THE PROJECTIVE DIFFERENTIAL GEOMETRY OF RULED SURFACES

A THESIS SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR

THE DEGREE OF MASTER OF SCIENCE

BY WILLIAM ALBERT JONES, JR.

DEPARTMENT OF MATHEMATICS

ATLANTA, GEORGIA AUGUST 1949 t-lii f£l ACKNOWLEDGMENTS

A special expression of appreciation is due Mr. C. B. Dansby, my advisor, whose wise tolerant counsel and broad vision have been of inestimable value in making this thesis a success. The Author ii

TABLE OP CONTENTS

Chapter Page I. INTRODUCTION 1 1. Historical Sketch 1 2. The General Aim of this Study. 1 3. Methods of Approach... 1 II. FUNDAMENTAL CONCEPTS PRECEDING THE STUDY OP RULED SURFACES 3 1. A Linear Space of n-dimens ions 3 2. A Ruled Surface Defined 3 3. Elements of the Theory of Analytic Surfaces 3 4. Developable Surfaces. 13

III. FOUNDATIONS FOR THE THEORY OF RULED SURFACES IN Sn# 18 1. The Parametric Vector Equation of a Ruled Surface 16 2. Osculating Linear Spaces of a Ruled Surface 22

IV. RULED SURFACES IN ORDINARY SPACE, S3 25 1. The Differential Equations of a Ruled Surface 25 2. The Transformation of the Dependent Variables. 26 3. The Transformation of the Parameter 37

V. CONCLUSIONS 47

BIBLIOGRAPHY 51 üi

LIST OF FIGURES

Figure Page

1. A Proper Analytic Surface 5 2. The Locus of the Tangent Lines to Any Curve at a Point Px of a Surface 10

3. The Developable Surface 14 4. The Non-developable Ruled Surface 19

5. The Transformed Ruled Surface 27a CHAPTER I

INTRODUCTION

1. Historical sketch.--Projective Differential Geometry- deals with the invariant differential properties of a con¬ figuration under a group of transformations. This geometry is largely a product of the first part of the twentieth century.

It has been compiled in five or more different languages.

Part of this work has been published in journals not readily accessible to all readers of mathematical works. Wilczynski in 1906 published in book form his projective theory of curves and of ruled surfaces in ordinary space; five years before, he had devoted about ten memoirs to this work, published in the Transactions of the American Mathematical

Society. This was a significant contribution to projective differential geometry. Wilczynski also gave to the world at this time a new method in geometry and established himself as the founder of a new school of geometers. 2. The general aim of this study.—In preparing this thesis no new theories were introduced, but through careful study the author has completed much of the computation omitted by Ernest

P. Lane in his work on the projective differential geometry of ruled surfaces.

3. Methods of approach.—For the benefit of the reader chapter II has been devoted to the fundamental concepts pre¬ ceding the study of ruled surfaces. It will be understood that

1 2 the study of ruled surfaces will imply the projective differen¬ tial geometry of ruled surfaces throughout this thesis. Founda¬ tions for the theory of a ruled surface in a space of n-dimen- sions will he introduced in chapter III, and a space of n- dimensions will he defined.

Wilczynski's theory of ruled surfaces is based upon a consideration of the invariants and co-variants of a system of two ordinary linear homogeneous differential equations of the second order in two dependent variables, under a suitably chosen group of transformations. This analytic basis will be used in chapter IV on ruled surfaces in ordinary space. Wilczynski's notation as is used by Lane will be used through¬ out this work and some of his most fundamental geometrical results included. In conclusion, chapter V gives briefly, a summary of all the accomplishments of this study. CHAPTER II

FUNDAMENTAL CONCEPTS PRECEDING THE STUDY OF RULED SURFACES

1. A linear space of n-dimensions.—A space which has the property that a straight line joining two distinct points in it lies entirely in the space is called a linear space, and is denoted by Sn. The coordinates of any point of a linear space of n-dimensions are represented by n + 1 homogeneous coordinates. A point x is represented in a space of n-dimen- sions by Px. The coordinates of the point x are (x^»x2,.. .xn<(>^)

x Sometimes the set (x^,Xg,...* nV^) is thought of as a vector x with n ♦ 1 components. A scalar may be thought of as a vector with only one component.

2. A ruled surface.—Just as a curve can be thought of as the path of a moving point, a ruled surface can be defined as the locus of a moving straight line, and may be described as a single infinity of straight lines which are generators of

the surface. The essential characteristic of a ruled surface

is that through each point of the surface there passes one

straight line that lies entirely on the surface. A quadric is an example of a ruled surface. A non-singular quadric surface

in ordinary space S3 is the only example of a ruled surface having precisely two generators through every point of the

surface.

3. Elements of the theory of analytic surfaces.--An

3 4 analytic surface may be defined as follows: If the n * 1 homogeneous coordinates x of a point Px in a linear space of n-dimensions are given as single-valued analytic functions of two independent variables u,v by equations of the form

(1) Z x1(u, v) (i Z 1,2,3, ,n+l), then the locus of Px as u,v vary is an analytic surface S.

Equations (1) are spoken of as parametric equations of the surface S. If we make x a general one of the x^ and drop the subscripts, we can replace (1) by (la) x Z x(u,v) which is called the parametric vector equation of the surface

S. In order for a surface to be a proper analytic surface, the coordinates of x must satisfy no linear homogeneous partial differential equation of the first order of the form

(2) ax^j +■ bxv + cx z 0, in which the subscripts indicate partial differentiation with respect to u and v, and a,b, and c are scalar functions of u, v, not all zero. If the coordinates x of a point Px are given as analytic functions of u,v and are suppose to satisfy an equation of the form of equation (2), to integrate this equa¬ tion would show that the coordinates could be expressed in the form

x± : p f^t) (i r 1,2, n+1), wherein ^ is a function of u,v which is a particular solution of the equation 5

Pig. 1. A Proper Analytic Surface 6

atu + btv = °» while the f^ are functions of t. The locus of Px might de¬ generate into a fixed point or a curve. We shall confine our discussion to a proper analytic surface, and avoid cases where equation (2) may happen to be satisfied conditionally. When the parameter u in the equation

x I x(u,v) remains fixed and v varies, the point Px describes a curve on a surface S for which v is the parameter called the v-curve upon which u is a constant, and hence du Z 0. This curve is denoted as Cv. Similarly when the parameter v remains fixed and u varies the locus of the point Px describes the u-curve on the surface, and is denoted as Cu. On the u-curve, v is a constant, and hence dv a 0. These curves are called the para¬ metric curves on the surface S, and the tangents to the u-

curves and v-curves are called u-tangents and v-tangents re¬

spectively. All of these tangents are spoken of as parametric

tangents, (See Pig. l). Two one-parameter families of curves on a surface S are said to form a net if through each point of

S there passes just one curve of each family, the two tangents of the curves at the point being distinct. We shall suppose

in our discussion that the parametric curves form a net, namely the parametric net, on the portion of the surface under

consideration. Each pair of values of the parameters u,v locates a 7 point Px on a surface S. In a region on S where points Px and pairs of values of u,v are in continuous one-to-one corre¬ spondence, u,v are correctly called coordinates of Px. We refer to these as curvilinear coordinates to distinguished them from the projective homogeneous coordinates x.

