An elementary treatise on solid geometry

10Equationof theplane throughthree given points 11 Equation of a plane through the line of intersection of two given Conditionsthat three planesmayhaveacommonline of intersection.. 113 T

GIFT F

Qr

SOLID G-EOMETBY.

WORKS BY CHAKLES SMITH, M.A.

Elementary Algebra. Thirteenth Edition. Revised and

A Treatise on Algebra. Ninth Edition Crown 8vo

An Elementary Treatise on Conic Sections Twen

An Elementary Treatise onSolid Geometry. Eleventh

Edition Crown8vo 9s. Gd

Geometrical Conic Sections Fourth Edition Crown8vo 6s. Key 6s.

Euclids Elements of Geometry. Second Edition.Books I IV, VI and XI Globe 8vo 4s Gd.

Book I, Is.

Books I and II Is. Gd

Books III and IV 2s.

Books I to IV 3s.

Books VI and XL Is. Gd

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MACM1LLAN AND CO., LIMITED, LONDON.

ELEMENTAEY TREATISE

ON

DY

CHARLES SMITH, M.A.

MASTEB OF SIDNEY SUSSEX COLLEGE, OAilBBIDQK

ELEVENTH EDITION

Hontion

MACMILLAN AND CO., LIMITED

NEWYORK: THE MACMILLAN COMPANY

1907

i/w/i, 1 899. ^//MEdition, 1 90 1 M/A,ff^Vw,1 903.

TenthEdition, 1905. EleventhEdition, 1907.

A

^y, :

book onSolid Geometry, and I have endeavoured to presentthe elementary partsof the subject inas simple amanner as

possible Those who desirefuller information are referredtothe more complete treatises of Dr Salmon and Dr Frost, toboth ofwhichI am largely indebted.

represented by the general equation of the second degree at

an earlier stage than is sometimes adopted. I think that

thisarrangement is for many reasons the most satisfactory,and I do not believe that beginners will find it difficult.

The examples have been principally taken from recent

I am indebted to several of my friends, particularly to

Mr S.L Loney,B.A.,and to Mr R H. Piggott, B.A., Scholars

of Sidney Sussex College, fortheir kindness in looking overthe proofsheets, and for valuable suggestions.

SIDNEY SUSSEX COLLEGE,

Distancebetweentwopoints . 4

CHAPTER II.

THE PLANE

Equationof aplanein theformIx+ my +nz=p 9

Equationofa planeintermsof the interceptsmadeonthe axes 10Equationof theplane throughthree given points 11

Equation of a plane through the line of intersection of two given

Conditionsthat three planesmayhaveacommonline of intersection 11

Equationsof a straight line 14Equationsof a straight line contain fourindependentconstants 14

Vlll CONTENTS.

PAOKEquationsof the straight linethrough twogiven points 16

Angle between twostraight lineswhosedirection-cosines are given 1C

Conditionof perpendicularity oftwostraightlines 17

Angle between twoplaneswhoseequationsare given 18

Perpendiculardistance of a given pointfroma given straight line 19

Conditionthattwostraight linesmayintersect 19

Equationsoftwostraight lines in their simplestforms 25

in arangeofconstantcross ratio 26

Distancebetweentwopoints intermsof their oblique co-ordinates 28Angle between twolineswhosedirection-ratios are given 28

Volume of a tetrahedron in terms of three edges which meet in a

Transformationof co-ordinates 29

CHAPTER III.

SURFACES OF THE SECOND DEGREE

Numberof constants in the generalequationof theseconddegree . 37

Polarplane ofanypointwithrespecttoa conicoid 39Polarlineswithrespectto a conicoid 40

A chord of a conicoid is cut harmonically by a point and its polar

Thehyperboloidofonesheet 50

Thehyperboloidoftwosheets

Conditionsfor a surface of revolution 66

CHAPTER IV

CONIOOIDS REFERRED TO THEIRAXES

semi-diametersis ofconstantvolume 76

Equation of conicoid referred to conjugatediametersas axes 78

Locusof intersection of threetangentplaneswhichare at right angles 80

Conditionthat aconemayhavethree perpendicular generators 85Conditionthat aconemayhavethree perpendiculartangentplanes 86Equationoftangent cone fromanypoint to a conicoid 86

Equationofenvelopingcylinder 88

CHAPTER V

PLANE SECTIONS OF CONICOIDS.

