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MATHEMATICS
LIBRARY
as
%e
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Latest Date stamped below.
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ANALYTICAL GEOMETRY
,
THREE DIMENSIONS.
B
A TREATISE
ON
ANALYTICAL GEOMETRY
THREE DIMENSIONS,
CONTAINING THE
THEORY OF CURVE SURFACKS,
rpAND,OF. ,,
CURVES OF DOUBLE CURVATURE.
BY J. HYMERS, D.D.
FELLOW AND TUTOR OF ST JOHN’S COLLEGE, CAMBRIDGE.
THIRD EDITION, ALTERED AND REVISED.
CAMBRIDGE:
PRINTED AT THE UNIVERSITY PRESS,
FOR
J.& J.J. DEIGHTON, AND MACMILLAN & CO., CAMBRIDGE.
WHITTAKER & CO., LONDON.
M.DCCC.XLVIII.
eats (Hild Ad
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CONTENTS.
a
i
t
‘of
é
d
SECTION LI.
ON THE PLANE AND STRAIGHT LINE.
ART, PAGE
1—16. Equations to a plane and straight line............... 1
-17—41.. Problems on the plane and straight line............... 18
SECTION ILI.
ON SURFACES OF THE SECOND ORDER.
42—-52. Equations to a sphere, and to a cylinder, cone, and
BH TIRCe COTE YUM ence ae rr iste ate dng soa e eaten pin tee cats 32
53—65. Equations and forms of surfaces that have a center.... 39
66—72. Equations and forms of surfaces that have not a
EEC ee eee OCT AG. coe oiet Sth een EN ha Ot ay CRE ol 55 cA 50
SECTION III.
ON THE PROJECTIONS OF LINES AND PLANE SURFACES, AND ON
THE TRANSFORMATION OF CO-ORDINATES.
79—82. Length of the projection of a line, and area of the
Rrorccholm Ole Ay DIANE sSUtt acts eee cats eo eit ce vb ps as SRR
Beene) PL OU CUCL CO-OLUINALCS niwerer: svar copes Scr cle ss a's se 64
88—98. Transformation of co-ordinates. Polar co-ordinates... 68
99—106. Plane sections of surfaces............. S25 snk Some: a2..0070
vi CONTENTS.
SECTION IV.
ON TANGENT PLANES AND NORMALS TO CURVE SURFACES, AND
THEIR VOLUMES AND AREAS.
ART. PAGE
107—111. Equations to the tangent plane and normal......... 86
112—120. Differential coefficients of the volume and area of a
curve surface in rectangular and polar co-ordinates ....... 90
SECTION V.
ON TANGENT LINES AND NORMAL AND OSCULATING PLANES TO
CURVES, AND THEIR LENGTHS.
121—130. Method of assigning the form of a curve in space
by equations. Tangent line, and normal and osculating
planes..... eR eer ea roe ee re a 96
131—132. Differential coefficient of the arc of a curve in rect-
angular or polar co-ordimates...c24 ae. 2 eens oe 104
SECTION VI.
ON THE DISCUSSION OF THE GENERAL EQUATION OF THE SECOND
ORDER.
133—138. Center, and diametral planes of a surface......... 106
139—144. Reduction of the general equation of the second
Oe Ee hac Fe ers Pre cat A RE Nya ony eee eee 112
145—148. Mode of determining the species and form of the
surface represented by a proposed equation...... Ait AcioN 118
149—150. Case of surfaces of revolution.............c.cc00- 2 ie
151—155. Properties of conjugate diametral planes............ 1
156—157. Position and magnitude of paraboloids, when repre-
sented by the general equation of the second order....... 133
CONTENTS. Vil
SECTION VII.
ON CYLINDRICAL, CONICAL, AND CONOIDAL SURFACES, AND ON
SURFACES OF REVOLUTION.
ART, PAGE
158—159. Number of constants in the equations to surfaces.
Mode of representing the intersections of surfaces...... 137
160—172. Finite and differential equations of cylindrical, coni-
cal, and conoidal surfaces ; and surfaces of revolution...... 141
SECTION VIII.
ON SURFACES HAVING MORE THAN ONE ARBITRARY CURVILINEAR
DIRECTRIX, AND ON DEVELOPABLE SURFACES, AND ENVELOPES.
173—177. Twisted surfaces generated by a straight line that
constantly passes through two curves and remains parallel
to a fixed plane, or which constantly passes through three
BIVVCUACUL VES ty giaoate fect ne clei stain tare) Ge ae sib aries oe it'6 6 148
Ps =10l-sDevelopable Surtaces eras ta ess ake te ecko sine ass « 153
TOS eaol OG MORVElOPes at te hears -reres pety a icler ahr ssa «amie fee 167
SECTION IX.
ON THE CURVATURES OF CURVES IN SPACE.
197—201. Intersection of consecutive normal planes of a curve. 172
202—206. Radius of spherical curvature ; radius of absolute
CLE CULES oon ee oa Scone te ee Pe ot en iets eee Ti elchs a bhe alate te eNetete oles 174
907—210. Evolutes. of a curve in*space.....:.....2..se00s.0: 178
SECTION X.
ON THE CURVATURE OF SURFACES.
211—214. Conditions for a contact of any order between two
THEIACESUR. ties she < BPO be DOC OOOO Pare Pees ecatet a a's . 181
Vill CONTENTS.
ART. PAGE
215—218. Radii of curvature of oblique and normal sections... 184
219—226. Normal Sections of greatest and least curvature... 189
227—229. Intersection of consecutive normals of a surface... 197
230—237. Lines of curvature, Principal radii of curvature, and
RMpilict of surlaces 4s: ine acre ksl. daahs Mein 5 2 AG 200
PPPODIEINS cts ciate hk 8a he lin pa ce ete hie ales 3 ae ee 209
Students reading this subject for the first time may confine their
attention to the following Articles.
1—41, 42—52, 57—71, 79—84, 107, 109—115, 121—126, 131.
ERRATA.
PAGE LINE ERROR. CORRECTION.
107% 3 YR LL
21 22 before this assertion supply the converse of
109 15 L,Y, z h, k,l
118 ] vy HS
121 9 +75+38 +255 -9
so that the surface is an ellipsoid.
133 12 2A’ x 2 A’, x
136 5 (> 1) ey kh)
7 a
dr dr
19] last line Peps h tan? 6 i7;:?
205 24 +4uv +4p?g?uv
which causes the two succeeding equations to be incorrect; the
equation at the top of p. 206 should be
2p? g° (1+ Raa +92)—p292
lip)e-(149%=" a, (L+p?)(1 +9°)-p°¢"
p”) ( +9° 1+ p? u f+ (l+p?)? 4u*p?9?=0,
231 li (rrr?) (rr'r’)2
251 1 : a gn
ANALYTICAL GEOMETRY
OF
THREE DIMENSIONS.
SE GDTON I.
ON THE PLANE, AND STRAIGHT LINE.
1. Iw order to determine the position of a point in space,
some fixed point is taken for the origin of co-ordinates, and
through it are drawn three fixed planes, called the co-ordinate
planes, at right angles to one another, and intersecting one
another two and two in straight lines, which are also at right
angles to one another, and are called the awes of the co-
ordinates. Then, if the perpendicular distances of a point
from each of the co-ordinate planes be given, its position will
be completely determined.
Let O (fig. 1) be the origin of the co-ordinates, and y Oz,
Ov, wOy the co-ordinate planes; M any point, and ME,
MF, MN the perpendiculars let fall from it upon the co-
ordinate planes; these perpendiculars are called the rectan-
gular co-ordinates of M, and, as their values change for the
different points of space, are denoted by the variables a, y, z.
The point WM will be determined in position, if we know the
values of its three co-ordinates; that is, if we know that for
that point
Oe 0.8 SC,
For, if along Ov we measure OA =a, and through A draw
an indefinite plane parallel to yOz, this plane will contain all
points whose distance from yOx is a, or for which w=a,
1
2
and, therefore, the point in question. Similarly, if we mea-
sure along Oy, Ox, the distances OB=b, OC =c, and
through B, C, draw indefinite planes respectively parallel
to Ox, wOy, each of these must contain the point in
question; therefore the three planes will, by their intersection
in M, determine one single position for the point whose
co-ordinates are w =a, y=b, x =c; which position coincides,
as we see, with the angular point, opposite to the origin, of
the rectangular parallelopiped constructed upon the edges
OA, OB, OC, equal to the three given co-ordinates.
2. The feet of the perpendiculars L, F, N, are called
the projections of the point M upon the co-ordinate planes.
Of these if any two be given, the third can be found;
thus if E, & be given, draw EB, FA, parallel to Oz;
then BN, AN, drawn respectively parallel to Ox, Oy, will
by their concourse determine JN.
Instead of the perpendiculars ME, MF, MN, the three
lines OA, AN, NM, which are respectively equal to them,
are commonly called the co-ordinates of M, and denoted by
@, Y, 3; also the axes of the co-ordinates Ox, Oy, Oz,
are often called the axis of w, the axis of y, and the axis
of x; and the co-ordinate planes vOy, yOx, xOw, the
planes of wy, yx, xa, respectively.
3. The determination of the point M will not however
be complete, unless we take into account the signs of the
quantities a, 6, c, in the equations
U =10e Yi Ds AAC
in order to measure these distances, when they are positive,
along the positive parts Ow, Oy, Ox, of the co-ordinate axes ;
or along the negative parts Ow’, Oy’, Oz’, of the axes pro-
duced in the contrary direction, when they are negative; as
is explained in Plane Geometry. For since the co-ordinate
planes, which must be supposed to be prolonged indefinitely,
form among themselves eight solid angles, there are eight
positions in which M might be situated at absolute distances
3
a, b, c from the co-ordinate planes; and it is only by attending
to the signs that we shall be enabled to select the true one.
Without going through the combination of signs which belongs
to each of the solid angles, it may be sufficient to observe
that, if a point be situated in the compartment wys, all its
co-ordinates will be positive; if in the compartment a’y’x’, all
negative; and that for points in the compartments vy'z, a’y2’,
we must have, respectively,
t= dy, Y=—b, S=C¢; #=—a, Y=b, sie,
4, Sometimes it is requisite to take the co-ordinate
planes not at right angles, but inclined at given angles, to
one another, in which case the system of co-ordinates is
called oblique.
Thus (fig. 2) if wOy, yOx, sOw be three planes drawn
through the point O, and intersecting one another two and
two at given angles, in the lines Ov, Oy, Ox; and if from
any point M, MN be drawn parallel to Ox meeting the plane
vw Oy in N, and NA be drawn in that plane parallel to Oy ;
OA, AN, NM, are the co-ordinates of the point M referred
to the oblique axes Ox, Oy, Ox; and WM is called the pro-
jection of M made parallel to the axis Ox.
In the present chapter we shall confine our attention to
the case of rectangular co-ordinates; noticing however, as
they arise, several relations of lines and planes to one another,
the analytical expressions of which, and mode of investigation,
are the same, whether the axes of the co-ordinates be rect-
angular or oblique.
5. To find the distance of a point from the origin in
terms of its co-ordinates.
Let IW (fig. 1) be the point,
OA=2, AN=y’, NM =» its given co-ordinates ;
join OM, ON, and let OM = d.
Then from the triangle ONM, right-angled at N,
OM? = ON’? + NM’;
ra
4
but the triangle OAN, right-angled at A, gives
ON* = OA’ + AN’;
“. OM? = OA? + AN’? + NM’;
Cor. Let a, 8, y be the angles which OM makes with
the axes of a, y, x; then, joining AM, since angle OAM is
a right angle,
OA x
cosa = —— = —;
OM da’
.. # =dcosa; similarly, y’ =dcosB, zx’ =d cosy.
n r2 12
Also (cos a)’ + (cos (3)? + (cos vy)’ = a = 1.
This is the condition which the three angles, made by
a line through the origin with the co-ordinate axes, must
satisfy ; they cannot, therefore, be assumed at pleasure, but
two being given, a, §, for instance, the third is determined
by the equation
cosy = + 4/1 — (cosa)* — (cos 3)”,
and will therefore have two values, y and t—-y; the value
a — yy corresponding to a line OM’ in the plane OM, which
is inclined at an angle y to Oz’, and visibly makes angles
equal to a, B with Ow, Oy, the same as OM does.
6. To find the distance between two points in terms of
their co-ordinates.
Let M’ (fig. 3) be a point whose co-ordinates are x’, y’, 2’,
and NM any other point whose co-ordinates are wv, y, #3; join
. MM’ =d, and upon MM’, as diagonal, describe a rectangular
parallelopiped having its edges parallel to the co-ordinate axes.
Then from the triangle M’MK, right-angled at K,
MM” = M'K? + KM? = M’K? + (2 - 2’).
5
But the triangle NW’LN, right-angled at L, gives
M'K? = N'N? = (#- af + (y-y')’;
MM? = (a#-2')? + (y-y) + (@- 2’),
or d=/ (wv — a’)? + (y—y)? + (#—8')
Both in this formula, and in that of Art. 5, we take the
radical with a positive sign; as the question only relates to
the absolute distance of the points.
Cor. Let a, 8, y be the angles which MM’ makes with
the edges of the parallelopiped, and which are manifestly the
same as those made by a line OP, drawn through the origin
parallel to MM’, with the co-ordinate axes. Then from the
triangle MM’H, right-angled at H, we have
MH «v-«
Mis adi i
cosa =
similarly, cos 3 -2? (Sttrigh Z—-z
7. To find the equation to a plane.
We may consider a plane as a surface generated by a
straight line which moves so as always to remain parallel
to itself, and to intersect a straight line given in position.