Any curve on a surface can be represented by a curvi¬ linear equation in curvilinear coordinates of the form,

(3) C:

u z u( t), v r v( t).

The slope of the tangent of curve C at the point Px is dv< du*

To write the differential equation of a single infinity of curves on S through Px, set

~ = À (u,v) ( À s a scalar function of t) du we have

dv I A(u,v) du, then

(4) dv - A (u, v) du z 0 which is the desired equation. The equation of any net of curves on a surface S may be found as follows: Prom metric differential geometry we know that the differential of an arc of a curve may be expressed in the form of non-homogeneous coordinates as

ds^ - dx^ + dy^ + dz^, 8 where x,y and z are functions of s. In dealing with the n + 1 homogeneous coordinates of the point Px, where

x - x(u,v)

and u z u( t), v z v( t) we have the vector equation dt2 s dx2 but

dx I xudu i xvdv,

and xv satisfies the equation of the form

xudu + xvdv s 0. Putting the value of dx in equation (4) and setting dx equal

to zero, we have

2 2 xu du + 2xuxvdudv •+- xv dv Z 0.

Letting

2 2 A = xu , B = xuxv, G = xv ,

we have the desired equation

(5) Adu2 + 2Bdudv + Cdv2 =0 (AC - B2 ^ 0)

which is the curvilinear differential equation of any net of

curves on a surface.

We shall now prove that at an ordinary point Px of a sur¬

face, all the tangent lines of all the curves on the surface through that point form a pencil of lines with its center at

the point Px. We know from the properties of straight lines

that two points determine a line. This is true of the tangent 9 line of a curve at any point. The tangent line of a curve at

a point P_ is determined by x and one other point on the tan- gent, x'. x' is spoken of as the derived point. Prom the

equation

x = x(u,v) where u,v are functions of t, we get

dx - c) x du 5 x dv dt“ç)udt ôvdt which can be written as

(6) x' r xuu* + xvv*

The tangent of the u-curve at x is determined by two points,

?x and x^, and the tangent of the v-curve at Px is determined by x and xv. If the point Px remains fixed while the curve C

takes on all possible directions, then xu and xv remain fixed while dv varies. The locus of the point x* becomes a straight du

line joining the points xu and xv. This we could detect from equation (6), seeing that x' was a linear combination of the

points xu and xv. Therefore the locus of the tangent line xx' is a pencil of lines lying in the plane determined by the

points x,xu,xy with its center at the point Px as was to be proved. (Fig. 2)

We shall now introduce the classical definition of the

tangent plane. The tangent plane at a point Px of a surface

S is the plane determined by all the tangent lines to all the

curves through P at P . At a point P of a surface where A X X 10

Pig. 2 . The Locus of the Tangent Lines to any Curve at a Point Px of a Surface. 11

x r x(u,v), the tangent plane is determined by the points x,xu,xy. If the surface is in ordinary space S3 the equation of the tangent plane may be written in the form (7) XL “*2 *3 *4

X1 x2 x3 X4 : 0, x x x x U1 u2 u3 U4

S % % *v4 where X is any variable point on the plane.

We have shown from a curvilinear equation of curve C that the position of the tangent at a point Px on C on a surface is determined by dv. Then we can say that this derivative is the cFü direction of the curve G at the point Px.

Since our definition of sin asymptotic curve on a surface involves the osculating plane of a curve on a surface, we shall consider first the osculating plane. The osculating plane of any curve C on a surface is the limiting position of a plane through a point Px and two neighboring points on G as the two neighboring points approach Px independently and sequentially along curve G. As any plane can be determined by three non- collinear points, we can say that the osculating plane at P A is determined by three points of a curve on a surface where u and v are functions of t. These points are x,x', and xM. x’ is given by the equation 12

x' r xuu• + xvv *, and x' ’ is given by

x* ' r x^u»' + u'fx^u' + x^v») + xvv’ ' + v'(xwv' + xvuu’^

2 , t 2 1 (8) X* ' = XUUU’ + 2xuvU V 4 Xy-yV* + xuu’ + xyv’ *.

If the surface is in ordinary space S3, the curve

u z u(t), v Z v(t) is an asymptotic curve if not only x,x' but also x'' satisfy the equation of the tangent plane. Therefore a curve is an asymptotic curve on a surface if the tangent plane of a curve on a surface coincides with the osculating plane at each of its points. In order to get the curvilinear differential equa¬ tion of an asymptotic curve on a surface in ordinary space, we substitute x'1 for X in the equation of the tangent plane and the results will be

? X ', X, Xu, xv| = 0.

After expanding this determinant we have

2 2 ( 9) L du + 2 M dudv + N dv s 0, where L, M, and N are determinants of the fourth order defined as

(iO) L = I xuu,x,xu,xv I , M r I xuvx,xu,xv |,

N = I xvv,x,xu,xy I .

Equation (9) is the curvilinear differential equation of an asymptotic curve. In a brief statement we may say that, usually asymptotic 13 curves form a net. When they fail to form a net the case is exceptional, and this case will he omitted.

4. Developable surf aces.-»-In space Sn the locus of all the tangent lines to a curve form a developable surface which may sometimes be called a developable. The tangents are gene¬ rators of the developable and the curve on which they lie is known as the edge of regression. The contact point of a generator and the edge of regression is known as the focal point. Some special cases of the developable are as follows:

a. If the edge of regression is a straight line the

developable reduces to this line and is not a proper

surface.

b. If the edge of regression reduces to a fixed point, a

little thought will show that the developable can be

thought of as a cone with its vertex at the fixed

point.

Let us proceed to write the parametric-vector equation of the developable surface. Let the parametric-vector equation of the edge of regression be

cy: y = y(t), and the curve Gy Is being generated by Py. Since we know that the tangent to Gy at Py is determined by y and y', we can de¬ fine any other point Px on the tangent by writing the equation of x as a linear combination of y and y'.

(11) x » y* + uy (u is a scalar) 14

Fig. 3. The Developable Surface. 15 In this equation x is a function of u and t. If u varies and

t is fixed, Px describes the tangent line; but if u and t both vary the locus of Px is a developable surface. Therefore (11) is the parametric vector equation of a developable surface.

