cfo5} cos Ju sin2 |f,
of which Bonnet has made application, but 6 '32 7 is not as it stands
a Codazzi equation.
60
7. The Trilinear Rate of Change of a Function of Position
7'11. In the last section the trilinear rate of change played only
the subsidiary part of introducing to our notice the Codazzi function
and establishing its symmetry, and for this purpose the variable
directions were restricted to be tangential to the ^-surface. The
next task is to investigate formulae involving the ,same trilinear
rate of change with one or more of the directions normal.
712. From the elementary formula
7121 &NP = G1P,
since this implies for any variable t on which OP may depend
7122 O^ (dlpldt) = Gl (dlp/dt),
it follows that whatever the direction OQ,
7123 &ypQ + &p (dlN/dsQ) = GV
Here is a simple proof that the function <E>2P (dlN/dso) is sym-
metrical in the directions OP, OQ, a conclusion reached in 5'3, and
substitution from any of the formulae of 5'32 gives a corresponding
deduction from 7123. Thus 5'325 gives
7124 &N &NPQ + <&% ^ = GG*PQ + &P &Q
in which no restriction is implied on OP or OQ, and 5*321, 5'322,
5-323 imply
7125
7126
7127 &NNN + G KP* =
KP in 7127 denoting either value of the curvature of the orthogonal
trajectory.
713. The function ^S^PQ is involved not only in the rate of
change d<&2Np/dsQ but also in the rate of change d&pq/dn, and
deductions from the symmetry of the trilinear function are to be
expected. If however OQ is normal, nothing is to be anticipated
that is not deducible from the symmetry of the bilinear function
GPyp', in fact we have from first principles
7131 d&xpldn = <&NNP + &N (dlP/dn) + $>2P (dlN/dn),
and since
7132 ^2^ (dlp/dn) = Gl (dlf/dn),
61
the comparison of 7*131 with 7*123 yields only the identity
7133 d&p/dn = G2NP + G1 (dlp/dn).
But d^sT/dn repays examination.
7*21. Expanding d<&2ST/dn in the usual way and substituting
from 2*741 we have
7-211 d&ST/dn
+ rgly) + 3>2S (<TNT1E
on the other hand, from 4*212,
7'212 d^STjdn = — d (GicST)/dn = - G (dKST/dn) - GINKST.
Hence from 4*212 and 5*114
7*213
&NST + GPffiKsT + # {(die8T/dn) - <rNsKDT - O-NTKSE - 2r5rr} = 0,
or in a form analogous to that of 6*118,
7'214 (dtc8T/dn) —
7*22. When 7*213 is compared with 7*125 the function <i> itself
disappears, surviving only in the slope :
7*221 G2ST + G1NKST
+ G \(dKST/dn) — O-NSKDT — O-NTKSE — 2rsrT — KS*KT*\ = 0.
An algebraical transformation reduces the number of terms in
this equation. Let T denote l/6r, which is the arc function of the
trajectory with respect to the variable O, and has been called the
spaciousness of the family; then identically, for any directions OP,
Oft
7-222 T1P = -^-2^p,
7-223
so that
7*224
But for tangential directions OS, OT,
7-225 GlsG1T
Hence
7'226 (d/cST/dn) — <rNs KDT —
62
or in another form, involving <l> but separating completely the
geometrical from the analytical terms,
7-227 (dx8T/dn) - (y
7'23. Particular cases of 7 '2 14 are
7"231 (dfcTT/dn) — %CTNT<>TT
= ~ {&X&1TTT ~ &NN&TT ~ 2
7-232 (dfcET/dn) - 2cr
2TT
and the corresponding cases of 7 '226 are
7-233 (d/cTT/dn) - 2<rNT<>TT - KT^ = (T2
7-234 (dKST/dn)- ^NT^ET-KE^KT^ = (T*ET
to appreciate the last two formulae, we must recognise tcT#2 as
KTI? + KET*y the square of the amount of the spin of the tangent
plane along OT, and KE^KT^ as KETKTT 4- KEEKTE, that is, as 2B/cET.
7'24. Two symmetrical bilinear functions in a plane GAB are
identical if they are equal for the pair of directions OA, OA, for
the pair of directions OA, OS, and for the pair of directions OB, OB.