Let AC (fig. 5) in the plane of xa be the line to which
the generating line PQ is always parallel, and BC in the plane
of yz, intersecting the former in C, the line along which it
moves. Let s = 4v+ec be the equation to the line AC
referred to the axes Ov, Oz, in the plane in which it lies;
similarly, let s = By +c be the equation to BC referred to
the axes Oy, Ox, where in both cases c= OC. From P any
point in the generating line, and from Q where it meets BC,
draw PN, QM, parallel to the axis of z, and join MN. Then
the plane QN is parallel to zw, and therefore MN is parallel
to Ow (Euc. xt, 15, 16); and consequently PQ is inclined
to MN at the same angle that AC is toOwv; and since MN,
NP, are the co-ordinates of P in the plane QN,
PN=A4A.MN + QM.
6
But, Q being a point in the line BC, QM = B.MO+ OC,
. PN=A.MN+B.MO+0C;
or, if v, y, s denote the co-ordinates of P,
s=Av+ By +e,
the equation to the plane.
Cor. The lines in which a plane intersects the co-
ordinate planes are called its traces on those planes; thus
AC, BC, are two of the traces of the plane BCA, the third
being the line in which, if prolonged, it would intersect the
plane of wy. Hence, we perceive the meaning of the constants
in the equation to a plane,
s=Av+Byt+e;
for A, B, are the tangents of the angles at which the traces on
#&,Yy%, are respectively inclined to the axes of w and y pro-
duced in the positive directions; and ¢ is the portion of the
axis of x produced in the positive direction, intercepted
between the plane and the origin.
It is important to observe, that the foregoing method
of finding the equation to a plane, applies equally to the case
where the co-ordinates are oblique, and leads to a result of
the same form; so that whether a plane be referred to rect-
angular or oblique axes, its equation may be represented by
x= Awxv+ By+c; but in the latter case, A will signify the
ratio of the sines of the angles at which the trace on xa is
inclined to the axes of w# and x; and B the ratio of the sines
of the angles at which the trace on yz is inclined to the axes
of yand x. Hence, all results which involve no other as-
sumption than s = 47+ By +e, for the form of the equation
to a plane, will be equally true for oblique and rectangular
co-ordinates.
8. To investigate the equation to a plane under the form
ve 41 x
weet AE as)
a C
7
Let AC (fig. 4) in the plane of za be the straight line to
which the generating straight line is always parallel, BC in
the plane of yz that along which it moves; Ga point in the
generating line DE in any position, ON=a, NH=y, HG =z,
its co-ordinates.
Draw DF, FE parallel to OC, OA, and let OA, OB, OC,
which are called the intercepts of the co-ordinate axes be re-
spectively denoted by a, b,c.
Then since G is a point in DE,
& wv
p+ at erin
DF'F (1)
but ere Se aie ye
Cc b a b
5 ae DF
therefore, multiplying the first term of (1) by ——, the second
c
FE
, and the second member by the equal quantity
term by
a
1 = and transposing, we get
—$- 4 5=]
hig tal ok ;
a relation holding between the co-ordinates of any point in the
generating line in any position, and consequently the equation
to the plane generated, and expressed by the intercepts of the
co-ordinate axes produced in the positive directions to meet the
plane.
Cor. It is easily seen that this result might have been
deduced from Art. 7. For if we take s = dv + By +c to be
the equation to the plane ABC (fig. 4), and in it make z = 0,
y=0, «=a for the point 4, we get 0= Aa+c; and making
s=0, w=0, y=b for the point B, we get 0= Bb+c; there-
fore, substituting for A and B their values, the equation to the
plane becomes, as above,
a
8
9. To investigate the equation to a plane under the
form wcosa+y cos + x cosy =p.
Let OQ =p (fig. 4) be the perpendicular from the origin
upon the plane 4BC, making angles a, (3, y with the axes of
&, Y, ¥, respectively ; join AQ, then, OQA being a right angle,
OA = OQ sec AOQ, or a = pseca;
similarly, 6b = psec3, c =p secryy; consequently, by substi-
tuting in the equation
oy
cD
we get
xcosa+ycos 3 + x cosy =p,
the equation to a plane in terms of the perpendicular let fall
upon it from the origin, and the angles which that perpen-
dicular makes with the co-ordinate axes produced in the
positive directions.
It will be observed that neither in this article, nor the
preceding, are the co-ordinates required to be rectangular ;
only, in the case of rectangular axes we must have (Art. 5)
cos’ a + cos’ 3 + cos’ y = 1;
but in the case of oblique axes the cosines, which fix the
position of the normal to the plane, will be subject to a
different condition.
Cor. If with the above equation we compare the general
form of the equation to a plane
Ax+ By+ Cz=D,
im) b)
p cosa cosp cosy
which shew that the constant term bears the same ratio to the
perpendicular on the plane, that the coefficient of each variable
9
does to the cosine of the angle which its axis makes with the
perpendicular.
Hence the perpendicular on a plane from the origin
constant term
s/ sum of squares of coefficients of variables
likewise the square of its reciprocal equals the sum of the
squares of the reciprocals of the three intercepts of the co-
ordinate axes.
10. The particular cases of the equation to a plane,
when it involves only one, or only two of the variables,
require a separate notice.
Thus the equation #2 =a, since it belongs to all points
whose distance from the plane of yz, measured parallel to the
axis of #, is a, represents an indefinite plane parallel to yz.
Similarly, the equation w = 0, characterizes all points situated
in the plane of yz, or is the equation to that plane; and
y=0, x=0, are respectively the equations to the planes of
SL, LY.
Again, the equation — + 5 =1,
a
represents not only the line 4B (fig. 6) in the plane of ay,
but all points situated in the plane RA drawn through 4B
parallel to the axis of x ; for any one of these points P, what-
ever be the value of PM or x, will have the same wv and y
as its projection M, and therefore its co-ordinates will satisfy
the equation
which does not involve x. Similarly, the equations # = mz+a,
y =nx +, represent planes parallel to the axes of y and 2;
i.e. each equation represents a plane parallel to that axis
whose co-ordinate it does not involve.
This also appears from the equation z = da+ By+c;
for if the plane be parallel to the axis of y, then its trace on
10
yz will be parallel to the axis of y, and therefore B =0, or
x = Awv+cis its equation, the same as the equation to its
trace on yz; similarly, if the plane be parallel to the axis of
x, 4=0. In order to deduce from it the equation to a plane
parallel to the axis of x, the equation requires to be reduced,
as in Cor. Art. 8, to the symmetrical form
now let ¢ be infinite, then the plane becomes parallel to the
axis of x, and its equation 1s — + : =1, the same as that
a
to its trace AB. If the plane pass through the origin,
c= 0, and its equation is 7 = 4v+ By.
11. Every equation of the first order involving three
variables is the equation to a plane.
Let the equation be reduced to the form = Awv+ By + ¢;3
and to find the trace of the surface which it represents on yx
letv =0; .. x = By+ ce, which is the equation to a straight
line ; let this be CB (fig. 5), Next to determine the sections
made by planes parallel to va, let y=b,0’, &c. «1. x= Av+Bb+t+e,
z= Axv+ BY +c, &c. which represent straight lines QP, OP’,
&c. parallel to one another. They likewise all intersect the
trace CB; for making w«=0, we have Bb+c, Bb’ +c, &e.
for the values of QM, Q’M’', &c. and these are the ordinates
of CB corresponding to y=b, y= 6, &c. Therefore the locus
of the proposed equation is a system of parallel straight lines
all passing through a fixed line; i.e. it is a plane,
12. The equation x= Ar+ By +e,
where A, B denote numbers, and ¢ a line, has the defect of
not being symmetrical; to compensate for which, it contains
only three constants, each having an obvious signification so as
to enable us readily to interpret any results to which its use
may lead; and it furnishes the analytical expressions of the
more useful relations of lines and planes to one another, under
—
11
forms the simplest possible ; and it is readily adapted to planes
in all particular positions, with the single exception of a plane
parallel to the axis of x. But it may often be advantageously
replaced by the symmetrical form expressed by the intercepts
of the co-ordinate axes (which however cannot represent a
plane passing through the origin)
or by the equation depending on the magnitude and position
of the perpendicular from the origin upon the plane, which
may be written
lv+my+ns=D,
if we take 1, m, m to denote the cosines of the angles which
the perpendicular p forms with the axes of a, y, #3; and which
may represent a plane in every position. ‘The above three
forms of the equation to a plane, as has been observed, hold
equally for rectangular and oblique co-ordinates.
13. To find the equations to a straight line in space.
A straight line is the intersection of two planes, and con-
sequently is given when the equations to any two planes
which contain it are given.
Among the various planes which may be drawn through
a given straight line, those are employed, for the sake of
simplicity, to determine it, which are parallel to the co-
ordinate axes, since their equations will involve only two of
the variables.
From all the points of the line MM” in space (fig. 6, bis)
let perpendiculars MC, M’'C’, &c., be drawn to the plane of
xy, meeting it in the points C, C’, &c.; the assemblage of
these points is called the projection of MM” on this plane.
This projection will manifestly be a straight line, since all the
perpendiculars are situatad in the same plane, which is parallel
to Ox, and is called the projecting plane of MM”. Similarly,
if we project IM” upon the other co-ordinate planes by per-
pendiculars upon them from its various points, we shall have
12
the three projections 44’, BB’, CC’, of which any two are
sufficient to determine MM’. For suppose Ad’ and BB’ are
given; if through these we draw planes respectively perpen-
dicular to ga, yx, each of them must contain the line MM,
and will therefore fix its position in space by their intersection.
But the positions of the projections NA, QB in the planes
za, yx, will be determined by their equations 7 =mz +a,
y=nx-+b6, in which m and m denote the tangents of the
angles at which they are respectively inclined to the axis of
zg, and a=ON, b=OQ. Hence the system of equations
G=IMS+a, y=nsv4+,
which belong to the projections of the line, or rather to its
projecting planes, will completely determine it.
Cor. 1. Any values of x, y, x which satisfy the above
equations are co-ordinates of some point in the line; for sup-
pose wv’, y’, x’, were such values, and let M’ be the point in
MP for which x = 2’; then for each of its projections 4’, B’,
g =’, and therefore a’ = ON’, y’ = OQ’; that is, if x’ = C'M,
then a and y’, provided they be determined from the above
equations, will equal ON’, N’C’. Hence the equations =mz+a,
y = nx +b, may be called the equations to the line MP itself.
Cor. 2. Corresponding to the value M’C’ of x, the
values of a and y in the simultaneous equations 7 = mz +a,
y=nz+b, are ON’, NC’, the co-ordinates of the point C’
of the projection PC; and therefore the equation to the
third projection may always be deduced from the two others
by eliminating x, which gives
w-a y-b
= » or nv —- my = na—mb.
m
14. In the same manner it might be shewn, that if the
axes of the co-ordinates were oblique, the equations to a
straight line would still be of the form
L=ME+a, Y=nsv4+db;
the only difference would be that the projections would be
made by planes parallel to the axes of the co-ordinates; and
13
m would denote the ratio of the sines of the angles which the
projection on x# makes with the axes of x and w respectively ;
and similarly for .
15. The point in which a line meets any one of the
co-ordinate planes, is called its trace upon the plane. The
positions of these points are determined by putting, in the
equations to the line, a, y, x, successively = 0; thus the co-
ordinates of the traces. of the line whose equations are
v-a y-b
L=ME+a, y=nz4+b, = —_
m n
3
upon the planes of yz, zx, wy, are respectively
a b
a — =
m n rie
=b
yaa 48| re A
16. Asin the case of a plane, it is sometimes attended
with great convenience to have the equations to a straight line
expressed in symmetrical forms.
If a, 2, y be the co-ordinates of a given point in a straight
line, and w, y, x those of another point in the line at a distance
r from the former, and /, m, n, be the cosines of the angles at
which the line is inclined to the axes of vw, y, =, we have
(Cor. Art. 6)
and as these relations subsist amongst the co-ordinates of every
point in the straight line, they are its equations.
This will still be true when the axes are oblique, if J, m,
m are the ratios of the projections of a line on the co-ordinate
axes, to the line itself.
Problems on the plane and straight line.
These principles being laid down, we proceed to the reso-
lution of several Problems relative to the plane and straight
14
line, the results of which are of great use. As the equation
to a plane contains three disposable constants, a plane may be
drawn so as to fulfil various conditions; as, for instance, to
pass through three given points; to pass through a given point
and be parallel to a plane, or to each of two given straight
lines; or to pass through a given point and be perpendicular
to a given straight line. Similarly, a straight line, since its
two equations contain four disposable constants, may be drawn
so as to satisfy various conditions.
17. To express that a plane passes through a given point.
Let zs = da + By +c, be the equation to a plane passing
through a point av’y’s'; then the equation to the plane must
be satisfied by the co-ordinates of the point, .. s’= Aa’+ By'+c¢;
therefore, subtracting this equation from the former to elimi-
nate c, we have
w-2'= A(w-a') + BYy-y),
in which A and B remain undetermined, so that the plane
may still be made to satisfy two other conditions.
Similarly, if a, 8, yy be the angles which the perpendicular
on the plane forms with co-ordinate axes, and the plane pass
through a point w’y’s’, its equation will be
(w — aw’) cosa + (y—y') cos B + (x — 3’) cosy = 0.
18. To find the conditions that two planes, whose
equations are given, may be parallel to one another.
Let x= dv + Byt+c, x= A'x+ By +c’, be the equa-
tions to two planes; then if they be parallel to one another,
their common sections with any plane which cuts them are
parallel; therefore their traces on the co-ordinate planes are
parallel. But, making y = 0, the traces on the plane of sa#
have for their equations s = dv+c, s = Aa +c, which will
be parallel if 4 = 4’; similarly, since the traces on yz are
parallel, B = B’. Hence the conditions of parallelism of two
planes are, that the coefficients of w and y in their equations
are respectively equal.
15
Cor. The equation to a plane drawn through a point
a yx’, parallel to a given plane x = 4’7+ By +e, is
s—-x =A (w-wv)+B(y-y).
19. To find the equation to a plane which shall pass
through three given points.