On a developable surface the tangent plane is determined by x and two other derived points, xu and x^, since x is a function of u, and t. Our tangent plane on the analytic sur¬ face was determined by x,xu,xv when x was a function of u,v.

Calculation of xu and xv shows that the tangent plane is de¬

termined by yjySy’’» Prom the equation

x = y* + uy

in which y = y(t)

then

(12) Xu = y

and

(13) xt : y* ' ♦ uy’.

,, Here we can see a relationship between x,xu,xt and y,y',y > which tells us that the tangent plane on a developable de¬

M termined by x,xu, and x^ also depends upon y,y',y . There¬ fore the tangent plane is the same plane at every point of a generator, and the tangent plane is identical with the oscu¬

lating plane at the focal point of the generator along the

edge of regression. We can now show by using equations (9),

(10) that the edge of regression is an asymptotic curve from

the preceding proof. Let us replace v in equations (9), (10) 16 by t. We have then

(14) L du2 t 2M dudt + N dt2 : : 0 and

(15) L : 3WX»XU*Xt | » M = xut»x,xu,xt

N s xtt'x’xu’xt | *

Prom equation (11) we get

(16) ^ I Ji Xyxu Z 0, xtt - y’ " + uy'*•

xt = y» »-t-uy' Xut = y»,

We shall now calculate the determinants (15) by using values of x^t^uu^tt^ut’ ln ®quations (16)

0 yi + uyx yi y ï 0 + y2 uy2 y2 y2 25 - o, 0 + yi uy3 y3 y3 0 y ’ + ^4 Uy4 y4

,? y 4 U7 y u y n i 1 i ^y'i yi yi yl yi + Y" y2 y2 Uy2 y2 uyavy 2 y2 y2 y2 *2 + y" y3 73 Uy3 y3 U^Vy3 y3 y3 y3

4 y" 74 y4 Uy4 y4 MÂy l y4 y4 y4 y4

y{ yx yx J'{

y y y 2 2 2 y2 + U - 0, y' y_ y_ y" 3 3 3 3

y 7A y 4 4 17 y ’ * y ' f ’ + uyîjj y I * Uyi yi y",yi ’ y I + uyi yi y"Ji

yt » » 4 y" ,y2 ' * uy£ y 2 Uy2 y2 y2 y'y2 ’ ' y 2 * Uy2 y2 y”» N = yl ' * + uy" y » + uy y„ y" y" • y ’ ♦ uy y„ y" 3 3 3 3 3 3 3 3 3 3 3

y*1 ' + uy" y » + uy y y” y" ' y * uy y. y" 4 4 4 4 4 4 4 4 4 4 4

» y,i y{ * u7l yi yi I yi’ n yi yl » yS y2 + Uy2 *2 y£ y2 *2 y2 y2 + U "* y y y y t v”y y y 3 3 ♦ uy3 *3 3 3 ' 3 3 3 i y" y u y y y* y" 7 y i * y4 4 ; ^4 ^4 l 4 * < \ r î y y y 7V* * t 7 y ' l l i 1 1 i yl

y y v"y Y» ' r" y yi' ’ 2 2 2 y2 ' 3 2 y2 2 +u • ' y, y„ y’' * 3r" y' y. y’3’ y"3 3 3 3 3

y’ ' ' y„ y. y" y'' • y" y* y. 4 4 4 4 4 4 4 4

ce L : M r 0, N = iyi * '»y \y- »y j and the asymptotic curves consist only of the generators of the developable sur face, and the equation of the asymptotic curves on a develop able becomes

(17) N dt2 = 0 CHAPTER III

FOUNDATIONS OF THE THEORY OF RULED SURFACES IN Sn

1. The parametric vector equation of a ruled surface.—

Let us consider again that a ruled surface is generated by a moving straight line. It can also be defined as a one-para¬ meter family of straight lines. A developable surface is a

ruled surface generated by straight lines, but these lines are all tangents to a curve which is the edge of regression of the

developable surface. The general ruled surface does not have this property, and hence we may say that every developable

surface is a ruled surface, but not every ruled surface is a developable surface.

We can proceed to write the parametric vector equation of

a ruled surface by taking two distinct curves in space Sn with parametric vector equations

C : z = z( t) (i) z cy: y = y(t).

We shall suppose that the point ?z and Py, which generate the

curves Cz and Cy respectively, are distinct. Let us now draw

a line joining a pair of points Pz on Cz and Py on Cy having

the same corresponding values of t. The points on the curves

Cy, Cz are in a one-to-one correspondence with each other. As

t varies, the line lyz joining the points Pz and Py, generate

a ruled surface R (Fig. 4). Any point P x except P y on the 18 19

Pig. 4. The Non-developable Ruled Surface 20 generator lyZ can be defined by writing the representation of the point x as a linear combination of y and z. This can be done by setting

(2) x s z f uy (u is a scalar).

If in equation (2) t is fixed while u varies, the point Px is allowed to move along the generator of the ruled surface. If u and t are allowed to vary, P describes the ruled surface R.

Therefore equation (2) is the parametric vector equation of a ruled surface. From our discussion of the equation of develop¬

able surfaces in chapter II we are able to see that if z - y'

in equation (2) our surface is a developable surface. If z is

a constant the surface generated is a cone. There is a case

in which the surface reduces to a fixed line. The discussion

of this case will be excluded from this study.

If there are two distinct curves on a ruled surface such

that their tangent lines at the points where they cross each

generator are co-planar, the surface is developable. We shall

consider the tangents to Gy and Cz at points Py and Pz respec¬

tively. C and C are director curves, and P and P are V «/

points where these curves cross the generator lyZ. The tan¬

gent of Cy at Py is determined by the points y and y', and the

tangent of Cz at Pz is determined by z and z'. Since these

tangents are coplanar, the points y,z,y’,z* are coplanar.

These points being coplanar, there exists the relationship

(3) ay + bz + cy* + dz' z 0,

where a,b,c,d are scalar functions of t. This may be further 21 written as

(3a) (cy + dz)' - c'y - d'z + ay + tz : 0,

Transposing and collecting the coefficients of y and z, we have

(3b) (cy + dz)' z (o' a)y + (d1 - b)z z 0.

In this equation when t varies cy + dz describes a curve whose

tangent is the generator lyZ* Therefore all the generators are tangent to a curve, and hence the surface is developable.

If we consider the ruled surface

x = z + uy, we find that the tangent plane is determined by the points x,xu,xt, and hence by y,z,z’+uy'. The equation of the tangent plane in space S3 is

X I > x, x^ xt I = °, where X is any variable point on the tangent of Cx determined by the points x and x*. Prom our equation of a ruled surface

x = z + uy

*u = *

Xt = z' + uy'.