Hence any one group of three independent particular cases of 7*214
or 7*226 is equivalent to any other group.
If OT is a principal direction, /cr#2 is the square of the corre-
sponding principal curvature and ?rr ig zero; hence*
7'241. If OC is a principal direction on a ^-surface and x is the
corresponding curvature, then
where T is the reciprocal of the slope of <I>, and the rate of change
of * along the orthogonal trajectory of the 3>-family is given directly
* Since the truth of this theorem for one of the principal directions is not de-
ducible from its truth for the other, there are two independent formulae involved in
7*241. These are the formulae whose existence was inferred by Forsyth in 1903
(Phil. Trans. Roy. Soc. Lond., Ser. A, vol. 202, p. 333) from an enumeration of
invariants ; discovered by the methods described here and translated into a form to
require no explanation they were announced to the London Mathematical Society
(Proc. L.M.S., vol. 16, p. xxvii) in 1918.
63
The form taken by 7*232 and 7 '234 for a principal direction, that
is, by 7'214 and 7'226 when OS, OT are principal directions at right
angles, is quite different ; KIZ being zero on every surface, dK\TJdn is
zero, and K^K^ , a multiple of KIZ> is zero also. Thus only two terms
of 7'234 and three of 7'232 survive, and since ?I5 is A, the amplitude
of curvature, and in the swings aNl an(l ?jv8 is to be recognised the
magnitude described as the twist of the family,
7'242. The twist CT of the family of surfaces associated with a
function <1> is given in terms of <I> itself by
and in terms of the reciprocal of the slope of <E> by
Of the results of applying 7 '2 14 and 7 '2 2 6 to the asymptotic
directions OJ, OK the most elegant is
7 243 (dsa/dn) sin v - ?a2 cos v = - T*JK/T,
which can easily be verified from 7'241 and 7'242.
7'31. To deduce from 7*242 formulae for evaluating the twist
when <J> is given as a function of Cartesian coordinates x, y, z is a
simple matter, but requires some preliminary investigations to which
the theory of multilinear functions is not essential.
By its definition, the gradient G of the function <I> is the vector
whose projections are <&x, <&y, <&z. The slope G therefore satisfies
the equation
7-311 (? = -T-a 1 cos 7 cos/9 $>x
cos 7 1 cos a. <&y
cos ft cos a 1 <&z
®x ®y <$>z 0
and since points where &xy 4>y, O^ are simultaneously zero are
excluded, G is determined throughout the region under considera-
tion by combining this formula with a choice made at a single point.
The direction normal bo the <f>-surface is the direction in which the
vector G has the amount G, and therefore
7'312. The direction cosines of the normal to the ^-surface are
The ratios of the normal direction are &'0/6r, y^G, ^G/^> where
XG> 2/o> ZQ are the components of G and are therefore given by
formulae of which the first is
7313
G
>a; COS 7 COS @
>y 1 cos a
>z cos a 1
The direction whose ratios are XT, yT, ZT is tangential to the
<E>-surface if
7'314 $>XXT + ^yr 4- <£z^r = 0>
and from 0'45 it follows that if the directions OS, OTare tangential
and eST is an angle from the first to the second, then
7-315 TG
Also if OE makes a positive right angle with OT round ON,
then OT makes a positive right angle with ON round OE and
therefore by 0"45
7-316 G(1E, mE, nE) = f
x.