Let 2, 91215 %oY2%2y V3Y3%3, be the co-ordinates of the
given points, and s = dv + By +c the equation to the plane ;
then the co-ordinates of the points must satisfy it ;
2 =Av,+ By, +c,
%= Av, + By. +,
%3 = Axv;+ By; + ¢,
which are the three equations for finding A, B,c. Multiply
the second and third respectively by indeterminate coefficients
t, w, and add them to the first ;
°, 8 + t%. + U%3 = A (a, + FH, + UH5)...... Clie
provided y, +¢y,+ uy,;=0, 1+t+u=0O.
These two latter equations give
YWi-Ystt(Y2—Ys)=% Yr—Y2t U(Ys — Y) = 0,
and so determine ¢ and zw; and substituting these values in
equation (1), we find A; and then B may be obtained by
interchanging a,,7, and Y,Y2y3; so that
#—#,=A(w-4,)+ Biy-y),
the equation to the plane, will be completely determined.
20. ‘To determine the line of intersection of two planes
whose equations are given.
Let the equations to the planes be
s=Av+By+e, x=Av+By+e.
_ Then the line of intersection is, as we know, sufficiently de-
_ termined by these two equations taken simultaneously; that is,
by supposing w, y, x to receive only such values as satisfy
16
both at the same time. If, for instance, we substitute any
series of values for x, we shall have for each of these values
two equations to find the corresponding values of w and y;
and thus we may determine as many points in the line as we
please. But the line of intersection is more conveniently de-
termined by its projections on the co-ordinate planes. To find
these projections, let a’y's’ denote the co-ordinates of any point
in the intersection, then 2’, x’, are also the co-ordinates of the
corresponding point of the projection on za;
. &=Aa't+ Byte, # = Aa t+ By +c;
therefore, eliminating y’,
(B’ — B)s' = (AB — A'B)a' + Be - Be;
which is the relation between the co-ordinates of any point in
the projection on zw. Hence, suppressing the accents,
(B' - B)z = (AB — AB) & + Be — Be’, and, similarly,
(A’ — A)x = (BA'—- B’A)y + A’c - AC,~z
are the equations to the line of intersection of the planes;
which, as we see, result from separately eliminating y and x
from the equations to the planes. The projection on wy has
evidently for its equation
(A’— A)v + (B’- B)y+c-—c=0.
21. To find the equations to a straight line which shall
pass through two given points.
VE OF OP
Let a’y 2’, v’y”z”, be the co-ordinates of the given points ;
L=MS+a, y=nsx +b, the equations to the line,
which must be satisfied by the co-ordinates of the points ;
* v0 =ms +a, yons + D....., (1),
4} ” ” 4?
C=HamMs+a, y=ns +d,
the four equations to determine the four constants. First, for
the values of m and n, we have
uv’ med a” =m (x' rl 2"), y wr y” = n (2 2") ;
17
also subtracting equations (1) from the assumed equations in
order to eliminate a and J,
v-v =m(x-2), y-y =n(e-%);
therefore, substituting for m and n, the required equations are
av —- a” y y”
, ar ’ t “hal ,
Gig Bilan Ae = *); Sas Unc a ten Pig S ds
Cor. 1. It is easily seen that if we take abe, a’b’c’, for
the given points, and call their distance d, and the distance
of abe from wyz, r, the equations to the straight line may
be written
@-a y-b B8-C¢
Qls
eed ah Oe ah Oo
Cor. 2. If the line is required to pass through only one
point a’y’s’, its equations will be
e-al=m(z-2), y-y=n(s-2),
where m and m remain undetermined, as might have been
foreseen ; for this condition may be satisfied by an unlimited
number of lines. If, besides, it is to be parallel to a known
line of which the equations are
c=mzta, yans4+0,
since the projecting planes (uc. x1. 15) and therefore the pro-
jections of these lines must be parallel, we have m = m', n =n’;
therefore the required equations are
e-xv=m(e-2%) y-ya=an (x —-2).
22. To determine the point of intersection of two given
lines.
Let their equations be
L=Ms+a c=omz+a
eae eapeaety
For any given value of z, the values of w and y furnished by
these two systems of equations will in general be different ;
2
18
but if s be the ordinate of their point of intersection, the cor-
responding values of w and y furnished by them, will be the
same. Therefore the co-ordinates of the point of intersection
are the values of wv, y, %, which simultaneously satisfy the
above four equations ; and as the number of equations exceeds
the number of quantities to be determined, there will be an
equation of condition, without which the problem would be
impossible; since two straight lines do not, in general, intersect.
Hence, eliminating # and y separately,
O=(m—-m)zx+a-a, 0=(n-n')x+b—-8;
and equating these two values of z,
a-a 6b-D'
it,
m—m rm—-nN
3
an equation which must be made identically true by the con-
stants entering into the equations to the two lines, in order
that they may intersect. The two remaining co-ordinates of
the point of intersection, will be found by substituting the
value of
a-a 6-0
3 = = = eit
? Le 9
m—-m wu— 7
in either of the above systems of equations.
Cor. When m=m’, and n=7’, the equation of con-
dition is satisfied, and yet the lines do not intersect, being
parallel; therefore the above condition will not ensure the
intersection of the lines, without the restriction that the value
of zs is not to be infinite. Taken without any restriction,
it expresses that the lines are in the same plane and may ~
therefore intersect; without deciding at what distance that —
intersection will happen.
23. To find the conditions in order that a straight
line and plane whose equations are given, may coincide, or
be parallel to one another.
Let s = da + By +, be the equation to the plane,
V=MS+a, y=nz+),
a
19
the equations to a line coinciding with it. Since every point
of the line is in the plane, if x be the ordinate of any point,
and ... mz +a, nz +, the values of the other co-ordinates,
these values must satisfy the equation to the plane;
. O=(4m+ Bn-1)24+Aat+Bb+e;
and since this equation must be true for all values of z,
“~. Am+Bn-1=0, Aa+Bb+c=0,
which are the required conditions.
If the plane and line are parallel, then a plane and
line drawn through the origin respectively parallel to them,
coincide; therefore the above equations must be satisfied when
we suppose ¢, b, a, to vanish; hence the remaining equation
Am+ Bn-1=0,
expresses the condition in order that a line and plane may be
parallel to one another.
24, If we employ the symmetrical forms of the equations
to the straight line and plane, then in order that the straight
line
may be parallel to the plane da + By + Cz=D, we must have
lA+mB+nC=0;
and if it coincide with it, the further condition is
Aa+Bb+Cc=D.
It will be observed that the results of the preceding
Articles from (17) inclusive, are also true, when the axes are
oblique ; since the only assumptions made, are the forms of
the equations to the plane and straight line; but in that case,
the constants will have different significations.
25. ‘To find the conditions that a straight line and plane
may be at right angles to one another.
2—2
20
Let the equations to the straight line and plane be
S Gnt hg and te 4 wor Bane
y=ns + b,
then the equations to the straight line and plane respectively
parallel to them through the origin are
(1). s=Auv+By (2).
eC=MsZ
y= ns
Now let OA (fig. 8) be the straight line (1), then since it
is perpendicular to the plane (2), it is perpendicular to every
straight line meeting it in that plane, and consequently is
perpendicular to the trace OB of (2) on the plane of #z whose
equations are
z= Ax, y=0 (38).
In straight lines (1) and (3), take two points, 4, B whose
co-ordinates are respectively a’, y’, 3’; #”, 0, 2°; and join
those points; then by equating the two values of the hypo-
thenuse of the right-angled triangle 4OB so formed, we get
AB’ = AO’ + OB’, or
(a — a") + yy? + (eX — 2" Poa ey? + ev? te? ee”;
U0 +22 =0,
or since a’ = mz’, 2" = Av”, m+A=0.
Similarly, for the trace on yz, we get 2+ B=0; hence
the required conditions are
m+A=0 n+B=0.
Cor. 1. Ifit were required to find the conditions for the
line and plane represented by the equations
DONT oy oT hesom CNIS Say
l a ee
5) Auv+ By+Cz=D
being perpendicular to one another, since the line is parallel to
the normal to the plane, those lines must form equal angles —
with the co-ordinate axes; but the cosines of the angles which
the line forms with the axes of x, y, x, are proportional to 5
l,m, 2; and the cosines of the angles which the normal forms
21
with the same axes are proportional to 4, B, C (Arts. 9 and 16) ;
Lia ed
Cor. 2. Hence we can find the equations to a straight
line which shall pass through a given point, and be perpen-
dicular to a given plane.
Let a’, y’, x’, be the co-ordinates of the given point,
x = Aw + By +c the equation to the plane;
then » — a =m(z —), y-y =n(z% —2’) are the forms of
the required equations (Cor. 2. Art. 21); and since the line is
perpendicular to the plane,
m=-— A, n=-—B,
- 2—-v' + A(z—2) =0, y-y + B(z -2%) =0,
are the required equations.
26. Since the trace of the given plane on vw has for its
equation xs = Aw + ¢, and the equation to the projection of the
. e . av a
given line on sw may be written x =— — —, the result
mm
1
A+m=0, or Ax —+12=0, shews that the trace and the
m
projection on sa, and consequently on any plane whatever, are
perpendicular to one another. The geometrical interpretation
of the result of Art. 25 consequently is, that when a straight
line is perpendicular to a plane, the projection of the line and
the trace of the plane on each of two co-ordinate planes, are
perpendicular to one another. ‘The only exception which this
assertion is liable to, is when the plane is parallel to one of
the co-ordinate axes, as the axis of x; for in that case the
two projecting planes become coincident, and a line CQ (fig. 6)
may have two of its projections CS, CR perpendicular to AS,
BR the parallel traces on x@ and yz, without being perpen-
dicular to the plane AR.
b
Suppose that y = —— w + b was the equation to the plane;
: a
then every line perpendicular to it would be in a plane parallel
22
to wy, and therefore have « =c for one of its equations,
a ° : Ale
and y = ,et b’ for the other, since its projection on vy must
be perpendicular to the trace on that plane.
27. If through CO (fig. 7) the axis of z, a plane COQ
be drawn perpendicular to AB, the trace of the plane ABC
on wy, and OP be drawn perpendicular to CQ, it is evident
that OP is perpendicular to plane ACB and z CQO is the
inclination of the plane ABC to the plane of wy, and is equal
to the 2 COP, which OP makes with the axis of x, each being ©
equal to the complement of 2ZPOQ; in other words, the
angle between two planes is equal to the angle between two —
lines respectively perpendicular to them. If the perpendiculars
should not meet, then by the angle between them, and gene-
rally by the angle between any two lines that do not intersect,
we understand the angle formed by two lines drawn through
any the same point respectively parallel to them.
28. Having given the equations to a straight line, to
find the angles which a line parallel to it through the origin,
makes with the axes.
Let v= mz+a, y=nz +b, be the equations to the given
line. Through the origin draw a line OA (fig. 8) parallel to —
it; then the equations to OA are
V=EMZ YHNS,
Let a, 3, ry, denote the angles which OA makes with the ©
axes of x, y, #3 also let OA = 1, and let 2’, y’, ', be the co-
ordinates of the point A, then (Art. 5) a =cosa, y’ = cos P, —
% = cosy; also since the point is in OA,
Ud , , ’
Cz MB 5 Y = 72S.
But OA? = v7? +y? +2", or 1 = (m? +n? +1) 2”,
; 1 1
aa ES i ee COS SS
Ji +m +n y /1 +m +n?
n m
cos 3 = ——————., COs |
/1+m + n° Y/1l+m +n*
23
The radical in these expressions admits of a double sien
+ or —; but as it must be taken + in all, or — in all, we
have only two systems of values for a, 6, yy; viz., the angles
which OA makes with Ow, Oy, Ox; or their supplements,
which AO produced or OA’ makes with the same lines. It
is usual to take the positive sign; then cosy is +, and
therefore -y acute; and the angles determined are those which
the part of the line above the plane of wy makes with the co-
ordinate axes produced in the positive directions.
Cor. Conversely these equations give the values of the
coefficients m and nN,
cos a cos (3
m = | 7 ae e
cos y cos y
Hence the equations to a line passing through a given point
(v’, y’, 2’) and making angles a, 3, y with the axes, are, as
we have already found at Art. 16,
Uy , U
0-@ Y-Y 8-8
cosa cos B cos
29. To find the angles of inclination of a plane whose
equation is given, to the co-ordinate planes.
Let s = Av + By +c be the equation to the plane ;
then v=-— Av, y= — Ba,
are the equations to a line through the origin perpendicular
to it; and if a, 3, y denote the angles which this line makes
with the axes of a, y, z, then (Art. 27) a, 8, y are also the
angles at which the plane is inclined to the co-ordinate planes
of yx, xv, wy respectively ; and by the preceding article we
have
-—A -B
cos @ = cos: BP = ——
/1+4+ A? + B? /1+ 47+ B
24
30. Having given the equations to two straight lines, to
find the angle which two lines parallel to them drawn through
the origin include.
Let
,
y=ans+
a. v=ms+a’
‘ Y ?
lines to which OA, OB, (fig. 8) are respectively parallel ; then
L=ms, y=nsz are the equations to OA, and r=m sz,
y=n's, those to OB. Take OA, OB, each=1; and let
the co-ordinates of A and B be denoted respectively by
L,Y, %3 @,y,2%3 join AB, and lett AB=d, 2 AOB=8.
Then from the triangle AOB, ~
be the equations to the
ns +b
124+1?-2cos0=ad’ =(w — a’)? 4+ (y—-y')? + (x — 2’)
or, expanding, and reducing by the relations |
e+y+e2?a1, w+ y? 4+ 2% = 1,
cosO0 = wa’ +yy' + 22".
But since 4 and B are situated respectively in OA, OB,
L=Ms, y=ne; 2 =m, yY =n's;
1
J 1 +m +n?