Setting up our equation of the tangent plane we have

zn + uy. z' + uy'

X z + u z +U 2 2 ^2 ^2 2’ *2 (4) = 0. z, + uy. z' + uy,!

X4 z4 + uy4 y4 z| + uy^

We shall show that the tangent plane is determined by y,z, z'+uy'. To do this we may write equation (4) as the sum of

two determinants; + uy + z X1 ^l 71 z{ ^y{ i i

X Z + Z + Uy 2 72 y2 2 Uy2 2 2 + + z* uy« X3 73 y3 Zi Uy3 ♦

X + 4 y4 y4 ^y4 + uy£

X1 zi yl H + uy{ + X2 Z2 y2 Z2 uy2 = 0 X3 Z3 y3 Z3 Uy3

X Z y + u 4 4 4 Z4 y4

Thus we have seen that the tangent at a point of a ruled sur¬

face contains the generator through the point.

2. Osculating linear spaces of a ruled surface.—An os¬

culating linear space of a curve in Sn of a surface

(0

P and k neighboring points on C as each of these k points

approaches P independently and sequentially along C . X

Any curve Cx on a ruled surface Sn, except Cy and the

generators can be defined by placing

u Z u(t)

in the equation 23

Cj cannot be defined here for there is no value of u for which x can equal y in this equation. The osculating linear space

at a point Px of the curve Cx is determined by the points

(x, x’, x”, x ^ ), whe re

x’ - z’ + uy' + u*y,

x” = z” uy” + 2u,y’ + u"y, I 1 x» » • 2 z" i + uy'*' + u’y" + 2u*y" *♦- 2u”y + u"y' + u' 'y,

“ z» “ + uy' 1 ' 3u’y” + 3u”y' 4 u!,,y,

i=0

The ambient of a set of linear spaces is the space of least dimensions which contain a set of linear spaces. For example, a set of linear spaces would be made up by two points.

The ambient of these points would be the straight line joining them. We can consider another example of a straight line and a point not in united position, the ambient of which is the plane determined by them. From our examples we see that the ambient of a set of linear spaces may be defined as the join¬ ing space of a set of linear spaces.

The osculating space S(k,r) with respect to an element

Er of a curve at a point Px of a ruled surface R is the am¬ bient of the osculating space at the point Px of every curve on the surface that passes through P and has at P the A A 24

same element Er(r <> k). Let us consider an example, say the osculating space S(2,l). We say that the osculating space

S(2,l) of a curve C at a point P of a ruled surface R is the A

ambient of the osculating plane at a point Px of every curve on the surface R throxigh Px, containing the same tangent line

at Px. CHAPTER IV

RULED SURFACES IN ORDINARY SPACE S3

1. The differential equations of a ruled surface In S^.—

In this chapter we shall begin with Wilczynski's system of differential equations defining a ruled surface in ordinary space S3. The canonical forms will be obtained for the differen¬ tial equations by means of certain projective transformations of the dependent and independent variables which do not dis¬ turb the surface. We shall now take into consideration a non-developable ruled surface R with the parametric vector equation

Rs x = z + uy, immersed in space S3. On the^cîevelopable surface the determi¬ nant | y, z, y', z'| does not vanish identically, for y,z satisfy no equation of the form

f ay + bz + cy 4 dz’ Z 0, where a,b,c,d are scalar functions of t. It can be shown that y, z satisfy a system of differential equations of the form

+ y" Piiy' + P12Z' + qxlY + q12z = o, (1) z + y + p z + q y + q Z " P2l ' 22 ' 21 22 = °' in which the icanc^ qi k are ^unc‘ti°ns a parameter t,

(i,k 2 1,2), and y,z are differentiated with respect to t. In space S3 the coordinates of P^, Pz are y^, z^ (i 2 1....4).

Therefore equations (1) represent two sets of four equations,

25 26 each in four unknowns. The coefficients Pn»P^2*^11*q12* can be determined by substituting the four pairs of coordinates

^i,zl ^'3-rsl3 equations (l), also P2l>P22»^21» ^22 can be determined similarly from the second of equations (l). R is called an integral ruled surface of system (l). Conversely, when equations (1) are given, the theory of differential equa¬ tions tells us that these equations have four pairs of solu¬ tions y^,z^, (i = 1..4) forming a fundamental set. When these solutions are interpreted as the coordinates of the two points

Py,Pz, the equations (1) define a ruled surface in space Sj except for a projective transformation. 2. The transformation of the dependent variables.—Our dependent variables as we have already seen are y,z. We shall choose our transformation as

y : oT ^ + 0 X > (2) ( A = a 5 - 0r ) J 0, z : r y + 5 Ç, giving us our new dependent variables -y, Ç. In this trans¬ formation the coefficients o(, (3,^,8 are scalar functions of t. We shall now calculate the values of y",y*,y,z",z*,z, and get

7’ - qr : Of *»! + Ç’ + P' C » 1’*

7” = o( 77" + 2cv’ Y + a"rj 4 + 2 /?» Ç* t ,

z ’ y~ * 1 + z y~r r; ’ * 7 * SX' z" = \ ) » 4 2 y» y» 4 r» y 4â Ç " 4 2 &» ç» * . Substituting these values in equation (l) we get 27

1 , , ,, ( <* YJ " + 2 Of rj ' * of "y + +2/d’ Ç f■^ f)

+ P1;L( a y' + q 'j + /J Ç* + ft' £ ) 4 P12( rrj' + r‘y +

)= SÇ + fe’Ç ) * qn(*7 + &Ç) i qi2(ry + 5C °» (4) ' 1 ’ ir-y" * 2 y' y' 4 ^"7 + S Ç” + 2 s' Ç' + 6 " £ ) +

P21^ <* y' +~ °( '*J + /

S Ç» + S'Ç ) t q21U y + /3Ç ) + q22(-r j * S Ç )= o.

Let us now collect the coefficients of the dependent variables y, . Our new equations in J , Ç become 1 4 2 tr y" -t ■* (2 pr pxlpf «♦ P12) y’ t /

r-y» ^ 5Ç" * (2r* 4 p21cr + v22r) y ' * (2 s» * p21

+ P22S)Ç' + (r* ♦ pax-’ * P22r' * î21« * 422^7

+ 1 ! S " + P21 ? ' P22 S' * Ï21 fi * Ç22 S ? = °-

The transformation (2) changes the director curves on the sur¬ face from Cy, C2 to Cy , Cj- (Pig. 5.) respectively without affecting the surface. Therefore the systems (1) and (3) have the same integral ruled surfaces.