yT
where %G, yG, ZQ have the values typified in 7'313, and by 0'46
7-317
E, yE,
3>,
7"32. The bilinear curvature of the ^-surface in the pair of
directions OS, OT is shewn by the comparison of 4'212 with 3'311
to be given by
7-321
so that in particular for the normal curvature in the direction OT,
7 '322 GKU = — (<&xx, <&yy, <\>zz, <&yz, <&zx, ^xy^Xf, yT,
Combining 7 '321 with 7 -3 17 we have
7-323
nT
65
hence in terms of ratios alone
7-QO4 onr/^2
I O<a^r & 1 Cr 9cT>
^r + VT cos 7 + ZT cos £
#r cos 7 + ^/r + ^r cos a
#r cos ft + yr cos a + £r
^s + y^ cos 7 + £5 cos ft
xs cos 7 + i/£ + Zs cos a
#s cos fi + ys cos a + %
and the geodesic torsion in the direction 0 T is given by
7-325
• XT + yT cos 7 -\- ZT cos ft
XT cos 7 + yT + -% cos a
#r cos ft + yT cos a -f £y
It is convenient to write
7-QOA T2C7a;a;
I OZO J B =
cos 7
cos/3
cos {3
cos a
1
®
cos 7
1
cos a
and so on, and to use H2y, E-1'2, E^ as equivalent to S^2, Sza;, EJ
with this notation, 7*324 becomes
7-327
and 7 325 takes the form
7-328 G% - T(Hxa;, Ey", a22, 5"*,
7'33. There is no need of the theory of multilinear functions
in establishing the theorem that the principal curvatures of the
4>-surface are the roots of the equation
7-331
cos 7 +
cos 3 +
cos
COS O +
cos / +
cos a 4-
G» -f «>
0
=0.
N.
66
Applying to 7*321 the determinantal identity
7332
R
ss
TS
R
T}yx flzx
-L v -Li
Ryy Rzy ZSXT — XSZT
J_\/ JL\J vU Q t/'7T ^""" t/OtX/'/"'
*v C tXj nn """""" tv C & T* ds Q ti rfl " Li Q VUT* \J
• OJ. O A Ot/-*- t/ *^ -*•
where on the left-hand side .Rp^ denotes
2RuvuPvQ, u, v = x,y, z,
and the nine coefficients of the form Ruv are arbitrary, we have in
virtue of 7*315
lid j. \ji j.\. — — -*-a;rr ~^ur. ~^zx ~*^x
V
zz
0
which is also a corollary of 7 '331, and applying the same identity
to 7*327, we have similarly
7-334 (7M2 = KXX Eyx E,ZX <&
in making these deductions we may take an arbitrary pair of tan-
gential directions and appeal to 4*468 and 4*472, or we may take a
pair of principal directions and remember that KIZ, 9n, Szz all vanish.
7*34. For the calculation of the twist, or indeed of the value
of any symmetrical bilinear function when its arguments are the
principal directions of a ^-surface, it is not necessary to calculate
the individual ratios of the principal directions ; it is sufficient to
discover the values of the six combinations x^z, y^yZ) z^zz, y^zz + z^yz,
z^xz-\- x^, x^yz + yiXz, and this we proceed to do.
It is easy to find five linear functions of these six combinations
which necessarily vanish : since KIZ is zero, 7*321 gives one such,
and because the principal directions are perpendicular,
cos a
cos
67
and similarly
also multiplying the conditions of tangency
<^xxl + ®yyl + <Mt = 0, 3>xxz + <&yyi + ®zzz = 0
by xz , XT and adding we have
;8 + <£z (stff8 + #ttf8) + <&y (a?Ty8 + yza;8) = 0,
s + $* (yi*B + *i3fc) + <£* Oi2/s + yTa?8) = o,
+ 3>y (yizz + Zlyz) + $>x (z^ + a?t2r8) = 0.
Hence
7 '341. With any Cartesian frame, the six expressions
z\%z, y\Zz + z\yz, Zi®z + %\Zz, %\.yz + y\®z are proportional to the five-
rowed determinants of the matrix
yy
^zz ^yz
1 cos a
0 0
0 $>,
cos 7
0
0
0 0 2<£z <£>y <&x 0
To find the factor which enables us to replace the proportionality
by equality, we remark that 7 '341 implies that
7*342. For arbitrary values off, g, h, the determinant
f2 92 h* gh hf fg
1 cos a. cos /3 cos 7
00 <&Z <&y
0 <J>z 0 <&x
<£„ <l>~ 0
1
0
0 24>,
0 0
is a multiple of the product (fxl + gyl + hz^) (fxz+gyz +
and we evaluate this product in another way.