Natureof aplanesectionfound byprojection 96

Areaofanyplanesection of a central conicoid 98

Areaofany planesection ofaparaboloid 99

Angle betweentheasymptotesof a plane section of a central conicoid 101

Conditionthat aplanesectionmaybe a rectangularhyperbola . 102Condition that two straight lines given by two equations may be at

CHAPTER VI

GENERATING LINES OF CONICOIDS.

Conditionsthat all points ofagiven straight linemaybeona surface 113

The tangent plane to a conicoid at any point on a generating line

CONTENTS. XI

PAGE

Twosystemsof generating lines 11G

Conditionthat four non-intersecting straight linesmaybegenerators of

thesamesystemof a conicoid H?

Equationsofgeneratinglines through any point ofan hyperboloidof

Equationsof thegenerating lines through anypoint of an hyperbolic

l"-1

Locusof the point of intersection of perpendicular generators 124

CHAPTER VII

SYSTEMS OP CONICOIDS. TANGENTIAL EQUATIONS RECIPROCATION

Fourconespassthroughthe intersections oftwoconicoids 129

Rectangular hyperboloids .131Locusof centres of conicoidsthroughseven given points 132

Xll CONTENTS.

PAGE

given planes are

CHAPTER VIII

CONFOCAL CONICOIDS. CONCYCLIC CONICOIDS. Foci OP CONICOIDS.

Confocalconicoids defined 144

Focalconies [See also 158] 145Threeconicoids of a confocalsystem pass througha point 145

The tangent planes through any line to the two confocals which it

touchesare at right angles 148

Correspondingpointsonconicoids 151

Locusof pole of a givenplane withrespect to asystemof confocals 152

Axesofenveloping coneof a conicoid 153

Equationofenveloping conein its simplestform 153Locusof vertices of right circularenveloping cones 155

CHAPTER IX

QUADRIPLANAR AND TflTRAHEDRAL CO-ORDINATES

Envelopeof asystemof surfaceswhose equationsinvolveonearbitrary

Envelopeof asystemof surfaceswhoseequations involvetwoarbitrary

Functionalanddifferential equations of conical surfaces 184Functionalanddifferentialequationsof cylindrical surfaces 185

Equationof developable surfacewhichpassesthrough twogiven curves 190

Linesof striction 191

Functionalanddifferentialequationsof surfaces of revolution 192

XIV CONTENTS.

CHAPTER XI

CURVES

PAGEEquationsoftangentatanypoint ofacurve 197

Equationof osculatingplaneatanypoint of a curve 201Equationsof the principalnormal 202Radiusof curvature atanypoint of a curve 202

Measureof torsion atanypoint of acurve 203Conditionthat a curvemaybeplane 204Centreandradius of sphericalcurvature 206Kadiusof curvature of theedgeof regressionofthe polar developable 207Curvatureandtorsion of a helix 208

CHAPTER XII

CURVATUBB OF SuKFACES

Curvaturesofnormalsections of a surface "," . 213

Thenormalstoanysurface at consecutive points of a line of curvature

Linesof curvatureona surface of revolution 218Linesof curvatureona developable surface * .218

Linesof curvatureonacone 219

22

CONTENTS XV

PAGELinesof curvature ofaconicoid are its curves of intersectionwithcon-

Curvatureofanynormalsection ofanellipsoid . 228

SOLID GEOMETRY.