. Ll=(m'? +n’? +1)2", or v=
1
Sia larl ve ver es ap pe
ae /1+m? +n?’
, ,
. cos@ = (mm'4 nn’ +1) 23 = nT DE OES :
Si tm +n /1 +m? +n’
This expression, on account of the two radicals it con-
tains, will furnish for 6, four values equal two and two,
corresponding to the four angles formed by the indefinite lines
Ad’, BB’. It is usual to take the denominator positive, —
i. e. to take the radicals with the same sign, so that the
points 4, B, are both above or both below the plane of wy;
and the angle determined is AOB or its opposite 4’OB,
contained between the two portions of the lines which form
25
each an acute angle with Ox or Ox’; this angle, moreover,
will be acute or obtuse according as 1 + mm’ + nm’ is positive
or negative.
Cor. By means of the formula sin = rv gee (cos)? we may
shew that
/(m — m')? + (n— n+ (mn' — m'n)?
sin 9 = fa
Si tm +n? /1 +m? +n?
31. The angle between two lines may also be ex-
pressed in terms of the angles which each makes with the
co-ordinate axes.
Let OA make with the axes of w, y, x the angles
a, 8, y; and OB make with them the angles a’, 8’, y’;
then, proceeding as in the last Art., we may shew that
cos 0 = wa’ + yy + 22".
But vx=cosa, y=cos, % = cosy;
, , ? ‘ , : ,
@ =cosa, y =cosf, % =cosy ;
*. cos@ = cosacosu + cos 3 cos 3’ + cos y cosy’.
32. Hence the condition in order that two lines may be
perpendicular to one another is cos @ = 0, or
mm +nn +1=0;
at which we may arrive immediately by observing that the
equation to a plane through the origin perpendicular to the
first line is 7 + maw +ny=0, which must necessarily contain
a line through the same point parallel to the second whose
equations are w= m'z, y=;
“St mms+nn'z=0, or mm+nn'+1=0.
This condition expressed in another form, is
cos a cos a’ + cos $ cos B’ + cosy cosy’ = 0.
26
Also, in order that the lines may be parallel to one another,
we must have
sinO=0, or (m—m’)? + (n—17')?+ (mn' — m'n)’= 0,
which can only be satisfied by m=m’, n=n’'; this agrees
with Cor. 2, Art. 21.
ee ————
33. To find the angle of inclination of two planes whose ~
equations are given.
Let s=Av+ Byt+e, s= Ax + By +c, be the equa-
tions to the two planes. Then the equations to two lines
respectively perpendicular to them through the origin are
(Art. 25)
e=-Ax, y= —-Bz; w= -— A's, y= —- B82;
and (Art. 27) the angle of inclination, 0, of the two planes
is equal to the angle between these two perpendiculars ;
therefore (Art. 30)
9 1+ 44+ BB
COs = ee ee ee
af/1+ + B/1 +A 4 BE
the double sign belonging to the acute and obtuse angles
between the two indefinite planes.
34. The angle of inclination of two planes may also
be expressed in terms of the angles at which each is inclined
to the co-ordinate planes.
For if a, B,y3; a, By’; denote the angles at which
the planes are inclined to the co-ordinate planes of ys, za,
xy, these are also the angles which the perpendiculars on
them from the origin respectively make with the axes of
#2, y, #3; therefore (Art. 31)
cos 9 = cosa cos a’ + cos 2 cos (3 + cosy cos ¥’,
35. Hence the condition that two planes may be per-
pendicular to one another (since in that case cos @ = 0) may
be expressed in the two following ways:
27
1 a AALS BB’ = 0,
cos a cosa’ + cos 3 cos 3’ + cosy cosy’ = 0.
We may arrive at the former of these results immediately
by observing that v= — dz, y = — Bx, are the equations to
a perpendicular through the origin to the first plane which
_ must coincide with a plane through the origin parallel to the
second plane whose equation is s = 4’x + By;
* AA t BBA Ie 0:
If the second plane be parallel to the axis of x so
that its equation is y=atan@+ 6, the condition becomes
B= Atan ¢.
36. To find the angle between a straight line and a
plane, of which the equations are given.
The angle, 0, here meant, is the angle between the
line and its projection upon the plane, and is therefore equal
to the complement of the angle made by the line with a
perpendicular to the plane.
Let s = dv + By +c be the equation to the plane; then
the equations to a line through the origin perpendicular to
it are
e=—-Azx, y= — Bz.
Let v=ms+a, y=nx+b, be the equations to the
: : uu : :
given line; then since aoa is the angle between it and
~
the perpendicular to the plane,
mute ae 1—-Am— Bn
cos (= _ ) or sing = /1 tne + 8 Viepeee Be °
37. To find the length and the co-ordinates of the
extremity of the perpendicular from a point upon a plane.
Let s=Axv+By+ec be the equation to the plane,
x, y’, x the co-ordinates of the point from which a perpen-
28
dicular is to be dropped upon it; then the equations to the
line passing through the point and perpendicular to the plane |
are (Art. 25, Cor. 2)
e-av =—A(xsx-2), y-y=-B(s-2);
and combining these with the equation to the plane, which
may be written
s-s'=A(ea—-v)+B(y-y) -(' -Aa' - By -0),
in order to deduce the co-ordinates XY, Y, Z of the point
where the perpendicular meets the plane, and putting
x — Aa’ — By’ -—c=P, we get
Z—-2# =-A(Z-3') -B (2-2) -P,
075 — is a
AP ; BP
Xe pete Ee igs == ——________ :
= ieee ee J 1 + A424 Be”
values from which it is easy to deduce X, Y, Z, the co- —
ordinates of the foot of the perpendicular.
If we denote by D the length of the perpendicular inter- —
cepted between the point and the given plane, we have
P*
1+ A? + B’
2’ — Aw’ — By -c
£4/14 4° +B”
the radical being taken with that sign which makes the whole
expression positive.
D?=(Z- x)? +(¥ -y)’+ (X-2)’ =
. D=
Cor. If we use the symmetrical form of the equation to
a plane
Av+ By+Czs=D,
D
/ A? + B+ C?’
and a plane parallel to it through (abc) has for equation
Axvx+ By+ Cz =Aa+ Bb+ Ce,
the perpendicular upon it from the origin =
29
the perpendicular upon which from the origin equals
Aa+ Bb+Cc
Aa+Bb+Cc~D
which is the perpendicular from (abe) upon the proposed
plane. :
the difference of these perpendiculars =
?
38. To find the length and the co-ordinates of the
extremity of the perpendicular from a point upon a line.
Let the equations to the given straight line be
V=ME+a, y=nsth,
and a’, y’, x’ the co-ordinates of the given point from which a
perpendicular is to be dropped upon it; then the equation to
a plane passing through (a’, y’, ’) and perpendicular to the
given straight line is (Art. 25)
#—s'=—m(x-a)-n(y-y);
and combining this with the equations to the line in order to
get the co-ordinates X, Y, Z, of their point of intersection, we
have, putting
x +m(a —a)+n(y -b) =P,
Zax’ —m(mZ+a)+ma—n(nZ +b) + ny’;
x +m (uv —a)+n(y —b)_
» 2
1 +m? +7? H
Reelily sno. i'd.)
1+m’ +n?
m P
aC nr cae oO)
m n
FNC aii py Die fists Ss el Va Ga
—Y =NnN4+ Pe hie rapes reece)
Consequently, if D denote the length of the perpendicular
intercepted between the point and the given straight line,
D = (Z - 2%)? +(X -wv)’?+(V-y’)’,
30
ie CA ae
lt m+n 1+m+n
S24 m (a —a)+n(y'— b)}?
Hy 1+m+n
+? 4 (a -— a)? + (y' - 5)’,
ae re) ae)
39. The perpendicular distance of a point from a straight
line may also be expressed in terms of the angles which the
line makes with the co-ordinate axes.
Let AB (fig. 10) be the given line passing through a
point 4 whose co-ordinates are a, b, c, and making angles
a, 3, y with the axes of a, y, and x; also let AP make with
the same axes, angles a’, 9’, y’, P being the point with co-
ordinates a’, y’, x’, from which the perpendicular PB is to be
drawn. ‘Then
cos PAB = cosa cos a’ + cos 3 cos 3’ + cosy cosy’
_ (#—a)cosa+ (y'—b) cosB+-(z"
W AP
. BP? = AP* — (AP cos PAB)?
= (wv — a)? + (y'— b)’ + (x — e)?
— {(w -— a) cosa + (y' = 6) cos B + (2’— c) cosy}*
Or, if we denote the cosines as usual by /, m, n, the distance
of the point w’ y’ x’ from the line
neyeoay SC Coe
©-a y-b w-e
— —_
—_—_—____
is »/[(#—a)’+ (y’—b)? + (x'—c)?— {1(w'—a) + m(y'—b) +n(x’—c) }?].
If the line passes through the origin, this becomes
J w+ y? +8? — (la’ + my’ + nz’).
40. As two straight lines in space, although not parallel,
may never intersect, it is a Problem which sometimes arises
to determine the shortest distance between them; it may be
solved by means of the following proposition.
The shortest line which can join any two points in two
straight lines in space, is the line which meets both of them
at right angles.
31
Take one of the lines OC (fig. 11) for the axis of #, and
let the other 4B meet the planes of sw, vy, in A and B, in
which planes draw AO, Oy perpendicular to OC, and let
these be the axes of = and y. Draw By parallel to Ow, join
Ay and draw OM perpendicular to it; draw MN parallel to
By, make Ow equal to MN and join Nw. Then MN being
parallel to Ow is perpendicular to MO; therefore OM is
perpendicular to each of the lines WN, Ay, and therefore to
the plane 4yB; also MN being equal and parallel to Oa,
the figure VO is a rectangle, and Nw is equal and parallel to
MO; therefore Nw is perpendicular to Ow, and also to the
plane AyB and therefore to 4B; hence Naw meets each of
the given lines at right angles. It is also their shortest
distance; for take any point K in AB, draw KL parallel to
By and join OL; then the distance of K from O# is equal to
OL, because a perpendicular from K on Ow would be parallel
and equal to OL, zLOw being a right angle, and OL is
manifestly greater than OM.
41, Any plane passing through the shortest distance will
be perpendicular to the plane to which both the given lines
are parallel; for since Na is perpendicular to the plane Ay B,
any plane through Nw is perpendicular to dyB; and AyB,
since it contains one and is parallel to the other, is itself
parallel to every plane to which both the lines are parallel.
Also if through OC we conceive a plane to be drawn parallel
to AyB, it will be parallel to 4B; and MO = Ne will be its
perpendicular distance from the plane 4yB; i. e. the shortest
distance between two lines not in the same plane is the per-
pendicular distance of two planes, each of which is drawn
through one of the lines parallel to the other.
The complete determination of the shortest distance be-
tween two lines, by finding the position and magnitude of the
shortest distance when the equations to the lines are given, is
effected in Probs. 4 and 5. (Appendix, Section I.)
SECTION II.
ON SURFACES OF THE SECOND ORDER.
42. Tue foregoing section comprehends all the more
useful results relative to lines and planes referred to rect-
angular co-ordinates. We shall now proceed to investigate
the equations, and forms, of surfaces of the second order,
which, after the plane, are next in simplicity; reserving for
the subject of the following section (after the reader has
become familiar with the mode of representing surfaces by
equations, and thence deducing their forms) one of the prin-
cipal means of modifying and simplifying those equations,
viz. transformation of co-ordinates. We shall next have oc-
casion to introduce the projections of lines and plane sur-
faces; and the more useful results, relative to the line and
plane referred to oblique co-ordinates, may be then more
conveniently given than at present. It will be necessary
however to make some preliminary observations on the geo-
metrical meaning of equations containing only one or two of
the variables wv, y, x, when we embrace in our enquiries the
three dimensions of space.
43. It was remarked (Art. 10) that the equation
=a, when three dimensions of space are regarded, repre-
sents an indefinite plane parallel to yz; similarly, the equa-
tion y? + py+q=0, which gives for y two constant values
y=a+b, represents two planes parallel to sw; and in
general, every equation containing only one of the variables,
represents a system of planes parallel to the two axes whose
co-ordinates it does not involve.
Again, the equation f(#, y)=0, which, in Plane Geo-
metry, belongs to a curve’ CC" (fig. 12) in the plane of vy,
in Geometry of Three Dimensions, represents a cylindrical
surface formed by drawing lines through all the points of CC’
ae
ae ee
33
parallel to Ox; for any point P in this surface, whatever be
the value of PC or x, will have the same 2 and y as its
projection C; and therefore the co-ordinates of every point
in the cylindrical surface will satisfy the relation f(a, y) = 0,
which does not contain x; and any point whose projection
is not in CC’ cannot by its co-ordinates satisfy the relation
Sf (@, y) = 0. The same observations apply to equations of
the form f(a, x) =0, f(y, x) =0. Hence, an equation con-
taining only two of the variables represents a cylindrical
surface parallel to that axis whose co-ordinate it does not
involve; and of which the trace upon the plane containing
the other two axes, is in Plane Geometry given by the same
equation.
44. The locus of an equation f(a, y, z) = 0, containing
all the variables, is a surface; that is, if all the points be
taken whose co-ordinates satisfy it, they will not comprehend
all the points of a solid figure, but only those situated in a
surface. In the equation f(x, y, s) = 0 we may assume any
values for two of the co-ordinates; and, deducing the value
of the third from it, we have the co-ordinates of the cor-
responding point in the locus; and similarly, we may de-
termine as many points in the locus as we please. Instead
of this, however, let us assume the value of only one co-
ordinate w=a; then (Art. 43) f(y, x, a) =0, represents a
cylindrical surface parallel to the axis of w; but since we
must only take the points in this surface which satisfy the
condition vw =a, it follows that by this assumption we obtain
a curve, namely, the section of the cylinder made by the
plane =a parallel to yx. Let this be PQ (fig. 13),
where ON =a; then of all the points in the indefinite plane
PNQ it is only those in the curve PQ which are deter-
mined by the equation f(y, x, a) =0; similarly, if we take
«= ON'=a, we shall have f(y, x, a’) = 0, which equations
determine the curve P’Q’; and, proceeding in this manner,
we may obtain an infinite number of curves all situated in
planes parallel to yx, and succeeding one another at intervals
as small as we please, the assemblage of which will form a
surface which is the locus of the equation f (7, y, x) = 0.