If we write temporarily m, n for fxl -f gyl + A,^, fx
then identically
I UtAS'T 1 1 1/tA/Y "^~~ (,/ \ t/XT L/C' "'"" L/T 1/^7 / "™^ At- \ t&T vU^ ~~~" 1//T ^"* I.
tyxl-j/O (/*•"/ \L O L 6/ J
and therefore by 7 '3 15
T(r (?2^r — 7/l^'5) = gtf>z —
similarly
TG (nyl - myz) = h<&x- f®z , TG (nzl -
-f 5 +
5—2
68
But because $ST is symmetrical and bilinear, to replace &T, yT,
ZT by nxl — ra#5, ny^ — myz, nzT — mzz on the right of 7*325 is to re-
place 90 by ?i25n — 2nmslz + ra29S5 on the left of the same equation ;
recalling that ?TI, 9TS, 9sg have the values 0, A, 0 and replacing the
product nm by its value we have the equation
7-343 - 2T3 G*A (fa, + gy^ + hz^ (faz + gyz + hzz)
cos 7
(f&y ~ 9®*) &** + (f®v ~ 9®*) cos
- <y cos 7
v ~ 9®*) ®*y + <f®v ~ 9®x) cos a
cos /3
It follows that the determinant in 7*342 is a multiple of the
determinant in 7*343, and once attention is drawn to the existence
of a connection between them it is a simple matter to reduce the
former to the product of the latter by — 2. We conclude that
7'344. The value of the determinant in 7*342 is
4T3 G4A (fa, + gyl + hzj (faz + gyz + hzt\
and further that
7'345. The value of the symmetrical bilinear function RPQ of
which the expression in terms of Cartesian coordinates is
> i-fWD i/ 01 11 /ji — /v> ti *y
z* £i UpVQ, u, v — x, y, z,
when the arguments of the function are the first and second principal
directions of the <&-surface, is given by
R** Ryy Ezz R«z Rzx
1 1 cos a cos/3 cos 7
00 0 <l>z <E>y
0 2^y 0 4>0 0 <&x
0 0 2<l>? OM <l>a; 0
69
Determinants of the form occurring here were first used by
Darboux, who discovered them in his re3earches on triply- orthogonal
systems, and we shall call them Darboux determinants. We may
express 7 '345 briefly by saying that the value of the Darboux de-
terminant which has Ruv for the typical element of its first row is
In passing we may mention another expression involving the
product (fx^ 4- gyl 4- hz^ (fxz 4- gyz 4- hzz), interesting in itself but
as ill adapted as that in 7'343 to giving the value of a bilinear
function that is not a product. From the identity
7-346
7? 7? f \ ' \ 1 '
-ftco -ft/TCi / XQ 4" Cf lie "r" fl Zo
-to J Oc/t/O1 o
RST RTT f'xT + g'yT 4- h'zT
ZgHOf """ ^S^T ff
U/Tpy ~T)tiy I yyy 7 /
Ai"*"^ rf"6 rs^Z /yi /i/ .__ /i/ /y» f»'
JLv JL v JL v vu C W T1 ' / Cfv 'T* * ^
ZSXT — XSZT ®syT~ysxT o o
/" g" h" 0 0
since HTI = 0, TETS=GU, HS5 = 0,
we have
7-347 2G*A (fx, + gy, + htj (fxz + ^5 + hz<)
= — T3
E8* Hy2/ Hzy 4>y gr
<&X <&y 4>Z 00
/ ^ • A 0 0
7-35. The application of 7'345 to 7'242 is immediate :
7'351. At a point which is not an umbilic of the 3? -surf ace through
it, the twist -or of the ^-family is connected with the derivatives of <I>
by the formula*
' The formula was first given, in terms of curvilinear coordinates and without
proof, to the Fifth International Congress of Mathematicians (Cambridge, 1912;
see Proceedings, vol. 2, p. 31). Eesults equivalent to 7'351 and 7g352 in terms of
rectangular Cartesian coordinates were proved subsequently by Herman (Quarterly
Journal of Mathematics, vol.46, pp. 284 et seq.). Algebraical transformations of the
determinant involved are to be found in Darboux's treatise on triply-orthogonal systems.
70
T T
J- -*-
xx
yy
T
•*-
T
-*-
zx
T
J-
0
0
1
0
0
cos a cos /3 cos 7
0
o
0
*
0
where
cos 7 cos /3
cos 7
cos /3 cos a
cos a
0
-4 ?s ^Ae amplitude of curvature of the ^-surface.
The value of A" in terms of derivatives of <E> is given in 7'334
above.