We shall now use the transformation (2) to simplify equations (1) which are the fundamental differential equations. From equations (5) we have o(, Y' and , £ as two pairs of solutions of the equations

2 e< * Pllf ♦ p12

Fig. 5. The Transformed Ruled Surface. 28 in p , 0-, The first derivatives of fj, Ç will not appear in our new equations. Therefore our equations (5) become

4 (p 4 p 4 4 o< rj" * 0 C" ll/°' 12 /°" P+ ^12 *

4 4 4 4 (

4 (p21/°’ 4 P22

By making use of our equations (6) we shall reduce our equations to a canonical form by eliminating , g- ", (°'» 0~ * from equations (7). We shall first compute the values of ", (T " >

x (° 0~ from equations (6) (8) ' . - Pix P - Pia

(9) P 11 ?' “ P!l/° I P12^' - P12CT . 2 Next we get

» P (10) 22tT (T I

and / _ _ v “ p21 " P21 ” p22

Since we desire to eliminate all derivatives of (° and

shall substitute the values of f> ’ and (7-’ in equations (9) and

(11) so that our equations will not involve /o' and

P P + P P P P P + P

+ P P P P p p °t 7" P Ç"-*£lip * 11 12 (T . 12 21 . 12 22 <7 ' ”2 + 2 + 2 2

P P P P _ £ll p _ 11 12 o' + £llp _ 12 21 p

P P q - 12 22 (T + £l2

P P P P P P - - 12 21/Q 4 12 - 2 2 2 2 .2 P P p* P P 11 12 rr 11 P _ 12 21 4 4 4 2 " 4

p;„P ’ q„_ q’" cr)C ; 0, £12 g- + ' ' (14) 4 2

rP21P] P12P21 P00P0-.22 21 p_ 122P (p T 2 2 2 2

Pr p2ip: P21P12 _2J P P + 22 21 4 4 2

k y v 9r ; _ £22cr + £22O- _ £gl p _ £22 (T >7 4 2 '

(POT Pi POTP-IO^ PooPr 4 21 11 p + 21 12 (T + 22 21 p + 122 2 2 2 2

p p + p + y * + -gg 2.i ,Q 4 4 2 2 30

-Skcr q2i p ^(p.r =o. 2 }

We shall further simplify our equations hy multiplying our equations by 4. Equations (14) become

4( o( rjn+ <3 Ç-") = (2pll/° 4 2pllp12

V11P “ PllP12

P12P22

4‘2p!lf° +2P11P12«- + 2pl2P2lP 42P12P22

Plll° ' PllP12

- P12P22

* P21P11P - P21P12(P 4 2HlP ' P22p21 P

P22 ^ + 2p22

4(2p21Pll/° * 2p21P12°" 4 2P22P21? 4 2pL'r

‘ p21pll/° - PsiPlgT * 2p2lP 4 P22P21 P

” p22ir 4 2p22ff " 4q2lP _ 4q22

We shall proceed to simplify our coefficients of y and .

4(01 4 4q 4 4 4 (2p 7" - n P11 P12P2I> <^ i2-

4q 4 p p 4 p p 12 12 ll 12 22^ ^(2pix - 4qxl

4 P 4 P P } 4 (2p 4q 4 P P (16) 11 12 21 CT i2 " 12 12 11

4 p12p 22-^7Ç = °» 31

4 4 + r(s ’< rrj"+ - 4q21 P21P22 P2iPn> < P22

4< 4 4 (2p 4 - l22+ P22 V21V12H7 ^ 21 - «21 n 4 p p 4 p p ) 4 ( 2 p * - 4q 4 p P21P22 P21P11 0 *22 H22 ^22

+ Pen21 FP1 2 ?C = We shall let

(17 2pil- 4qil+ PL + P12' P2l'

U : 2p 4 P (P 4 P K 21 k “ 4q21 21 11 22 & _ 4< + p + p p U22Z Sp22 ^22 22 21 22» 4 U12~ 2pl2 4ql2 P12(pll 4 P22>* and write our canonical . form thus

(18 > ) 4(^ y ”t 0C)Z (4 un 4 rM12 7 4 1 ^ *11 4 «12 )C

M W 4 r W 4 (/? M + 4()~ rj "4 5Ç )= (* 21 22>7 21 £ ^22 >Ç

We have already imposed the condition upon our transformation (2) that o( § - (3 Ÿ* 0. Therefore we may solve equations (18) for y”, and get our equations (18) in the same form as the

system of equations (1), with y', £■' missing. This will he shown by Cramer's rule. The right member of equations (18) will be treated as a constant. For this purpose we shall write equations (18) in the form

(19) 4 ofy" + 4/3 : h.vj 4 k Ç ,

4 r y " 4 4 & Ç" = I»; 4 mÇ , in which h, k represent the coefficients of y , £ in the first of equations (18) respectively and 1, m represent the

coefficients of 7 . t in the second equation of equations (18). 32

We shall write formulas for h, k, 1, m thus;

(20) h - or VI;L + y U.12, k = * & Z^i2'

i =-IM +r-u , + 21 22 = 0 ^21 ^ ^22*

Upon solving equations (18) we have

(21) (h7*kÇ) 4*| j> - ( 17 tm £ ) 4SI w &(h^+kC) ^(1*7 +mC) 7 - 16o 45 4c

(4* (h1? +k C )| r. - I 4r (17 tmt )l - qf ( ry MC) _ r(h7 -k < ) C - 16c 4c 4c in which * P A = 4c s 4 r b 7 o. Collecting the coefficients of y , we have

Y]" - (hS - 1? )„ (k & - mfi) Y

- ( 1<* - h r ) y, , (mg' - k r ) >-

S> A / A S •

We shall determine our new coefficients "by the formulas

hi k S - mA . 11 1 A A i— *12’

1* h r m °r - k r _ A -- *21- A **•22' in which the bars indicate the transformed functions. Let us proceed to interpret our new coefficients of ~J , C in terms of the coefficients of the equations (18) as follows: Prom equations (21) and (23)

A ^H=(hS -\

ûals=(ks-.

A =(m0< Kr) U rU r U ^22 “ - U21- <* à 22~ ^ ll~ & 12' Our equations (18) can now be written

(A a + AU ) + lA (25) ?" " il 12 7 M.xi+ A W12) £ C" 7 * (A *21*A *22>C, as was to be shown. If we consider the formulas f or A A we we can set the right members of these formulas equal to zero and obtain the equations

2 2 6 tc (27) ’ll* S “l2- ^ 21 ^22- 2 £ °< ri^ir r II CO 21+ ^^22“ H Dividing the first of equations (25) by Ç and the second by 2 - °C , equations (27) become

(28) W12 * ^11 " lX'22) " U21 ~ °»

1^12y / a + ^11 ” ^22} ^ ^ 21 “ °*

If the equation

(29) ^u12 + (U u - U 22) % - u S1 =0 ( A = , V< ) has distinct roots so that its discriminant Q defined by the formulas «• (30) Q Z {IX -Z^)2+4Ua 11 22 12 21 is not zero, we may take one of these roots for Y'/o{ , and the other V (3 . Then = A^-21 = °* Therefore it is possible by means of the transformation (2) to reduce our system of equations (1) to a canonical form for which 34

•^12= tX. g^= 0. The canonical form becomes

ta i" : A Üu»7 "uC- (31) 7 7 4 A ç in which A^ = A V z o. 2 21 It is possible to make any transformation (2) for which

P ~ Y' z. 0, of ^ 0 without disturbing the canonical form.