3
34
45. We conclude, therefore, that every equation, whether
containing one, two, or three of the variables, represents a.
surface; if however the equation can be satisfied by no
system of real values of the co-ordinates, the surface will
be altogether imaginary; or, if it can be satisfied only by
breaking it up into two or three others, the surface is reduced
to a limited number of lines or points. In other cases the
equation will enable us to determine the figure and properties
of the surface which is its locus, as will be seen.
In discovering the figure from the equation, the traces
on the co-ordinate planes, determined by putting #, y, #
separately = 0 in the equation to the surface, and the sections
made by planes parallel to the co-ordinate planes, determined
by putting the co-ordinates each separately equal to a constant,
are the principal means. Thus (fig. 13) 4B, BC, CA are the
three traces of the surface; and PQ is a section made parallel
to yx, the equation to which is f(y, x, a) = 0, where ON = a,
and QN, NP are to be regarded as the axes to which the
curve PQ is referred.
As in Geometry of ‘wo Dimensions a curve may have
several branches, and an ordinate may have several values
corresponding to the same value of the abscissa; so a surface
may have several sheets, and points in different sheets may
have the same projection on the co-ordinate planes. As many
real values as x has for given values of w and y, so many
sheets will the surface have; if the value of x is imaginary,
there 1s no point in the surface corresponding to those values
of wv and y.
46. Surfaces, in the same manner as lines, are divided
into orders, according to the degree of their equations; the
degree being determined by the sum of the indices of the
three variables in that term of the equation (supposed to
contain no fractional or irrational term) where it is greatest.
The plane is the surface of the first order, being the locus
of the equation of the first degree between three variables ;
the sphere, the common cone and cylinder, the ellipsoid, &c.,
are surfaces of the second order, because their equations are
35
of the second degree. We shall at present obtain the equation
to each variety of the surfaces of the second order from some
known properties of them, and determine their figures; re-
serving the discussion of the general equation of the second
order, and the closer consideration of the properties of the
surfaces which it represents, to a more advanced part of the
work. ‘This arrangement, which corresponds to that usually
followed with regard to curves of the second order, will
gradually introduce the student to the more difficult parts of
the subject, and put him at once in possession of the more
important results.
47. 'To find the equation to the surface of a sphere.
The characteristic property of this surface is, that every
point in it is at the same distance from the center.
Let #, y; x be the co-ordinates of any point in the sur-
face, a, 6, c the co-ordinates of its center, and 7 its radius;
then
(vw -— a)? + (y— by +(e - cP? =7”",
is the relation which the co-ordinates of every point must
satisfy, and therefore the equation to the surface.
Cor. This equation will assume different forms accord-
ing to the position of the origin; thus if the origin be in
the center, a=b=c=0, and fie equation becomes
Pty tear;
if the origin be in the surface of the sphere,
a+bP+e=r",
and the equation becomes
v+y +s? =2 (av + by +c).
48. To find the equation to the surface of an oblique
cylinder, the base being circular.
This surface is generated by an indefinite line which is
carried round the perimeter of a given circle, always remain-
ing parallel to a given straight line,
3—2
36
Let r=mz, y=ne, be the equations to a line OR
(fig. 14) through the origin, to which the generating line
is always parallel; and r the radius of the circle OPA
described in the plane of wy with its diameter coinciding
with the axis of x, along which the generating line moves.
Let PQ be the generating line in any position, and wv, y,
the co-ordinates of any point M in it, and therefore in the
cylindrical surface. Then the equations to PQ are
X-wv=m(Z—2), Y-y=n(Z—-s), (Cor. 2, Art. 21);
therefore, making Z = 0, we have the co-ordinates of Pjror
X =ON=2-mz, Y=NP=y- nz.
But, P being a point in the circular base,
Y?=2r X'— X”";
therefore, substituting for Y’ and X” their values,
(y — nx)? =2r(# — mz) -—(# -— mz)’,
a relation among the co-ordinates of any point in the surface,
and therefore its equation.
Cor. We may suppose the curve OP, instead of a circle,
to be any curve of the second order determined by the equa-
tion Y°= 1X’ — (1 — e?) X”; we shall then obtain the equation
to any cylindrical surface of the second order, viz.
(y —nz)?=1(e — mz) —- (1 -— e&) (@—- mz)’.
And it is easily seen that whatever the curve OP be, if its
equation be Y’ = f(.X’), the equation to the cylindrical surface
generated by a straight line carried along it, and of which
it is called the directrix, will be (y — nz) = f(#@ — mz).
49. To find the equation to the surface of an oblique
cone, the base being circular.
This surface is generated by an indefinite line which is
carried round the perimeter of a given circle, always passing
through a fixed point.
37
Let a, b,c be the co-ordinates OR, RQ, QV, (fig. 15) of
the vertex V, and r the radius of the circle OPA described
in the plane of wy with its diameter coinciding with the axis
of xv, along which the generating line moves. Let PV be the
generating line in any position, and a, y, x the co-ordinates of
any point M in it, and therefore in the conical surface ; then
the equations to VP are (Art. 21)
=f;
2.33) We ea rm ay 7A Sc)
z—C€
xv
X-a=
x
therefore making Z = 0, we have the co-ordinates of P, or
z Sf
PON ae ere VON oe ee
wot C bs a
But, P being a point in the circular base,
Y= or XX";
therefore, substituting and reducing,
(bz —cy)’ = 2r (x — c) (ax — cx) — (ax —c2)?,
the equation to the conical surface.
Cor. We may suppose the curve OP, instead of a
circle, to be any curve of the second order, determined by
the equation Y? = 1_X’— (1 — e) X”; we shall then obtain the
equation to any conical surface of the second order, viz.
(bz —cy)’=1(% —c) (az -— ca) — (1 - &) (az - € 2)’.
And it is manifest that whatever the curve OP be, if its
equation be Y’=f(X’), the equation to the conical surface
of which it is the directrix will be
bz—cy aS— Cx
ra alereces
50. To find the equation to the surface of a spheroid.
This surface is generated by the revolution of an ellipse
about one of its axes.
38
Let CA (fig. 16) be a quadrant of an ellipse which, by
revolving about its axis OC coinciding with the axis of x,
generates a spheroid. Let A’C be any position of the gene-
rating curve, and ON =7, NM =y, MP’==3z the co-ordinates
of any point P’ in it; AO=a, CO=c; then from the right-
angled triangle ONM,
OM = a aa y’.
But, since P’ is a point in an ellipse with 3 axes a and ec,
OM? 3
weg e om opt.
o
the equation to the surface; which is called an oblate or
prolate spheroid, according as the axis 2c, about which the
ellipse revolves, is the less or greater of its axes.
Cor. Similarly, if the hyperbola 4Q (fig. 18) revolve
about its conjugate axis OC coinciding with the axis of gz,
the equation to the surface of the hyperboloid generated,
consisting of one continuous sheet, will be
2 2
very
Pe:
and if it revolve about its tranverse axis OA (fig. 19) co-
inciding with the axis of aw, the equation to the surface
generated, consisting of two disunited portions or sheets,
will be
| yt
DB) °
a~ Ce
Ya
51. To find the equation to the surface of a paraboloid.
Let OQ (fig. 20) be a parabola which by revolving about
its axis, coincident with the axis of w, generates a paraboloid ;
OP any position of the generating curve,
ON =2, NV HW, eM Lee,
39
the co-ordinates of any point P in it; then
PN’ =1.0N = la,
1 being the latus rectum of OP; but from the right-angled
triangle PMN,
PN? me y” a 2",
YY te = le,
the equation to the surface.
52. When, as in the preceding instances, a plane curve
revolves about an axis coincident with one of the co-ordinate
axes, the equation to the surface generated may be readily
obtained, whatever be the generating curve. For let OC
(fig. 16) the axis of x be the axis of revolution, CPA the
generating curve, and PR =f(OR) its equation; and let it
revolve about OC into the position CP’4’; ON =a, NM = y,
MP’ = x, the co-ordinates of any point P’.
Then by the equation to the curve, MP’ =f(OM),
or x= f(a" + 9"),
the equation to the surface.
53. The surfaces which we have hitherto considered,
are the simplest cases of surfaces of the second order; their
equations are all contained in the general equation of the
second order between three variables, which is
av +by? +e 42a yr+20 sx4+2cuyt2a v4+2b y+20"s+4+d=0.
Tt will be shewn hereafter that this equation, by giving
a proper position and directions to the origin and axes of
the co-ordinates, can always be reduced to one of the forms
Av? + By + C2? =+ D,
By + C2? =2A'e,
which represent two distinct families of surfaces; the former
40
those which have a center, the latter those which have not
a center; meaning by center, a point such that all chords
of the surface drawn through it are bisected in it.
54. The origin is the center, and the co-ordinate planes
are the principal planes, of the surface represented by the
equation
Aa’ + By? + Cz’ = + D.
For let P (fig. 22) be a point in the surface whose
co-ordinates are v=h,. y=k, x =1; then the equation is
satisfied by these values, and therefore it is also satisfied
when —h, —k, —/ are written for w, y, and x. But if we
produce PO to P’, and make OP’= OP, the co-ordinates of P’
are evidently equal to —h, —k, —1; therefore P’ is a point
in the surface, and PP is a chord, and it is bisected in O;
i.e. every chord is bisected in O, therefore O is the center of
the surface.
Also the surface is situated symmetrically with respect
to the co-ordinate planes. For if in the equation we make
v«=ON=h, and y= NM=k, it will give for = two equal
values with contrary signs PM, QM; so that for every point
of the surface, situated above the plane of wy, there will be
a corresponding point situated at an equal distance below it.
Similarly, the other co-ordinate planes may be shewn to bisect
all their ordinates at right angles. Planes which have this
property are called the principal planes of the surface, and
their intersections (which in this case coincide with the axes
of the co-ordinates, and with respect to each of which also
the surface is symmetrically situated) the aves of the surface.
Also every chord passing through the center is called a diameter
of the surface.
55. In the equation to surfaces that have a center
Aw’ + By’ + Cx2? = +D,
having taken care to make the second member positive, since
the coefficients of the variables cannot be all negative together,
41
we can only have three varieties of form, (1) all the coefficients
positive, (2) one negative, (3) two negative; so that the equa-
tion may assume the three following forms :
2 2 2
Lv yo
nee + F + re ! 9
oO 5
ian Lene
x’ y 2
ates POR a
pa: y? oe?
The surfaces represented by them are called respectively the
ellipsoid, the hyperboloid of one sheet, and the hyperboloid
of two sheets; the two latter surfaces being-also sometimes
called the continuous, and discontinuous hyperboloid.
56. In the family of surfaces, represented by the equation
By’? + Cz? = 242,
the origin is not the center, since the equation does not remain
unchanged when w, y, x are changed into —w, —y, —#; and
it will be seen hereafter that no other point can be its center.
Also since only even powers of y and x enter the equation,
the planes of sw and wy are principal planes, but the plane
of yx not a principal plane; consequently the surface has
only one axis, coinciding with the axis of vw. In the equation,
if Band J’ are not both positive, let the coefficient of y? be
made positive; and then, if necessary, change the sign of &
in order to make the second member positive, for the change
of vw into —@ will merely alter the position of the surface ;
the equation will then offer two varieties according as C is
positive or negative, under the forms
2 2?
Tipe
y” 2
1 ie v 3
and the surfaces represented by them are called respectively
the elliptic and hyperbolic paraboloids.
42
57. To find the equation to the surface of an ellipsoid.
This surface is generated by a variable ellipse which
moves parallel to itself with its axes in two fixed planes,
and vertices in two ellipses in those planes having a common
axis coincident with the intersection of the planes.
Let BC, CA (fig. 17) be quadrants of the given ellipses
traced in the planes yx, sv”; OC =e their common 3 axis
coinciding with the axis of zy, OA =a, OB =b, the other
4 axes; QPR a quadrant of the generating ellipse in any
position, having its plane parallel to wy, its center in OC,
and two of its vertices in the ellipses AC, BC, so that the
ordinates QN, RN are its 4 axes; also let ON=x, NM =a,
MP =y be the co-ordinates of any point P in it; then
x” y
2
in ere ae 1
NQ? NR ()
NQ? zs NR Pte
but AG es ——=1--—.
a’ Ce : Cc
TL)?
Multiply the first term of (1) by
» the second term by
ae and the second member of the equation by the equal
Pei
quantity 1 aKa and transpose and we get
2 2 2
(ite ATs ie
=+3+-
=},
‘Tie 8 SSO pe
the equation to the surface.
58. To determine the form of the ellipsoid from its
equation.
Since in the equation, w can only receive values between
a and —a, y between 6b and — 6, and z between ec and —e,
the surface is limited in all directions. If we put x = 0,
we obtain
43
for the equation to the trace on wy, which is therefore an
ellipse AB; also, from the mode of generation, all sections
parallel to vy are ellipses, and since their axes are in the
ratio of a to b, they are all similar to AR.
If we make x = +h, we have
aphTe Bigs® h?
~=+—sl--
bh? Ce a’
for the equation to any section parallel to yz, which is an
ellipse similar to the trace BC (since its axes are in the
ratio of b to c whatever be the value of h) and which becomes
imaginary when h>a. In the same manner it may be shewn,
that all sections parallel to xsw-are ellipses, similar to the
trace AC. The ellipses, of which 4B, BC, CA are quadrants,
in which the surface is intersected by the co-ordinate planes,
which are its principal planes, are called the principal sections
of the surface; and the parts of the co-ordinate axes inter-
cepted within the surface, are called its axes. Hence a, b, c
represent the J axes of the ellipsoid, and also the 4 axes of
its principal sections. The extremities of the axes such as
A, B, C are called the vertices of the ellipsoid, of these it has
six, one at the extremities of each axis.