The alternative expression for the twist given by 7 '242 is more
complicated, but gives the result explicitly in terms of third
derivatives of 3>. If #Q, yQ, ZQ have the meanings assigned in 7*31,
then G<&*yp is a linear function of OP in which the coefficient of
xp is
and G<&3NPQ is a bilinear function of OP and OQ in which the
coefficient of xPXq is
7*352. -(/" the typical element in the first row of a Darboux
determinant is
1
cos 7
cos 8
_ 2
cos 7 cos /3 <&x
1 cos 7 cos /3 <&xuv
1 cos a 3>y
cos 7 1 cos a ^^p
cos a 1 '5>z
COS y8 COS « 1 <l>ZMy
3>y $>z 0
^^ ^^ ®z 0
1 cos 7 cos /3 O^
1 cos 7 cos/3 <&xv
cos 7 1 cos a <&yu
cos 7 1 cos a O^
cos/3 cos a 1 <&zu
cos ft cos a 1 <&zv
<bx <&„ $>z 0
<&x $>y <&z 0
the value of the determinant is 8T7GSA2'&.
71
It is to be remarked that the expression given in this enuncia-
tion is not in general simply the product of the second derivative
TUV by — T4(r5; the difference between the two is a function which
is such that its use as a typical element of the first row produces
a Darboux determinant that vanishes.
8. Functions of Direction on a Surface
8'11. The fundamental difficulty in applying the theory of
multilinear functions to problems connected with any single surface
other than a plane is due to the absence of genuine gradients.
The directions forming the arguments of such functions as the
bilinear curvature and the Codazzi function are essentially tan-
gential, and if the current point varies these tangential directions
are necessarily affected.
Suppose the surface to be referred to curvilinear coordinates
u, v and let a standard tangential direction 0 W be associated with
each position of 0. Then a function F(Q, R, ...) of the tangential
directions OQ, OR, ... may be described explicitly as a function
F(u, v, eWQ, 6WR, ...), and if the directions OQ, OR, ... vary in a
given manner with the position of 0 on a curve in the direction
OT, the rate of change of F ' (Q, R, ...) along the curve is given by
dF dF du dF dv vdeWQ , p,deWR
-j — = 3— -j — i- — j— + aaQJf —j— - + aasJf —j
dsT ou dsT dv dsT dsT dsT
that is, by
. n dF (dF du dF dv
8112 -j-=] 3— +3r;j -- (d
dsT [du dsT dv dsT
+ (crTQdaQF+a-TRdaRF+ ...).
Since u, v are merely particular functions of position on the surface,
the rates of change du/dsT) dv/dsT are the linear functions ulT, v1T
of OT, and since OW is assumed to depend only on the position
of 0, the swerve aTw also is a linear function of OT. On the other
hand, neither the rates of change dF/dsT, ddqF, daRF, ... nor the
swerves <rr^, q-TR, ... depend in any way on the actual choice of
coordinates u, v and initial direction 0 W. Thus
8 '113. Associated with any function of direction F (Q, R, ...) on
a surface there is a function gdbTF which is linear in the direction
72
OT and is such that the rate of change of F along any curve in the
direction OT is
gdbTF + o-TQdaQF+o-TRdaRF + ....
The function gdbTF will be called the Darboux gradient of F.
From 8112,
8114. In terms of coordinates u, v and an initial direction 0 W,
the value of the Darboux gradient gdbTF is given by
gdbTF = (dF/du) ulT + (dF/dv) vlT - (daQ F + daRF + . . . ) <JTW.
8*12. For a multilinear function PQRS...> the angular derivatives
daQP, daRP, ... have the values PBRS...> PQCS...> ••• where OB,
OG, ... make positive right angles with OQ, OR, . . . ; hence
8121 dPQRSJdsT = gdbTPQRS... + PBRS.°TQ + PQCS...VTR + ••••
There is another route, open only in the case of multilinear func-
tions, which leads to a similar formula and therefore shews a
different aspect of the Darboux gradient. If the function PFGH
was defined for all sets of directions, there would be a gradient
PFGH...K also defined for all sets of directions, and for tangential
directions OQ, OR, OS, ... OT we should have
8122 dPQES JdsT = PQBS...T +
+ PNRS... KQT + PQNS... KRT + —
If the functions P^Rs..., PQNS...I •-• which might, of course, be
different functions, were known, the function PQRS T could be
determined for tangential arguments by this formula; if PQRS... was
given in the first place for tangential arguments only, the functions
PNRS...> PQNS...> ••• could be assigned arbitrarily, and by comparing
8122 with 8121 we see that the Darboux gradient is the gradient
found by supposing the functions PXRS..., PQNS...> ••• a^ to be
identically zero.