Since this satisfies our previous condition that oi S - fîY f 0, our transformation (2) becomes

(32) y = oj y , z = b C •

We can reduce equations (4) to a form of equations (1) by solving equations (5) for yjn and Ç". For convenience we shall write equations (5) in the form

(33) qy" + fiç" = -a^«-bÇ'-cj7-dÇ,

Yrj" f 8 £” - ~ e YJ ’ - f £ » - g Yj - hÇ , in which a, b, c, d, e, f, g, h represent the coefficients of

, £■’, ^ , £", in equations (5). We shall write formulas for these coefficients by letting

a ; 2 or' H + P b = 2 P' * p (3 + p 0 11* 12 ■ xl 12 *

c I + P 1 q + q r - 11* ' 12 r * ll* 12 (34) 11 * ' + P12 S’ + qll? * «12

1 + P f = 2 g 21 * 22 r * + V21fi + 22

21^' + p22 y* + q2i°( + «22 r. 35

+ q h = S" ♦ P21 (j> + P22 g’ t q21(J 22S>

By Cramers rule we find

1 (a -ÿ'+b C '«*07 +d £7 Pi 7 I (e *7*+f jrUgy 6(a y'tb +dC ) -

l(e «y »»f çtev+ht )j (35) • A

°< (a 7 »+b Ç'+c7 +d C )| tl r (e jy'tf g *+g y +h £ )l _

Collecting the coefficients of ^ , Ç’•, ÿ, we have

J' - ( S a-/? e) Wf ( 8b-^f) /-'*( £ c- P g) rt +( à d- Ph) j ~ A / A / ^ ( 36 ) - (^e-ra)yi1 (* f-^b) ;-' . ( gfg-r-c) ^ ( oTh- rd) ^ . l=>” A / ' A ^ * A /Tr A

We shall now define our new coefficients of the dependent variables by the following set of formulas.

II Sa - @ e, û7r c( e - h~a, aX S c - A g H 2i = 11 =

“ T c, £b - /?f, A1T o

aX s d - /? h, 12 = aX22 " <*h - T d.

The values of the coefficients k» X i k terms of the coefficients of equations (5) are

£ a- /?e “ 2 of' 5 + S + P12 a r -2 P y* ~P2101 @ ~P22 $h»

A = 2( )+ +P < r V V r * * ^ll r' Pn^ ^ 12 * ~ 21*^ ~ 22? '

2 p p & b~ @ f — 2 & /^^Pll ^ ^ +P12 & “ 5* ^ “ 21^” 22^ ^ *

• * A ^12 “ ^ ^ ^+PU;^k +P12^ "P21^ ”P22 @ ^ *

i 36

,, + r &c-£g=pn 5 f £ oC + qll0<6 ^12 S

-P22/3 r'- ^r"-q21« 0 -q22 ^r.

b «'+P126 r +q12ré

~v21«'0 -p22 ^ >"'"q 21rf/y _q22^'r •

Sd- /3h;Pll 0

"P22 ft 8*“ /^s"-q2i -q22 & & >

=( 4 5 i .‘^2 " - ^«"J+Pu/J'S 4P12 g’f *qlx/SS +q126

(38) 'P>21 -q22 ^ •

e + of c r p 2 * “ ra-2 P22 r -*2 or»r "P11 *? - 12 y- ,

lT =2 + 0f + c • ^ 2l ( * T’- qr’r ^ P21 P22 ^ r—Pn<^r -P12y?.

f 0< 2 ^f-rb=2 PTS*^^ P22 ^ “ (9’r -P13.^r -P125^»

’•ATT s2 1 • * 22 ( ** S’ /s’ r )+P2i°f^ •♦P22 °f 5 -Pn^r -P12 S y~ .

2 «g- ro=p21o( '« +pgg « y-' + »ir" + q21q< 4q22«r-pl:Lc<,r

■p12 "* l"1- «fV "^12 K-2'

Y 2 • ^^2i~( ^ "~ of" r )+P2i **' ** +P22 °f r’+q2i or +q22 «*r

p q 2 - nofV -P12 of r'^u^r “ i2>' *

,+ +( p dh-rd:p21^ /

-p12ré’- ^"-qn fir -q12 r/ ,

z( , + q • ^22 - 1^|8")<-P21 « (3 * *P22 ^ £ 21°^ **22 « $

“Pii^ ^’“Pi2 r i'-Qn/rr -q12 r 5 .

Now that we are acquainted with the new coefficients of our 37 dependent variables we can write our canonical form of equation

(1) thus; 7" = A 7' + û1T12 Ç'+ + aX12 C > (39) C" I ATT >7* + Air T'*-AX -n Ç . S 21 / 22 S 21 I 22 S

3. The transformation of the parameter,—We shall con¬ sider in this section, effects on the system of equations (1) when the parameter is changed from t tot . Thus our trans¬ formation of independent variables become

(40) X = T(t), ( x' / 0).

We must calculate the first and second derivatives of y,z with respect to t, changing the parameter from t to X • Therefore

âx d -v - L dz - dz dx _ _ ’ dz t'^L > dt dz dt dZ* dt “ dt dt “ t d-e (41) ,2 d2z » 2d2z ^”dz éEX = t' ü 4 t'*, z -x ^ + X —. dt2 dt2 dî dt2 d-C2 dt

We shall now substitute these values in equations (1) for y",z",y,,z' and get the equations

,y'+P t'z'+q. y+q. Z = 0, ( T* y" + T'V)+P lit 1120 iri 120 z , ( z' " + t"y )+P21 -p'y’+Pgg ^'z'+q^y+qggZ Z 0» in which the primes on y,z indicate differentiation with re¬ spect to X , and x’ or î” indicates differentiation with re¬ spect to t. We shall collect the coefficients of the dependent variables. 7 p z q y q z y"*( T " Pix t') ' 12 T' | ll , 12 « 0. 2* »8 +^-2’-p (43) 38

tt » p p q q ai £ 2.2 -c a.i + &2^ + + zt 4 0, z” * )y' 2,y £| 2Z x'2 X*2

Our next step will be to write the formulas for our transformed functions, p , q, ,, {i,k z 1,2). i,k' qi,k » T> X , P11 X ’ _ . P12 . P21 P22C\ ; p p P p ll * 12 “ 21 " 22 2 2 x,z Vs ■ Î' C'