The whole surface consists of eight portions precisely
similar and equal to that represented in the figure.
Cor, If a=b, the equation becomes that to a spheroid
generated by revolution about Ox; similarly, if any other two
of the semiaxes become equal, the ellipsoid becomes a spheroid
generated by revolution about the remaining axis.
59. To find the equation to the surface of a hyper-
boloid of one sheet.
This surface is generated by a variable ellipse which
moves parallel to itself, with its axes in two fixed planes,
and vertices on two hyperbolas in those planes having a
common conjugate axis coincident with the intersecticn of
the planes.
44.
Let AQ, BR (fig. 18) be the given hyperbolas traced
in the planes xa#, yx; OC =c their common 4 conjugate
axis coinciding with the axis of xs, OA4=a, OB=b, the
4 transverse axes; QPR the generating ellipse in any posi-
tion, having its plane parallel to wy, its center in OC,
and its vertices in the hyperbolas 4Q, BR, so that thie
ordinates NQ, NR, are its 4 axes. Also let ON = 2,
NM=.2a, MP =y be the co-ordinates of any point P in
the generating ellipse; then
x y”
NO}? NR (1)
NQ? SONAR ed
but =i eas Sie it ye
2
Multiply the first term of (1) by zu , the second term by
a”
2
, and the second member of equation (1) by the equal
b?
2
quantity as and transpose and we get
a y”
a iit pestered
the equation to the surface.
60. To determine the form of the hyperboloid of one
sheet from its equation.
Since the equation will visibly admit values of a, y, z
positive and negative however large, the surface is extended
indefinitely on all sides of the origin, If we put #=0,
we obtain
2
| &
y
4
+21
©
w
g
for the equation to the trace on wy, which is the ellipse 4B;
and from the mode of generation all sections parallel to wy are
similar ellipses, the dimensions of which increase indefinitely,
45
the least being that of which AB is a quadrant, and which
forms the interior limit of the surface. For the sections
parallel to yx, putting # = +h, we have
which, as long as h<a, and therefore the second member
positive, represents a hyperbola similar to the trace BR, with
its vertices in the ellipse AB, and conjugate axis parallel to
Ox, since the coefficient of x’ is negative. When h=a, the
equation represents two straight lines; and when h >a,
making the second member positive, the equation is
y’ h?
ee at
which also represents a hyperbola, but in a new position,
namely, with its vertices in AQ, and conjugate axis parallel
to Oy. In the same manner, the sections parallel to zx” may
be shewn to be hyperbolas similar to the trace 4Q. The
principal sections of this surface are the ellipse 4B, and the
hyperbolas 4Q, BR,
The quantities a, b, which denote the distances from the
origin at which the surface cuts the axes-of w and y, are
called the real semiaxes of the surface; the quantity ¢ is
called the imaginary semiaxis, because if we put 2 = 0, y = 0,
to find where the surface cuts the axis of x, we find
v= —c, or s=+0e\/-1,
so that c is the coefficient of the expression for the imaginary
4 axis of the surface. ‘The extremities of the real axes are
called the vertices of the surface; two of them are 4, B, and
the remaining two are at distances from O respectively equal
to a and 6, in AO and BO produced. The whole surface
consists of eight portions, precisely similar and equal to that
represented in the figure; and since it is continuous, that is,
since we can pass from one point in it to any other point in it,
without quitting the surface, it is called the hyperboloid of
one sheet.
46
Cor. If a= 8, all sections parallel to wy become circles,
and the surface becomes a hyperboloid of revolution about the
conjugate axis.
61. The hyperboloid of one sheet has an interior conical
asymptote.
Putting the equation under the form
3 x? y° x yy a’*b?
char ae ee pas) La 5 ap a
PA hace b- ad 6 ay” + bx’
and expanding the value of ‘a by the Binomial Theorem, we
c
see, since when a and y are very great the quantity
a’ b?
a*y* of b? a?
is very small, that the relation among the co-ordinates of
points very distant from the origin, is nearly expressed by
a” 2
Poteet ds
CG amos
this then is the equation to a surface, whose distance from the
surface of the hyperboloid, measured parallel to the axis of z,
diminishes indefinitely as w and y increase; and, as its
ordinate is always greater than the corresponding ordinate of
the hyperboloid, it lies within the latter surface.
This asymptotic surface is a right cone vertex O, and base
an ellipse A’B’, whose center is in C, and 3 axes equal and
parallel to 40, BO; for if x, y, be co-ordinates of any point
in a generating line of this conical surface, its equations will
be
Ye a7 ae Ze
& be
. the co-ordinates of the point where it meets 4’B’ are,
making Z=c,
47
and as these are co-ordinates of a point in an ellipse whose
4 axes are a and 6,
“ve\? yc” Pa Yad 2
Za 3b CF 5 ae be
62. To find the equation to the surface of a hyperboloid
of two sheets.
This surface is generated by a variable ellipse which
moves parallel to itself, with its axes in two fixed planes,
and vertices in two hyperbolas in those planes having a
common transverse axis, coincident with the intersection of
the planes.
Let AQ, AR (fig. 19) be the given hyperbolas traced in
the planes xv, wy; OA =a, their common 4 transverse axis
coinciding with the axis of 2, OB=b, OC =e, the 4 con-
jugate axes; QPR the generating ellipse in any position,
having its plane parallel to yz, its center in Ow, and its
vertices in AQ, AR, so that the ordinates QN, RN are its
semiaxes. Let ON=a2, NM=y, MP==2 be the co-ordi-
nates of any point P in the ellipse, then
3° y°
Qn? + RNeT ? M)
pay ahs RN* x
but Q —— Out 9 9 aay 2 1 2
a b- a
2
multiply the first term of equation (1) by
—, the second
Cc
2
RN°
term by aprrat and the second member of equation (1) by
o
-
ES ao
the equal quantity — — 1, and transpose and we get
me
the equation to the surface.
48
63. To determine the form of the hyperboloid of two
sheets from its equation.
The equation shews that all values of v, between + a and
—a, are inadmissible, therefore no part of the surface can be
situated between two planes parallel to yx through A and A’;
but the equation can be satisfied by values of wv, y, % in-
definitely great, therefore there is no limit to the distance to
which the surface may extend from O. If we put # =0, we
have
2
Dae ;
therefore the principal section by yz is imaginary; but all
sections parallel to yz, and at a distance from it greater
than a, are similar ellipses as appears from the mode of
generation. For the sections parallel to wy, putting x= +/,
we have
2 I?
—
eb Te?
which represents a hyperbola similar to the principal section
AR with its vertices in AQ and the opposite branch, and
conjugate axis parallel to Oy; and in the same way it may
be shewn, that the sections parallel to xv are hyperbolas
similar to the principal section AQ with vertices in AR and
the opposite branch, and conjugate axes parallel to Ox.
If we make at once y = 0, x = 0, to find where the surface
cuts the axis of w, we have 2 = a? or a = £a3 2a is the real
axis of the surface, and its extremities the vertices of the
surface of which there are but two; the quantities 6b, ¢ are
the imaginary semiaxes of the surface, because it does not
cut either the axis of y or x. ‘The whole surface consists
of two sheets perfectly similar and equal, and indefinitely
extended in opposite directions, but separated by an interval
in which exists no point of the surface ; it is therefore called
the hyperboloid of two sheets; each sheet consists of four
portions precisely similar and equal to that represented in the
figure.
49
Cor. If 6 =c, or the two imaginary axes become equal,
this surface becomes a hyperboloid of revolution about the
transverse axis.
64. The hyperboloid of two sheets has an exterior
conical asymptote.
Putting the equation under the form
x y° sig 3° : (4 =) ( b? Cc )
= 0 as tree Bad ede reer Witeustna y Can
i ON bans Ce ey? + b?2*)”
and expanding the value of — by the Binomial Theorem, we
a
have, for points very distant from the origin, the relation
among the co-ordinates nearly expressed by
this then is the equation to a surface whose distance from the
hyperboloid, measured parallel to the axis of w#, diminishes
indefinitely as y and s increase; and, as its. ordinate w is —
always less than the corresponding ordinate of the hyperbo-
loid, the latter surface lies within it. It may be shewn, as
in Art. 61, that the asymptotic surface is a right cone vertex
O, and base an ellipse B’C’ whose center is A and semi-axes
parallel and equal to OB, OC. In this case, as in the former,
we observe that the equation to the conical asymptote is ob-
tained by omitting the constant term in the equation to the
surface.
65. Since from the equation to the ellipsoid, the equa-
tion to the hyperboloid. of one sheet results by changing ce
into c\’—1; and the equation to the hyperboloid of two
sheets, by changing 0b into b,\/—-1 and ¢ into CAL sevais if
any result in terms of its axes be obtained for the ellipsoid,
the corresponding results for the hyperboloids may be de-
duced by writing
GA for c; or b /—1, ae at, for 6 and c
4
50
66. To find the equation to the surface of the elliptic
paraboloid.
This surface is generated by a parabola which moves with
its plane perpendicular to a fixed plane, and axis in that plane,
and parallel to the axis of another parabola along which its
vertex moves; the concavities of the parabolas being turned
towards the same parts.
Let OR (fig. 20) be a parabola traced in the plane of
wy, vertex at the origin, and axis coinciding with the axis
of «, J its. Jatus rectum; RP the generating parabola in any
position with its plane parallel to zw, vertex in OR, and axis
parallel to Ow, and let /’ denote its latus rectum, and ON = a,
JM=y, MP =z be the co-ordinates of any point P in it;
also draw RM’ parallel to Oy. Then
2
s=l'.RM =U (ON - OM’) =! (« _ =|
9
Neen aye
f 7a
the equation to the surface.
67. To determine the form of the elliptic paraboloid
from its equation.
Since only positive values of w are admissible, no part of
the surface is situated to the left of the plane of yx; also
since the equation can be satisfied by positive values of w, and
by positive and negative values of y and x however large, the
surface is extended indefinitely towards the positive direction
of w If we make y=0, 3° =/w is the equation to the
principal section OQ, and from the mode of generation all
sections parallel to xv are parabolas equal to OQ with vertices
in OR; similarly, all sections parallel to wy are parabolas
equal to the other principal section OR, and having their
vertices in OQ. If we make w =0, we find I'y? + /z*=0;
*. y=0, x=0, or the trace on yg is a point namely the origin.
If we make w =h, we have
y” 3?
TM
51
for the equation to a section parallel to yx, which represents
an ellipse whose semiaxes are the ordinates QN, NT' which
are to one another in a constant ratio \//' to \/1; therefore
all sections perpendicular to the axis of the surface, are similar
ellipses, and hence the name of the surface. It has only one
vertex O and one indefinite axis Oa, and consists of one sheet ;
and is reduced to a paraboloid of revolution when J =U’.
68. To find the equation to the surface of the hyperbolic
paraboloid.
This surface is generated by a parabola which moves with
its plane perpendicular to a fixed plane, and axis in that plane,
-and parallel to the axis of another parabola along which its
vertex moves; the concavities of the parabolas being turned
towards opposite parts.
Let OR (fig. 21) be a parabola in the plane of wy, vertex
at the origin, and axis coinciding with the axis of w; and J its
latus rectum; HP the generating parabola in any position
with its plane parallel to x7, vertex in OR, and axis parallel
to Ox, and let /’ denote its latus rectum, and ON=a, NM=y,
MP = the co-ordinates of any point P in it; draw RM
parallel to Oy, then
o
~
2 =I.MR =! (OM -ON)=0 (F- w)
2
l
the equation to the surface.
69. To determine the form of the hyperbolic paraboloid
from its equation.
The surface cuts the co-ordinate axes only at the origin,
and since the equation will visibly admit values of aw, y, %
positive and negative however large, the surface is extended
indefinitely from the origin. If we make y=0, we have
s* = —I'x for the equation to the trace on xa, which repre-
4 —2
52
sents the parabola OQ, with its concavity turned towards the
negative direction of a, since its equation shews that w must
be taken negatively; and from the mode of generation, all
sections parallel to s# are parabolas equal to OQ, with their
vertices in OR. The other principal section of the surface in
the plane of wy, is the parabola OR, forming the interior limit
of the figure, and all sections parallel to vy are parabolas equal
to OR, with their vertices in OQ. For the trace on ya,
putting wv = 0, we have EVA sabat AWA which represents two
straight lines passing through the origin; and for sections
parallel to yz, making x = h, we have
which represents a hyperbola with its vertices in OR, and
conjugate axis parallel to Oz. If we make h negative, the
equation to the section becomes
» 9
8" Ji
Uh th |
which also represents a hyperbola, but in a new position,
viz. with its vertices in OQ, and conjugate axis parallel to
Oy. The surface consists of one sheet, and has only one
vertex O, and one indefinite axis Ow; and all sections per-
pendicular to its axis are similar hyperbolas, whence its name.
It does not become a surface of revolution when 7 = 7’, nor in
any other case.
b)
79. The hyperbolic paraboloid has plane asymptotes.
Putting the equation under the form
\é ,
U Lx
v=-y—lea =F (2 - =) ;
and expanding the value of x by the Binomial Theorem, we
have the relation, among the co-ordinates of points very
distant from the origin, nearly expressed by
53
is the equation to two planes through the origin perpendicular
to the plane of yx, the distance between which and the para-
boloid continually diminishes. These planes contain the
asymptotes to all the hyperbolic sections parallel to yz, which,
as we have seen, have their centers in the axis of #, and axes
in the ratio of Vs to / le
71. The equation to the hyperbolic differs from that to
the elliptic paraboloid, in having — 7’ instead of J’; this re-
lation will enable us to modify all results obtained for one
surface, so as to be true for the other.