Nevertheless, the Darboux gradient is a disappointing function.
The Codazzi function is the Darboux gradient of the bilinear
curvature, and if <t> is a function of position on the surface and <&1T
is the linear function d<&/dsT, the Darboux gradient of <&1T is the
function (d$>ls/dsT) — <&IDO-TS, which is in fact symmetrical and is
valuable on account of its symmetry. But the Darboux gradients
of the Codazzi function and of the function (d^>ls/dsT) — <&IDCTTS
prove both to be unsymmetrical ; gradients with the symmetry
73
that is desirable are not yielded by any simple general method,
and all that is possible is to discover special devices effective in
particular cases.
8 21. To define multilinear rates of change of a regular scalar
function of position 4> on the surface, we extend the function to
the whole of space in the neighbourhood of the surface by associating
with every point on the normal at 0 the value of <3> at 0 itself. If
two or more normals meet at a point Q, the function so defined may
be many- valued at Q; if however the surface has only ordinary
points there is a region of space within which no two normals inter-
sect, and within this region <I> is not only single-valued but regular.
To assign the values of 4> outside the surface in the way suggested
seems at first no less arbitrary a proceeding than to construct the
successive Darboux gradients by defining the functions <&lx, ^Vr.
^3NST) -•• to be zero. But a number of considerations combine to
modify this impression : the multilinear rates of change formed by
extending the function in any regular way are necessarily sym-
metrical; throughout the whole of differential geometry the straight
line is much more than merely the simplest of curves; and the hypo-
thesis made is in fact equivalent only to the assumption that the
functions <&1N, &XN, ^SNNN> •••> functions in which no arbitrary
directions are involved, are all zero.
8'22. If attention is concentrated upon the distribution of <£ on
the surface, <&1N, ^2^N, ^V-zw* • • • figure as functions of position only,
^T) 3>2NT> ^NNT* - • • as linear functions of the one variable direction
OT, ^ST* 3>*NST> ^NNST'-- as bilinear functions of the pair of
variable directions OS, OT, and so on. The rate of change of any
one of these functions along a curve on the surface is expressible
by means of other functions in the set. the typical relation being
8-221
.,
— CD _ rr,
N..NNPQ...RST T* N...N*PQ...RS
p.,
°* ®N...NNAQ...RS
h+k , h+k
N..NNNP...RS
h+k
74
where OP, OQ, ... OR, OS, OT are k + 1 tangential directions and
0^., 05, ... OD make positive right angles with OP, OQ, . . . 0$. Com-
7, l I- -4-1
parison of 8'221 with 8*121 shews the relation of <I> N
7i+&
to the Darboux gradient of ®N_NNPQ ES :
N,..NNPQ...RST
k ,h+k
NNpQ RS + hKT*®
none of the functions <&ZNT, ^NNT> ^>4NNNT> ••• vanish identically,
and therefore unless the surface is a plane, h must be zero for
h+k
h*T*3>N N PQ RS to vanish, and k must be unity for the remaining
terms in the difference between the two functions to vanish. That
is to say, <&2ST is the Darboux gradient of <&1T, but there is no similar
relation between others of the multilinear functions with which we
are dealing unless either the surface is plane or the function <5> has
some special relation to the surface.
8'23. It is easy, accepting the assumptions
8-231 <£ = 0,
with the implications
8-232 d®lN/dsT = 0, d&NN/dsT = 0, d®sNNN/dsT = 0, . . . ,
to arrange the formulae included under 8'221 in such an order that
each of the multilinear functions is introduced without further
reference to space outside the surface than is implied in the occur-
rence of bilinear curvatures as factors. The first equation is
8-233 &T = d®ldsTt
from which <&1T may be calculated from a curve lying wholly in the
surface. Then since
8-234 d^NjdsT = &NT - KT* 4>V
identically, and the rate of change is zero,
8-235 ®-NT - KT* <&\ = 0 ;
also because 3>1N is zero,
8-236 3>2ST -I- <TTS®ID = d<bls/dsT.