,p p ,p P P = P p P + x n - “ * n’ x i2 = 12’X' 21 21 X* 22 = 22 (44)

q qi2 _ _ q21 _ q22 q _ H q ll - ^72’ ^i2 2* q2i “ x 72* 22 “ ^72*

.» 2q - q » q , ?q - q , ?q - q , ( uj _ X^\ - 11 “ 11 X^l2 - X *21“ 21 X 22“ 22 “ t*' Our equations (1) take the following form in terms of our transformed functions;

,+ Z h 2q 7+ ,2q z = y"+ T'PnyJll^ x'Vi2 ' V ll ^ 12 °» (45) 2_ 2 7 + Z,+ q 7+ . 7q z t’P2i ’ T’P22 x’ 2l *' 22 = °-

We shall now undertake to show the effect of our transformation

X a x(t) on the functions IXJ. ^ (i*k Z 1,2; k ^ i). We shall do this by taking a specific case in which i a 1,

Then we must write

UH = 2pll “ 4qll + P11 + p12p21* in which p^, q^, p^ are to be functions of X , We must cal¬ culate the first derivative of p^ with respect to X , and get the value of p2^ in terms of X • Prom our formulas (44) we have 39

(JÔ P11 = (- *P11

1 ,p TS T - u>' . P\i11= = -Cü "11’ +' P"lllf 7

But WJ = T"/ x' •

Then becomes ti2 • rf • + P XljtJZllL .—LIZ) 11 “ T *11 11 ri1 -.12 *»3 * or >»2 , t »T=i » • 11 (46) +F, li - <- il "il Tr 12 i3 * Prom the value of Pu»

,2 2 (47) p^ s T P^ - awt'Pn + «>

Now we can write in terms of the transformed functions as

, ,2 (48) H,= 2T (P{1X» * " -3LÜ1 + JL^) ~ 4? qX11- 11 12 >i 3 11 U“

(X'2P2 - 2*oT,:D + e*J2) - -T,2P P • 11 11 12 21

In clearing our parentheses we get the form

2— . j. o=r «Il __ o — t I I o«.tl 2 2 U1]L = Sx’^Pi! + 2?!^" - 2*c«", - 2^_f - 4 T » qn (49) *C' T'2 ' ^ ' 2—2 — 2 2- — + X' Pn -2eox’pn + Oü + ^ P12P21*

If we observe the terms 2p 11X" and we find by our

definition of OL) that they are negatives of each other and

vanish. Then we may collect the coefficients of -c'2 and omit

these terms. Therefore

2,^-,2 -2 + 2 (50) Lf1]L --ç’ ( P{i " 4q11 + plx + P12P21^ “ ^ » *

We shall replace the term_ 2x* * * by its corresponding value in X ' 40

terms of ai. If

(A3 Z

then 1 ..2 _ -£ f t I y w< w’ - T» X' ' _ _t z ,2 "C 2 X » 2 z'

and from this

1 1 2 (51) -C » _ + UJ # -€ »

Our equation now becomes

,2 2 4 5w2 2w3 2t 2 (52) t^i;L • -c (2p|1 - 4q1]L ■* p x + P12P21^ “ ' " ° *

This equation contains ln the form

(53) K, s 26* - 4q + b2 + p p 11 P11 *11 P11 P12P21

Therefore in collecting the terms of ou2 and transposing the

terms of m and wf, we have the desired formula 2 — Q (54) Z = L^ii - 2w' + U3C

We may now write a more general formula as

(55) i2 j Z t/u “ 2(v' + ui: X —ilA “ 1 i

in which tt^ equals ^^1 or ^22* ^»ut not both in the same

formula.

We shall now turn our attention towards the functions

*^i k = 1,2; k i- i). We shall show the relationship between and We shall consider a specific case

L^i2 and write the general form in conclusion. Prom formulas (44) -U 2 F 4 12 “ P^2~ ^q-jl12 Q PlC>(PlT12 11 ’ PQQ)*22 41

Before our substitution let us calculate keeping in mind our transformation

r= T(t) by formulas (44) - ^ t T) , 12 - <■ p12* therefore

(66) p*P - 'r’p* 4 pv _2L_. 12 " L p12 12 *

Then

7 2 4 2- (5 ) ^12 = T’( -C'Pi2 + “ ^'^12 * T'Pl2/^ *Pll

- <^) ♦ ( 'C'Pgg ~ t*i7*

After clearing of parentheses and brackets we have 2 2 2 1 (58) K - 2 t' p 4 2p T" - 4 T' q 4 'r'p P - T ' p «*3 12 12 12 '■* n12 o <■ fi129 rn11 t- 12 2 * 'TX' ’ ™PizPaa-

We shall now simplify the expression by collecting the terms 2 in "£* and (58) becomes

2 (59) = X’ Z2p{2- 4q124 p12(pxl 4 P22J7 * SPjiT" - « T'P12 u).

But -2 T'p^gUJ equals therefore

2 (60) vig ; x- ZSp{g - 4

Since

£7 a 2p*P - 4q4 4 pV V(pP 4 p ), **12 12 12 12 11 *22 '

We may write (60) as

1 (61) t X — >T 12 c w12 42

We may now write our general formula

2 (i k k (62) = T’ ^i,k » = 1>2> / *■>•

Under the transformation of dependent variables, 64 was seen to be absolutely invariant. In our transformation of the parameter we shall show that 64 will be transformed Into 6^.

The relationship between 64, 6^ will be shown by formula.

Prom equation (30) the discriminant was given;

2 4 U #4 = ( ^- 11 “ ^ 22) + 12 ^21*

Referring to formulas (56) and (62)

,2 + 2w 2 : ■'t ûLi ' - *> > and ik ik*

Therefore, using these formulas

(63) ,2^n + 2w' ~ w2) “ ( *2 ^22 * 2w,! "

4 + 4^ » Û12U21.