72. The elliptic and hyperbolic paraboloids are par-
ticular cases of the ellipsoid, and hyperboloid of one sheet
respectively ; viz. when the centers of these surfaces are
removed to an infinite distance.
In the equation
write vw —a instead of wx, then the resulting equation
2
(w-a)y ¥? ON ety
ceemeese e ee AT, gree oe fe O
a hous ct ei Ge i Ciden On Gi da
is reckoned from the vertex of the surface. Let p and p’
denote the distances of the foci of the principal sections in
wy and xa, from the vertex,
. P=ad—(a-p)=2ap-p, e=2ap sp’;
hence, by substitution,
a” 2a y -
re Fe 9 , 7 = 0,
a a 2ap —p 2ap) += Pp
a y x
or ——-2@+ = 5 m= OS
MP
AY 1 lara Ay UN ae
a a
therefore, making @ infinite, 7,e. supposing the center of the
surface to remove to an infinite distance from the vertex,
54
whilst the distances of the foci of the principal sections from
their common vertex remain finite, we have
2 2
Ee — ,—~2xH=0,
2p 2p
which coincides with the equations to the paraboloidal surfaces.
Cor. Hence if any result be obtained for the ellipsoid
or hyperboloid, it will be adapted to the paraboloids, by the
modification above indicated; viz. by transferring the origin
to the extremity of an axis, and making that axis infinite.
Also both families may be represented by the equation
Ax’ + By? + Cz? = 242,
the origin being at a vertex, and A = 0 when the surface has
not a center.
feectilinear generating lines of surfaces of the second order.
Surfaces of the second order admit of another division,
viz. into those which can be generated by the motion of a
straight line, and into those which cannot. This property,
which we have seen to belong to the cylinder and cone, we
shall now shew to be possessed by the hyperboloid of one
sheet and the hyperbolic paraboloid; it is easily foreseen that
it cannot belong to the remaining surfaces of the second order,
as their forms manifestly preclude their having a straight line
applied to them throughout its indefinite length; this will also
appear from our results.
73. The Hyperboloid of one sheet can have an infinite
number of straight lines entirely coinciding with its surface.
Since the equation to the surface may be written
2 2
v 2 y
a b
2
Ee)
w
it is evident that either of the following systems (each con-
sisting» of the equations to two planes and therefore represent-
ing a straight line) will satisfy it, viz.
v& R 4
or ~ = Fay (1 +4)
@*- Cc b
w being an arbitrary constant; since then the co-ordinates of
every point in either of the lines (1) or (2) will satisfy the
equation to the surface by causing its two members to become
identical, it follows that for every value of mu there are two
straight lines that lie entirely in the surface.
74. ‘To determine the equations to the projections of the
generating lines of a hyperboloid of one sheet.
Let the equations to the surface and to a line be respectively
x” y 3
erg rhe oh
Ta eb 6 :
vw=ms+h, y=nsr+k;
if the line coincide with the surface in all its points, the
equation
(ms+h)? (nz+hkh)
GOLgey es Bo peer 3
a” b Cc
will be true for all values of x; therefore, equating to
nothing the coefficients of z* and z, we have
m oi 1
ae =O
a” ieee Ca :
mh nk ta Mi ;
riggs sae ee ie eet
ae b- > ef BP
Eliminate & between the two latter equations, and reduce
by means of the former ;
h? mb h\? RENS
2 aa 4 — +e = te or ( = 1%
a~ na a VUaAC
b
Dis ie BEG eat Me mhe
>
56
therefore the two lines whose equations are respectively
nae NaC
C= MZ + — &=>M2Z — —
£ b
(1) += -Hh2)>
mbe mbe
y= NZ —- — Y = ns + —
a a
will coincide with the surface of the hyperboloid in all their
points, m and m being any quantities which satisfy the
equation
m 1
=r 5
Cc”
and if m be made to assume all values, and m be always
6 ; ae
taken = — \/a? = m2c?, we shall determine two infinite sys-
ac
tems of lines represented by the equations (1) and (2),
having the aforesaid property.
Cor. No two lines in the same system intersect; for
if we take two lines in the first system represented by the
equations
NAC fe NIE
L=MSs + — C= MS +
? ,
mbe f m be
y= ns — —— Y=NZ—
a J a
we shail find (Art. 22) that they cannot intersect unless
(m—-m’)?=0, that is, unless they become identical. But
any line in the first system intersects every line in the
second system; for let the equations to two lines, one in
each system, be
nac Welin ae
= MZ +—— L=EMZ—
b b
? , 2
mbe ; mbe
Y= 2S = a Y= 22. -.——
a
57
24 fo ry;
? 2 2
. ‘ : At Li:
then (Art. 22) these lines intersect if — call Ghia
a a
ay 1
which is always the case because each equals —.
c
Hence if we take any three lines in the second system,
and suppose them fixed, and make a line intersect them, it
must be a line of the first system, and by assuming all possible
positions will generate the surface of the hyperboloid; and
since consecutive positions of the generating line do not
intersect one another, the surface is what is called a twisted
surface. The surface evidently admits a second mode of
generation, in which a line belonging in every position to the
second system, moves along three fixed lines of the first
system.
75. The projections of the generating lines of the hyper-
boloid upon the principal planes, are tangents to the traces of
the surface on those planes.
Suppose w = mz + h to be the equation to a line touching
the principal section on sv whose equation is
9 2
ri fa &
then cial SAAN as We
must give two equal values for x, and
Pi ms +h)?
5 ae ME Ok: +1 must be a perfect square ;
1 mM? h? m? h? . b Peps
es Spaete ort FET = 4 , or h*=a° — mc’;
cb a* a a
e w=me+/ae—m ec,
which coincides with the equations to the projections of the
generating lines on zw. Similarly, the projections on the
58
other principal planes may be shewn to be tangents to the
traces on those planes.
Cor. If through the origin we draw a line parallel to
the generating lines, its equations will be v=ms, y=nzx3
and the equation to the surface generated by it, eliminating
m and » by means of the equation which connects those
quantities, will be
which represents the conical asymptote; hence any line drawn
through the center parallel to a generating line of the hyper-
boloid lies in the surface of the conical asymptote.
76. The hyperbolic paraboloid can have an_ infinite
number of straight lines entirely coinciding with its surface.
It is evident that either of the following systems (each
consisting of the equations to two planes, and therefore repre-
senting a straight line,) will satisfy the equation to the
surface; viz.
oo. ¥ % y x
VLR tte an ae
Ore Ae UD Pike keer eae es (2)
DRT ARE 5 Ao A a
Since then the co-ordinates of every point in either of the
lines (1) and (2) will satisfy the equation to the surface by
causing its two members to become identical, it follows that
for every value of the arbitrary constant p, there are two
straight lines that lie entirely in the surface.
77. ‘To determine the equations to the projections of the
generating lines of a hyperbolic paraboloid.
Let the equations to the surface and to a straight line
be respectively
59
e=met+h, y=nsr+k;
then, if the line coincide with the surface in all its points,
the equation
(nz+k)y
eo eae tae
ms+-h=
ti i j
is satisfied for all values of x;
are pa} i Qnk ; toe
=a TFT =U N= = <5
aoe Tieaee ttre
t 2
caper Ty fe py
2n 4,
will coincide with the surface of the paraboloid in all their
. 1
points, m being any quantity, and ” = NES and if m be
made to assume all values, we shall determine two infinite
systems of lines represented by equations (1) and (2), having
the aforesaid property.
Cor. In the same manner as for the hyperboloid, it
may be shewn that two lines in the same system never inter-
sect, and that two lines in different systems always intersect.
Hence if we suppose three lines in either system to become
fixed, the surface may be generated by making a line move
so as always to intersect them. Or, since the generating lines
in the two systems are respectively parallel to the fixed planes
yY=Nn2%, y= —nz, the paraboloid may be also generated in
two ways by a straight line which moves so as always to
Intersect two fixed lines, and to be parallel to a fixed plane.
60
In this case also the projections of the generating lines are
tangents to the principal sections of the surface.
Thus (changing the sign of w in both equations)
—x=mz+h will bea tangent to x* =/'a,
if 2° + Imzx+Uh=0 bea perfect square,
ay!
or 40h =1?m’®, orh= aa
(+7)
. -—-& =m - —
fe
is the equation to a tangent; which, measuring # in the
positive direction, coincides with the equation to the projection
of the generating lines on zw.
78. We shall terminate this Section with demonstrating
the following general and important property of surfaces of
the second order.
If two surfaces of the second order have a plane section
in common, their other curve of intersection, if it exist, will
also be a plane curve.
Let the equations to the two surfaces be
Aa’ + By? +C2?° 4+ 2A y2z4+2B' s24+2Cay+2A"v+2B’y
+20°2+ D=0,
au + by +cx? + 2a'yx 4+ 2b’ sax + Qcuxy 42a" v4 2b'y
+ 2c"24d=0,
and suppose them to have a common section in the plane
of wy; then making x= 0, the curves represented by the
equations
Ax’? + By? +2Cay+2A"v+2B"y+D=0,
ax+by+2ceary 42a a+ 2b y+d=0,
61
are identical; if therefore m be a constant multiplier,
A=ma, B=mb, C=me', A” =ma", B'’=mb", D=md.
But in order to determine the complete intersection of the
surfaces, we must combine their equations. ‘Therefore, mul-
tiplying the latter by m and subtracting it from the former
and having regard to the above relations among the co-
efficients, we find
(C—mc)2°+2(A'— ma) yz +2(B’— mb’) z7+2(C”’—me")z=0;
the equation to a surface which contains all the points com-
mon to the two proposed surfaces. But this may be decom-
posed into
%=0, (C—mc)x +2(A’— ma’)y+2(B'—mb')a +2(C”—mc")=0;
the first belongs to the assumed curve of intersection in the
plane of wy; the second is the equation to a plane, and
cannot therefore, when combined with either surface, give
any thing except a plane curve of the second order for the
other curve of intersection.
SECTION III.
ON THE PROJECTIONS OF LINES AND PLANE SURFACES,
AND ON THE TRANSFORMATION OF CO-ORDINATES.
Tur meaning of the projection of a point, and of a line,
upon any plane has already been explained, (Arts. 2, and 13.)
Moreover if from the extremities of a limited line we drop
perpendiculars upon any indefinite line either in the same
plane with it or not, the part of the latter line intercepted
between the feet of the perpendiculars, is called the projection
of the former line upon the latter.
Also, if the sides of any plane surface be projected upon
a plane, the figure bounded by these projections is called the
projection of the given surface upon that plane. Between
the length of a finite line and the length of its projection
upon any plane or line, and also between the area of any
plane surface and the area of its projection upon a plane, a
remarkable relation exists, which we shall now exhibit.
79. The length of the projection of a limited line upon a
plane, is equal to the length of the line multiplied by the
cosine of the acute angle which it forms with the plane.
Let CD (fig. 25) be the line produced to meet Gd K, the
plane on which it is to be projected, in H. Let DHd be
the projecting plane, intersecting GdK in Hd; and in the
projecting plane draw Cc, Dd, perpendicular to Hd, and CD’
parallel to it; then ed is the projection of CD, andz DHd =i
is the inclination of CD to the plane GdK; also CD’ = CD
xX COS 23
« ed= CD = CD cosi.
63
80. The length of the projection of a limited line upon
any other line, is equal to the length of the line multiplied by
the cosine of the acute angle which the two lines form with
one another.
Let CD (fig. 26) be the line, 4B the line upon which
it is to be projected, Cc, Dd perpendiculars upon AB, then
ed is the projection of CD upon 4B. Let Ne, Md be planes
through C, D, perpendicular to AB, and of course containing
the lines Cc, Dd; draw CD" parallel to AB, meeting the
plane Md in D’, and join DD’; then the triangle CDD’ is
right-angled at D’, and DCD’ =i is equal to the angle
formed by the two lines, .. CD’ = CDcosi, but ced = CD’,
each being the perpendicular distance of parallel planes ;
*, ed = CD cosi.
81. ‘The area of the projection of any plane surface upon
a plane, is equal to the area of the surface multiplied by the
cosine of the acute angle which the planes form with one
another.
Let ACB (fig. 27) be a triangular area traced in a plane
which intersects the plane Gd, upon which the projection is
to be made, in GK. Through the angular points draw planes
AGa, BKb, CHc perpendicular to GA; then these planes
will contain the perpendiculars Aa, Bb, Cc let fall from the
angular points upon the plane of projection, and their inter-
sections with each of the planes 4H B, a Hb, will be perpen-
dicular to GK; also the angle CHe will equal the inclination
of the planes =7. Join the feet of the perpendiculars; then
acb is the projection of ACB; also join Dd, which is parallel
to Ce.
Then the triangles acd, ACD, since they have a common
altitude GH, are to one another as their bases ed, CD; simi-
larly the triangles bed, BCD, having a common altitude HK,
are to one another as cd to CD;
ed
CD?
= triangular area ABC, cos?,
" triangular area abe = triangular area ABC.
64
Next suppose the figure to be projected is a polygon; then it
can be divided into triangles, each of which will be to its
projection as 1 to cosi; and therefore the sum of the triangles
will have to the sum of the projections the same ratio, or
area of projection = area of polygon . cosi ;
and as this is true however much the number of sides of the
polygon be increased, it is also true when the figure to be
projected is bounded by a curve ;
.". area of projection of any plane surface = area of surface. cos i.
82. The square of the area of any plane surface is equal
to the sum of the squares of the areas of its projections on
three co-ordinate rectangular planes.
Let the given area be denoted by 4, and its projections
on the planes of wy, yx, xu by A,, A,, A,, respectively.