75
Next come
8-237
8-238
8239
in which the only fresh functions are
the process can be continued to any desired extent.
8'24. Emphatically the formulae of the last paragraph and their
successors are neither definitions of the multilinear functions nor
aids to their calculation. For the former part they are unsuitable
because neither the symmetry nor the multilinearity of the functions
is in evidence in the formulae, for the latter because the rates of
change and the swerves contain parts that are not multilinear which
it is superfluous to evaluate. To discuss the expression of these
multilinear functions by means of curvilinear coordinates on the
surface requires an analytical foundation which is beyond the range
of this pamphlet, and we must content ourselves with the observa-
tion that rather than calculate the functions directly from 8"221
we should combine 8"222 with 8*114 and use the formula,
8-241
N...NNPQ...RST
w h+k
N...NNPB...RS
but this is not the method actually to be recommended.
The very lack of symmetry which renders the formulae covered
by 8*221 unfit to serve as definitions implies that significant rela-
tions which do not themselves involve multilinear rates of change
are deducible from these formulae. To work out details is interest-
ing — it will be found for example that 8'235 and 8'238 together
imply the symmetry of the Coclazzi function — but here we will
76
confine our attention to the simplest problem of the kind, and
examine only 8'236.
831. The bilinear function <£2sr being symmetrical, we have
from 8-236,
8-311 (d&s/d8T) - aTs^D = (d&Tldss) - a/4^;
involved in 8'311 are really two families of curves and their
orthogonal trajectories, and the equality may be written in the
form
d*<5> sd<$> d*3> Td3>
O 6IZ -j - = -- 0>* - — = -^ - = -- <7</ -=— ,
asTass asD dssdsT dsE
or in a different notation as
d*$> /. dco\d<Z>
8616 -jTry-. — «« — -jr. I -j-r = TT^T — ( KQ + -p ] TTT ,
dsds \ - as) dm dsds \ " ds/dm
where d/dm, d/dm indicate rates of change along the orthogonal
trajectories of the families. If the families of curves are everywhere
orthogonal, 8'313 becomes
d*3> d<S>
dmds Kgm dm
where tcgs, /tgm denote the geodesic curvatures of a typical member
of a family and of an orthogonal trajectory of the same family.
8'32. We must not fail to observe that if what is being discussed
is the variation on a particular surface of a function already defined
throughout space, the formulae of 8'23 and the transformations of
8'31 are not usually valid. For example, in general when the function
and the surface are defined independently of each other,
8-321 d$>ls/dsT = <E>26T + o-Ts$>lD + ^r^1^,
and for the last term to disappear either <&1N must be zero or OS,
OZ'must be conjugate directions. Of the cases in which the latter
condition is satisfied the most important is that in which the two
principal directions occur: without any hypothesis as to a relation
between <& and the surface,
d*3> d<t>_ d23> d<$>
O o2t2t <P [5 — -j - j -- KqZ -^ — 7 —j~ T Kqi ~T ,
ds^dsl ' dsz dS[dsz ' dsl
and so in particular the function Tlz which was shewn in 7'242
to be connected with the twist of the <&-family is expressible
77
as (d^T/dszds,) - Kgz (dT/dsz) or as (d^T/d^ds^) + Kgl (dT/ds,)
although in general T*N is not zero and T2ST differs from
(&T/d8Td8a)-(rTs(dTld8D) by KST(dT/dn).
8"33. The cases of 8'311 which are most easily appreciated are
found in the application to two special families of curves on the
surface, the <E>-curves and the <E>-orthogonals. The gradient of O
is a tangential vector G which is at right angles to the ^>-curve.
Points where the gradient is the zero vector being excluded, the
amounts of the gradient are separate single-valued functions of
position on the surface, and one of these is chosen to be called the
slope of <3>; the slope will be denoted by G. The direction in which
the gradient has the slope G is defined to be the standard direction
of the 3>-orthogonal and of the tangential normal to the 4>-curve, and
will be denoted by OM. The direction OL with which ON makes
a positive right angle is the standard direction of the 3>-curve.