CO-ORDINATES

called the co-ordinate planes, and their lines of intersection

measured parallel to the lines of intersection of the other

and let LP, MP, NP, be the co-ordinates of P Theplanes

MPN, NPL, LPM are parallel respectively to YOZ, ZOX,

figure, we have a parallelepiped of which OPisa diagonal;

Hence, to find apointwhose co-ordinates are given, we have

onlyto take OQ, OR, OS equal to the given co-ordinates,

and draw three planes through Q, R, Sparallel respectively

Z

tivelybe a, b, c, then Pis said to be thepoint (a, 6, c).

3 To determine the position of any point P it is notsufficient merelyto knowthe absolute lengths of the lines

LP MP, NP, we must also know the directions in whichthey are drawn. If lines drawn in one direction be con

must beconsideredas negative

We shall consider that the directions OX, 0\, OZ are

The whole of spaceis dividedby the co-ordinate planes

OXYZ, OXY 2, OX YZ, OXYZ, and OX YZ

If Pbe any point in the first compartment,

straightlinejoiningtwogiven points in agivenratio

Let P, Q be the given points, and R the point which

2

Let Pbe fo, ^ ^), $ be (a?2,ya , *

Draw PL, QM, ENparallel to OZ meeting.XOFinL, M,

clearly all in one

4 CO-ORDINATES.

The above results are truewhatever the angles between

theirco-ordinates

(a?a , y2 , s

2 ).

Draw through P and Q planes parallel to the co-ordinate

z

Let the edges PL, LK, KQ be parallel respectively to

QKL, the angle PLQ is aright angle,

from the plane YOZ, so that we have PL =

the above byputting #2=0,

y^=

+ 2

+*2

CO-ORDINATES 5

l+x^+x^, i (yi+y*+y8),

and i(*i+*2+2a).

(1, 2, 3)and(3, 2,-1). Ans. x-22=0

-2).

with lines through P parallel to the axes of co-ordinates

Then,sinceinthe figuretoArt 5theanglesPLQ, PMQ, PNQ

Hence cos2a+cos2/? 4-cosfy=1.

with the positive directions of the co-ordinate axesare called

its direction-cosines, and we shall in future denote thesecosinesby the lettersI, m, n.

From the above we see that any three direction-cosines

are connected by the relation f + m2

+ n2=1. If the

easilyseen that those

of QPwill be I, m, n; and it is immaterial whether we

changed, as direction-cosines.

direction-cosines For we have - = T =

-; hence each is equal to

6 CO-ORDINATES.

where the line is met by a plane through the point perpendicular to the line. Thus,in thefigure to Art 2, Q,R, S

spectively

another straight line is the length intercepted between

In estimating these projections we must consider the

always have pq +qr+rs=

PS on any line is equal to the algebraic sum of the pro

P and ending at Q is equal to the projection ofPQ.

8. If we have any number of parallel straight lines, the

between planes through P and Q perpendicular to theirdirections Theseintercepts are clearly all equal; hence the

length by the cosine of the angle between the lines, we have

the angle betweenthe lines.

9. In the figure to Art 2,let OQ = a,OR =b, OS=c Then it is clear that x = a for all points on the plane

PMQN, and that = b for all on the plane PNRL,

CO-ORDINATES 7

along the line NP we have x =a, and y =b; and at the

So that a planeis determined by one equation, astraightlineby two equations, and apoint by three equations

in which the variables are the co-ordinates of a point,

; two equations represent a

weproceedto prove.

equation of the surface is of the form F (x)=0. Then theequationis equivalentto (x a) (x b) (x c) =0,where

whose co-ordinates satisfythe equation F(x) = are on one

Let one of the variablesbe absent, so that the equation

isof the form F(x,y) =0. Let P beany point in the plane

z= whose co-ordinates satisfy the equation F(x,y) = ;

parallel to theaxis ofz, are the same asthose of P, so far as*

x and y are concerned; it therefore follows that all such

the equation F(x,y)= is traced out by a line which is

always parallel to theaxis of z,and which moves along thecurve inthe plane z=Q defined by the equation

F(x,y] 0.