Upon simplifying this expression we have

,2 2( ?, 2 ,2 2 (64) 6>4 s ZT &n * * “ w» - X ^2 ” 2«* + c«r7

,4 + 4 'C Ü12^21*

We now have

(65) = (x ,2t(. ~ 'T.'2 t/ )2 t 4 /y* *4 7/ 7V 4 “’ll L 22' U12 ^21*

We may factor *£*f from the above expression and write

i 2 (66) ^ z z' /Tû - tl ) + 4 Ï2 Ü J. 11 22 12 21

Prom this expression we can observe that 43 2 U„J * 4^ U, «4 11 22 2 21

Therefore *4 = t'4 «4* as was to he shown. If we consider the transformations (32) and (40) which were (67) y s z = 5^,T=T(t) ( * bt* f °)»

t Z we can characterized by the conditions = ^21 ^>urt^ier

simplify the canonical form of our system of equations (l). The above transformation leaves these conditions invariant and

nto as we transforms the coefficients p^, VQ2 * Pu» P22

show by making use of our formulas (38) and (44). Keeping in mind that our transformations carry the conditions that

4 6 / °» we can realize the fact that - 0 does not

destroy the condition Imposed in our transformation (l) that

4 & - fir 4 0. It has also been shown previously that it is

possible to reduce system (1) to a canonical form for which

al rov W12 ^21 * P ided ^ / 0. Under the condi tions o( S ^ 0, fi,?* Z 0, the determinant of our transformation (2) becomes

(68) A * 0 Z - 0 0 h .

Prom our equations (38)

éK 77 2 (69) ^ 11 = ol'& + pxl o< 6 ,

therefore (70 tru , 4 p 11* 44 If we place for v-,-, in the formula 11n 11

r P11 " pn11"+ œ> we have

* -pp — p + + 2 * (71) T n - n * u)" * sr

By symmetry we may replace p2g by in the general formula for p ( i z 2) and get

(72) -T-* p _ p + O) + ——~ T P22 - P22 R We may write (71) and (72) thus

c*t,'Pn = (plx + «*W + 2 of’, (73)

^'P22 = (P22 + *26*.

Setting the right members equal to zero we have

1 2 rt * (p11 + ctfi) o( 2 0, (74) 1 2 & + (p2g -f oJ) 8 I 0.

We may write these equations in the form

2 <*» + P1;L®< + erf w = 0, (75) 2 S * i p2a£ » or 2 a»/ot t «> = -Pi;l, (76) 2 &*/& ♦ tP = “P22*

We have two linear homogeneous differential equations. We shall solve the equations for cf and $ . Beginning with the equation in qf in which ta z —— we have T (77) 2 -2Ll t -XII = -p-,. ( cK, T are functions of t) oi x* 11 45 Setting z 0, we may write

d« d(T') (78) 2 .âi- + —âj— = 0.

Multiplying equation (78) by dt we have

(79) d_± d(r') _ 0. X '

Integrating this equation we have

2> If d*_ + |f d( r 11L - c (c Z const.) J « J f Therefore 2 In o( f In 1 I In c,

Solving for o( we get

In a : In c, - lnt', or 2 c lnd -T In zl ». c*2 £l m ! *

Then

(80) <* - (* = 1/ ) • ( X* Is We shall now solve equations (76) for

a_L aLSji = 0 2 { + -c-

Integrating we have /"dj a( f) . 2 f C J 6 7 - 2 In S + In ( -c * ) 3 In

In 8 : In c, - In ( t ' i

In 8 s In 2%. T* 46 b (81) S = r- , = 1/~Cp). (t')*

Since °< , & are solutions of equations (53) with "c? arbitrary which makes w arbitrary, then s 152g * 0# and our trans¬ formations become (S8) T = t{t). y - : b £ (a,b - consts.). ( x' ) ®

By means of this transformation we can choose 'C so that

-jJ^ ^ 4» malcing ^ = 1. A second possibility is that we

2 may choose X so that x) r - t422> making ^11 - ^ 22 ” CHAPTER V

CONCLUSIONS

Early In this study we were given the fundamental con¬ cepts for the study of ruled surfaces. So we were prepared to go into the investigations made in chapter IV. Upon entering this chapter we began with Wilczynski's system of linear homogeneous differential equations of the second order, which were

+ z 3 y" ♦ y’Pn + Z'P12 + unJ qi2 °» (i) P 7 P Z q Z 21 ’ 22 ’ «217 22 = °* The ruled surface whose equation was

(2) Rî x s z + uy, was considered an integral ruled surface of this system of differential equations.

We first used the transformation

(3) y = + > z - y'YJ + £ t , (A ft)r 4 0), on the dependent variables to deduce the canonical form from our fundamental set of differential equations. The first canonical form became

4(4--J" * 0Ç«) z U u1:L* » |/uu* <£w12>£\ 41 4(r7» + iÇ") ; U u21* ru22y 4

47 48

Geometrically, the transformation was a change of director curves on an integral ruled surface from C^, Cz to , C^- ,

(Pig. 5) upon which y,z were replaced by y , Ç respectively, leaving the surface unchanged. By solving equations (4) for

y”, Ç” we found that equations (4) could be reduced to a canonical form for which °» , )r i 0,

This canonical form was

(6)

Prom the above conditions that ft- 0, of, 5 j. o, our trans¬ formation took the form

(7) 7 = ^7 , z'-iK .

By solving the equations (5) of chapter IV for y", we found that we could reduce them to a form the same as the form of our fundamental equations (1). The coefficients of equations (l) were replaced by 77^^, X i,k* Th© ©quation was of the form

(8) 49

11 The i k, 'X 4 ^are given in terms of k t>y formulas

(38) chapter IV.

Our next step was to verify some of the facts given to us under the transformation of the parameter. We were given the transformation

(9) t : X ( t) ix ’ f o) in which the independent variable was changed from t to t .

By means of this transformation we set up formulas giving the relationship between the original functions p^ These relations were given by formulas (44) of chapter IV. These formulas were

■t’Pn = <*> + Pu- -c'Pia = Pl2> x'P2l = P21> ^22 -~»22* "• (10)

12— 2 t 2- » q q q q 11 = qil' t qi2 : qi2* X ^21 “ 21* 22 " 22* (<*> r ).

It was also shown that the functions were transformed into the functions lA. ^ ^ (i,k srl,2; k jt i) by the transfor¬

an< mation X - X (t). We gave the relationship between 'U^>k ^

LCi kby the formulas

2 : 2 2 ■c ün - “>' + « . (11) 2- ” u (i,k : 1,2; k ^ i, w r *"/'£')• V “-i,k Ui,k

Under the conditions "U-^2 ~ ^21” ^ = 0; O). we found that our transformations became 1 (12 ) y - c

Finding that cf, <5 were solutions of the equations

2 on 4 (pi;L 4 a?) 4 - 0, (13) f 2 8 * (PI;L + w) 6 : 0, our final set of equations of transformation "became

(14) 7 - ——T» z - x/ T = "C (t), ( T ’ )* (H»)* in which a,b are constants. BIBLIOGRAPHY

Books

ZX7 Lane, Ernest P. Projective Differential Geometry of Curves and Surfaces. Chicago: The University of Chicago Press, 1932. /"2_7 Eisenhart, Luther P. Differential Geometry. Ginn and Company, 1937.

51