By definition
8-331 3^ = 0, &M=G,
whence
8-332 <£ *MT = dG/dsT = G1T,
8-333 <b*LT = - GvTL = -G {tcgT + (deTL/dsT)}.
Thus
8*334. The geodesic curvature of the <&-curve is — ^LL/^M*
and
8'335. The geodesic curvature of the <&- orthogonal can be expressed
both as — ^LM!^M and as-^d (log G*)/dsL.
It may be added that 8'335 is deducible from 2'831 and 5'224,
for with the convention by which <J> is extended into space a
<3>-surface is the ruled surface composed of the normals to the
original surface along a <E>-curve.
841. I have not succeeded in continuing satisfactorily the
sequence of geometrical functions of which the first two members
are the bilinear curvature and the Codazzi function. Differentiation
of 6115 gives
8-411 4>VsT + G*QTKKS + G'2BTKQS + G*STKQR + G2RSKQT
UN (KQTKRS + KRTKQS
78
where
8*412
and while 8*412 shews that ^QRST depends only on the form of the
•^-surface, 8'411 shews that the function is a symmetrical quadri-
linear function of tangential directions. Thus there is no difficulty
in the construction of the third member of the sequence of functions,
and none is to be anticipated in repeating the process again and
again, but no general rule is apparent under which the constructions
fall, and it is evident that the formulae rapidly become too com-
plicated to be intelligible without some clue to their composition.
79
INDEX OF DEFINITIONS
Amount of a vector ... ... Ol
Amplitude of curvature, A ... 4'42
Angular derivative, da ... O6
Asymptotic angle, v 4*54
Asymptotic torsion, su, sa 4 '52, 4*54
Bilinear curvature, KST ... 2*31
Bilinear rate of change, <&2PQ ... 3'31
Bilinear torsion, SST ... ... 4*32
Codazzi function, \RST ... 6'11
Core 1-31
Darljpux determinant ... ... 7 '34
Darboux function of
direction, \TET ... 6*13
Darboux gradient, gdbT ... 8'11
Defect of curvature,
B-Kn> SET 4-12, 4'32
Geodesic curvature, <ff, crTT 2 -21, 2-41
Geodesic torsion, sg, KET 2*11, 2*32
Gradient of a core 1-71
Gradient of a scalar function, Gr 3*22
Laguerre function of
direction, \TTT ... 6*13
Linear function ... ... I'll
Mean curvature, B ... ... 4'12
Multilinear function ... ... I'll
Multilinear rate of change, ^PQ...^ 3'41
Normal angle, w 2 '52
Normal curvature, <n, <TT 2 '11, 2-32
Projected product of two cores T53
Projected product of two
vectors, C£TS ... 0'2
Radial, 1^ O'l
Slope, G 3-22
Source 1-51
Spaciousness, T 7-22
Spread, rs 2*73
Swerve, <TTS 2-41
Swing, <TNS 2-71
80
INDEX OF SYMBOLS
1R 0-1 <TNS 2-71 B 4-12
a,/3 0-5 <r/ 2-41 6'T, Cz 4-42
a, fry 0-4 <rrr 2'41 D 0'5
A, B, T 0-4 s 2-52 da, daQ 0'6
€ST 0-5 Sa 4-54 E 0-5
£ 4-44 Sg 2-11 G 3-22
K 2-52 SET 4'32 G 3'22
Kg 2-21 Sn 4'52 £ 0-2
Kn 2-11 SST 4-32 gdbT 8-11
KET 2-32 trr 4-32 / 4^52
KSI 2-31 TS 2-73 J,K 4-54
KTT 2-32 v 4-54 K 4'46
»t, »g 4-42 Y 0-4 I, w, n 0'4
6-11 *1P 3-21 PAB...K I'22
7-32 *2pw 3.31 (Pj»)s 1-33
OT 2-52 ** v 3-41 ra 01
or 2-72 w '" 0-5 T
<TTE 2-41 A 4-42 x,y,z 0'4
PRINTED IN ENGLAND BY J. B. PEACE, 3M.A.
AT THE CAMBRIDGE UNIVERSITY PRESS
Neville, Eric Harold
Multilinear functions of
direction and their uses in
differential geometry
Physical &
Applied Sci.
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