Next let the equation ofthesurface be F(x,y,z)=0.

the equation F(x,y, z)=0, we put x a, and y b,the roots

the locus is met by aline through (a, b,0)parallel to theaxis

ofz.

Since thenumber of roots is finite, thestraight line will

locus, which is the assemblage ofall such points fordifferent

s CO-OEDINATES.

must be on both the surfaceswhichthoseequationsrepresent

tion ofthe two surfaces Whenthree equationsaregiven,we

havesufficientequationstofindtheco-ordinates,althoughthere

may be more than one set of values, sothat three equations

othermethods besidesthe one described in Art. 1.

Another method is the following: an origin is taken, afixed line OZ through 0, and a fixed plane XOZ The

between the planes XOZ, and POZ are given. These

noted by the symbols r, and 0, and thepoint is calledthe

point(r, 6, $)

IfOX be perpendicular to OZ, and Ybe perpendicular

tothe planeZOX, wecanexpresstherectangular co-ordinates

ofP interms ofitspolar co-ordinates

Draw PN perpendicular to the plane XOY, and NM

perpendicularto OX, and join ON Then

termsofthe rectangular. The values are,

and =tan

CHAPTER II.

THE PLANE.

13 To shew that the surface represented bythe general

equationofthefirstdegree isaplane.

Multiply these in orderby -?

, and -- andadd:

on the locus, any other point inthe line joining themis also

definitionofa plane

Let p be the length of the perpendicular ON from the

on the and let m, nbe the direction-cosinesof

10 THE PLANE.

PL perpendicular on XOY, and LMperpendicular to OX.

Hence ifP be (a?, y, z), we have

(i),the required equation

By comparing the general equation of the first degree

plane given by the general equation of the first degree are

Also the perpendicular from the origin on the j)lane is

15 To find where the plane whoseequation is

meets the axis of x we must put y=z =Q; hence if the

we have Bb + D =0, and Cc4-D =0. Hence the equation

ofthe planeis

This equation can be obtained

Eliminating A, B, 0, D from these four equations, we

x , y , i

-0.

17 If8=0 and 8 = be the equations of twoplanes,

S X S = will be the general equation of a plane through

S = and S =0,those co-ordinates will also satisfy S =X/S".

Hence, since \ is

arbitrary, S \S = is the generalequation of a plane through the intersection of the givenplanes

common lineofintersection

Let theequations ofthe planes be

The equation ofany planethrough the lineof intersection

of(i) and (ii) is ofthe form

+cz+ X (ax +b f c z+ d =Q

cis

12 THE PLANE.

Eliminating X and /A we have the required conditions,namely

0,

19 We can shew, exactly as inConies, Art 26, that if

Ax + By + Gz + D = be the equation ofa plane,and x,y,z

negative for allpoints on theotherside.

20 Tofind the perpendicular distance of a given point

from agivenplane

Let the equationofthe given plane be

Ix+ my + nz=p (i),

andlet x, y,z be theco-ordinates ofthe givenpointP. The

equation

is the equationofa plane parallel to the given plane

It will passthrough the point (x, y, z) if

*

THE PLANE 13

Now ifPL be the perpendicular from P on the plane (i),

(i) and (ii) respectively, then will

=p -p

=lx + my -f nz

p.

Hence the lengthof the perpendicular from anypointon

If the equation of the plane be Ax-\-By-\-Cz-\-D-Q, it

B

which isof the same form as (i); therefore the length of the

perpendicular from (x,y, z) onthe plane is

Ex 1. Findtheequationof theplane through(2, 3,

plane3os-4y+Iz=0 Ana 3oc-4y f7 +13=0.

Ex.2. Find theequation of the plane throughthe originand through

andlx

Ex.5 Shew that the four points (0,-1,-1) (4, 5, 1), (3, 9, 4) and

(

-4, 4, 4,) lieonaplane.

-ly-6z+3= ?

fromtheplane 5x+2y-Iz+9=0,andonoppositesides of it.

Ex.8. Findtheequationsof the planeswhichbisect the anglesbetween

Ax+By+Cz+ D _ Ax+B y+Cz+D

"

14 THE STRAIGHT LINE.

in aconstantratio,is a plane.

fromany numberof fixed planes is constant, is a plane.

Hence any two equations of the first degree represent a

X OY bylines parallel to OZ Thenthe co-ordinatesx andy

ofits

projection inpq

Hence if Ix-fmy I be the equation of pq, theco-ordi

Ix+ my 1.

plane YOZloe ny +pz =1, the co-ordinates of any point on

PQwill satisfythe equationny-\-pz =1. Hencethe equations

lx+ my =I, ny + pz-1.

Itshould be noticed that the equationsof a straight line

The above equations are unsymmetrical and are not so

THE STRAIGHT LINE 15

(ex,ft<y); and let I, m, n be the direction-cosines ofthe line.

Draw through A and Pplanes parallel to the co-ordinate

be edgesof this parallelepiped parallel tothe axes of x, y, z

respectively Then AL is the projection of AP on the axis

-4y=1,By-5z=2.

^ Ex.2. Findin a symmetrical form the equation of the line x-2y-5,

3x+y~7z=Q Am %(x-5)=y=z-If.

Ans.

^.^j.-^.

Ex.4. Writedownthe equation of the straight line through the point

x-2 = y-3 z-4

16 THE STRAIGHT LINE.

of ^4Pand AB on any axis is equal to AP : AB Hence

the equations ofthe line are

Let I, m, n and lt m, n be the direction-cosines of the

Let P,Q be any two points on the first line.

Draw planes through P, Q parallel to the co-ordinate

equal to the sum of the projections ofPL, LM, and MQ on

Hence PQcos = PL.1 + LM m + MQ.n.

But Pl = l.PQ, LM=m.PQ, and MQ = n.PQ;

THE STRAIGHT LINE 17

Ifthe lines are at right angleswe have

II + mm + nn 0.

Hence the anglebetween twolineswhosedirection-cosines

LL + MM + NN =0.

|=*and

^=

Ex 2. Shew that the line 4z=3t/=-Z is perpendicular to the line

Eliminating I, we have 2mn -(m +n)(2m-n)=0, or 2in 2-mn-nz=0.

Hence,if the direction-cosines of thetwolines bel^mvn^andl z ,m2,w2,wehave- 1 -

, , ,, x y z , x u z

betweenthe lines T= =

-, and -.= .=- .

I m n V m n

18 T&E STRAIGHT LINE.

the middle point is on the bisector, the required equations are

.

l+l m+m n+ri.

-;= m-m n-supplementaryangle are .,

{(mri mrij*+(nl rilf+(Imf Im)}.

aregiven.

The angle between two planes is

clearly equal to the

angle between two lines perpendicular to them. Now we

have seen [Art. 14] that the direction-cosines of the normal

tothe plane

betweentheplanes whose equations are

Ex.1. Find the equation of theplane containingthe line

Ans 15x+ y-7z+2=Q Ans. x+lly+14je=0.

fj v

= + +nv

THE STRAIGHT LINE 19

27 To find the perpendicular distance of a given point

Let theequations oftheline be

Let A be the point (a, /3,7),and draw through A and P

the axes

is equalto the sum oftheprojections ofAL, LM, and MP\

20 THE STRAIGHT LINE.

(iii),

Eliminating X, //,, vfrom the equations (ii), (iii) and (iv)

4 4-S4=0, the condition of intersection of the lines is the

which is found atonce by eliminating x, yy 2.

29 To find the shortest distance between two straightlineswhose equations aregiven.

LetA KB and GLD be the given straight lines, and let

KL be a line whichisperpendicular to both. Then KL is

planePABcutCDinL; thenifLKbedrawnparallel toPA it will be the