Also let a, (3, yy denote the angles which a perpendicular to
the plane of the given area from the origin, makes with the
axes of w, y, 8. ‘Then vy (Art. 27) is the inclination of the
plane of the given area to the plane of wy, and therefore
A,= Acosy; similarly A, = dA cosa, A,= A cos DB;
“. 42 + Ai + At = AP $ (cos yy)? + (cosa)’ + (cos B)?} = A?.
Cor. Hence in fig. 9, since the triangles 4OB, BOC,
COA, are the three projections of the triangular area ABC,
if OA, OB, OC be denoted by a, 8, c,
(AABC)? = ($.4b)° + (f bc) + (hea)? ;
». DABC =}/ (aby + (ao) + OO
Oblique co-ordinates.
83. To express the distance of a point from the origin
in terms of its oblique co-ordinates.
fig 2, AL the angles yOx, Ow, wOy, be denoted by
A; wy v3; and let OA, AN parallel to Oy, and NM parallel
to Ox, be the co-ordinates w, y, x of the point M. From
65
the points 4 and N drop perpendiculars 4m, Nn upon Ox;
then mv is the projection of 4N upon Oz, and = AN cos),
since AW is parallel to Oy (Art. 80); and Om = O4 cosy.
Now from the triangle ONM, since MN is parallel to Ox, we
have
OM? = ON? + NM? +2MN.NO cos s ON.
But ON? =a? + y° +2vy cos v, from the triangle OAN,
and ON cosz ON = On = Om +mn = 4 cosun+Ycosr;
. P= a +y’ +2 + 2rycosy + 2a cose + 2yX COSA.
Cor. Since d is the diagonal of the parallelopiped of
which a, y, s are three conterminous edges, the above formula
gives the diagonal of any parallelopiped in terms of its edges
and the angles which they make with one another. It is also
the equation to the surface of a sphere whose center is at the
origin and radius = d.
84. To find the distance between any two points in
, terms of their oblique co-ordinates.
Let vw, y, x; @, y’, x’, denote the co-ordinates of two
points referred to oblique axes; then as in Art. 6, if through
these points we draw six planes parallel to the co-ordinate
planes, we shall form a parallelopiped with its edges parallel
to the axes, their lengths being a —a, y’—y, and 2 —3;
and the distance of the points is the diagonal of this
parallelopiped ;
“ @=(v — 2) + (y’ —y) + (2 - 2)?
+2(a' —x)(y’— y) cosy +2(a’ —x)(x'—2)cosu+2(y’—y) (2-2) cosa.
Cor. Hence we have the equation to the surface of a sphere
,
whose radius is d, and co-ordinates of its center, a’, y’, 3’.
85. To find the angle between two lines, whose equations
are given referred to oblique axes.
Piet
L=Mz) vx =mMsZ ;
, » be the equations to two lines
y=ns) yYEns
parallel to the proposed ones, through the origin.
4
66
Then, as in Art. 30, if we take a point in each of these
lines, at a distance = 1 from the origin, and call their co-ordi-
nates v, y, 3 wv, y’, 2’, respectively; also their distance d,
and the angle between the lines 0, we have
2—2cos@ = d* = (a — x)? +(y'-y)* +(x’ —2)?+2(a'— x) (y'—y) cosp
+2 (« — x) (2-2) cosn + 2(y’— y) (2-2) cosa,
or, since |
l= a+ y? +2" + 2evy cosy +222 cosp + 2YX COSA;
1
wo? 4 y? + 2 4 20'y' cosy + 24a'2' cosm + 2y'X COSA,
cos 0 = wa! + yy + x2" + (ay + vy’) cosy + (a's + w2’) cosp
+ (yz + 2’y) cosa. (1)
But since w=ms, y=ne, & =m’'s’, y =n'",
1=3°(1 +m’? + n° +2mn cosy +2mcosu+2nCcoSd),
1=2?(1 +m? 4? +2m'n' cosy +2m cosu + 2 cosa);
* cos@ = |
1+mm’+nn'+(m'n+mn’)cosv+(m+m’')cosn+(n+n’)cosr
rie
i
0 é
a/ 1+m?-+n®+2mn cos +2 m Cos 1+2n cosry/1+m”?+n?+2 m’n’cosv+2m’cospr+2 n’Cosr
Cor. Hence the condition that the lines may be at right
angles to one another, is cos @ = 0, or
1+mm' +nn'+ (m'n + mn’) cosv + (m +m’) cos u
+(n +n’) cosr = 0.
86. To find the conditions in order that a straight line
and a plane, referred to oblique axes, may be perpendicular
to one another.
et a line and plane be drawn through the origin, re-
spectively parallel to the proposed ones, and let their equa-
tions be
\, and s= Aw + By.
Y= NB
67
Now this line, since it is perpendicular to the plane, is per-
pendicular to any line situated in the plane, and consequently
to the trace of the plane on wz, whose equations are
ae eel
9=—8, y=O; .*. (Art. 85) since m’=—, n =0,
A
ri ll i ( 7) + ent
—- —— COS MW —— We COS 7’ COS =
Fyne 4 vt+ uy, fs :
or A(1+mcosu+ncosr) + (m+ ncosy + cosp) = 0.
Similarly,
B(1+mcospu + 2 cosr) + (2 + mcosy + CosA) = 0.
Cor. Hence, we can find the angle between two planes
referred to oblique axes; for the above conditions enable us to
find the equations to two lines through the origin respectively
perpendicular to them; and knowing the equations to the
lines, we can compute the angle between them, that is, the
_angle between the planes, by Art. 85.
87. If we employ the symmetrical forms of the equations
to a straight line
we obtain immediately, for the angle 9 between these lines,
from equation (1), Art. 85, denoting by wy the angle between
the axes of x and y, and similarly of the others,
cos @ = Il’ + mm’ + nn' + (lm' + I'm) cos vy + (ln +1'n) cos vz
+ (mn' + m’n) cos yx.
If we now make the former line to coincide successively
with the axes of a, y, x, and call the inclinations of the latter
line to those axes a, (, vy, we get
5—2
68
,
cosa =l1' 4+ m'cosavy + n' cos az,
, , U
cos 3 = m' + I’ cosvy + n' cosy®,
cosy =n + I cos ax + m' cos ys;
-- cos 0 =l cosa + m cos 3 + n cosy,
where a, 3, y are the inclinations of one line to the co-ordinate
axes, and J, m, n the projecting ratios of the other.
Hence we obtain for the condition of two lines being
parallel,
lcosa + mcos 3 + n cosy = 13
and for the condition of their being perpendicular,
Icosa +m cos + ncosry = 0.
Likewise these equations express respectively the conditions
for a line whose equations are
OP PS
Lome ne
being perpendicular, and parallel, to a plane whose equation is
vw cosa + ycosP + % cosy = p.
Transformation of Co-ordinates.
The discussion of the nature and properties of surfaces
is much facilitated, when their equations are reduced to the
most simple form they are capable of, without being deprived
of any of their generality. This simplification is effected
by giving a suitable position to the origin, and suitable direc-
tions to the axes of the co-ordinates. We shall therefore next
proceed to investigate the chief formule for transformation of
co-ordinates.
88. To change the origin of the co-ordinates without
altering the directions of the axes.
Let a, y, x, be the co-ordinates of the point M (fig. 3)
referred to the origin O, and axes Ow, Oy, Oz; also let M'
69
be the new origin, OA’=h, A’N’=k, N’M’=1, its co-ordinates,
and a’, y’, s, the co-ordinates of M referred to M’ as origin
and to axes parallel to the original ones; then _
e= OA= MH + OA =a' +h,
similarly, y=Y th, sax 41;
and substituting these values of w, y, x in the equation to the
surface, we shall obtain the equation referred to the new origin
and to axes parallel to the original ones.
89. To pass from one system of co-ordinates to another
having the same origin, supposing the first rectangular, and
the second oblique.
Let Ow, Oy, Ox (fig. 28) be the rectangular, and O2’, Oy’,
Oz’ the oblique axes, having a common origin O. From any
point P draw PM, PM’, respectively parallel to Ox, Oz’,
meeting the planes of wy, a'y’ in M and M’, and draw MN,
M'N’ parallel to Oy, Oy’.
Then the co-ordinates of P in the two systems are
ON = aN Moen yy ALB i= 2,
ON’ =a, NM =y', MP=-2'
and our object is to express each of the first set in terms
of the latter. From N’, M’, drop perpendiculars V’n, M’m
upon Ow, and join PN which is also perpendicular to Ow;
then On, nm, mWN are the projections of ON’, N’M’, M’P
upon the axis of #; therefore (Art. 80)
,
On =x cosv# a, nm=y'cosya, mN =2' cosz'a, ,
denoting, by wx, the ZO contained by the axes of «” and
#2 produced in the positive directions, and similarly of the
others.
. @=ON = Ont+nm+mN
= a cosa’« + y' cosy’a# + 2’ cosx’a@3 (1)
that is, each primitive co-ordinate is equal to the sum of the
projections of the three new co-ordinates upon its avis.
70
Hence y= a'cosa’y + y cosy’'y +2 cosx’y,
/ , , , , re
=a cosvs +y CosyX + COS8R;
or, if we denote by (m, m, 7) the cosines of the angles
which the axes of a’, y’, x’, make with the axis of w; and
by (m', n’, 17’), (m", n”, vr”) similar quantities relative to the
axes of y and z, we have
v=me+ny + re’,
y
sama +n'y + 7's".
Pee Lad 2 ‘Seth
ML+NY +7TR,
Of the nine angles involved in these formule, six only are
independent, there being three equations of condition; for
since wv, wy, a's, are the angles which a straight line, viz.
the axis of w’, makes with the three rectangular axes of
Uy Y, &;
.*. (cos aa)? + (cos ay)” + (cos az)? = 1;
and similarly with respect to the angles which the axes of
y and x’ make with the primitive axes; therefore
m +m? +m”? =
n+n'? +n’
1,
BE AREDE
ye 2? 4 7’? = 1,
Cor. In the figure, the axes of a’, y’, x’ are supposed
to make acute angles with the axis of w; if this were not
the case, and if for instance one of them Oy’, made an
obtuse angle with Ow, m would fall to the left of », and
we should have
e®= On-nm+mN,
with which the general expression (1) still agrees, because
the term y'cosy'# is negative in the case supposed, but.
numerically equal to the projection of y’. If again, the
axes being as in the figure, y’ were negative, the point m
would fall to the left of , and we should have
e=On-nm+mNn.
Hence, we collect that the formula (1) is applicable to alk
wil
cases, provided we pay attention to the signs of the co-
ordinates, and of the cosines; the angles, as was before
observed, being those formed by the axes produced in the
positive directions.
90. To pass from one system of rectangular co-ordi-
nates to another also rectangular.
Here it would be sufficient to join to the expressions,
and equations of condition of the preceding Art., three new
equations of condition expressing that the axes Oa’, Oy’, O3'
are at right angles to one another.
_- We may, however, arrive at the expressions for x, y, %,
briefly as follows, Let d be the distance of P from the origin ;
then referring the lines OP, Ow to the rectangular axes Oa’,
Oy’, Ox’, we have for the angle between them
, ,
cos Px = cos Px’ cos x2’ + cos Py’ cos vy’ + cos PX cos #2;
hence, multiplying by d, and observing that dcos Pw = x,
dcos Pav’ = «’, &c., we have
, ? , , , ,
V=H COSA’ + Y COSYXR4% Cosza,
and similarly for y and x; hence denoting the cosines as
before,
L=me +ny + 72,
y=ma + n'y + 1'2',70.(3)-
s= malt n'y +72".
In this case there are only three independent angles. For
since the axes of a and y’ are at right angles,
cos wa cos y’#@ + cos vy cos y’y + cos ax cos y/’% = 0;
similarly, expressing that the axes of w’, x’, and the axes of
y', %, are at right angles, and substituting for the cosines their
values, we have three new equations of condition to be joined
to those of the preceding Article, viz.
mn +m'n' + mn” = 0,
mr +m’ + my" = 0,>)...(4).
Ld a A
nr +-n7r +n’r”’ =0.
72
91. Sometimes, in, the case of two rectangular systems,
it is required to find a’, y’, x in terms of a, y, x. This
may be effected by regarding a’, y’, 2’ as the primitive co-
ordinates, and recollecting that each is equal to the sum of
the projections of the co-ordinates 2, y, upon its axis;
“. @ = @ cosaxu’ + y cosya’ + 2% cosza’;
and similarly for y’ and x’; hence
v=maxe+my +m’,
ne + n'y + n's,>...(5).
v= rat ryt v's.
I
These expressions might also have been obtained from (3) by
adding them together after having multiplied them respectively
istly by m, m’, m”’; 2ndly by n, n’, n”; 3rdly by 7, 7’, 7;
reducing in each case by means of the equations of condition
(2) and (4).
Also the equations of condition, when «x, y, x are re-
garded as the new co-ordinates, and consequently their axes
referred to the axes of a’, y’, x, will be
m+n + 7. =1,
m”? +n”? + 7? =1,>...(6),
m+. 9? 4 9? = 1,
mm +-nn +r
0,
mm’ +nn" + rr” = 0,}...(7),
m'm" + n'n" + rr" = 0.
which are entirely equivalent to the relations between the same
constants obtained before, and may in every case replace them.
Indeed there is no difficulty in shewing that they are a neces-
sary consequence of (2) and (4). For since P is at the same
distance from the origin in either system,
(3) , Pe id
eP+y t+ Pax ty? t+ 2°75
and putting for «’, y’, x’ their values given in (5), and
equating coefficients on both sides, we obtain the equations of
condition (6) and (7).
73
92. In the preceding Articles, the means of passing
from one rectangular system of co-ordinates to another, have
been given in simple and symmetrical formule; they have
however the inconvenience of involving nine constants, six
of which must be eliminated by means of six equations of
condition in order to make the determination of the new
axes relative to the primitive ones depend upon three quan-
tities. This led Euler to invent a mode of expressing each
of the nine constants in terms of three others, which are the
inclination of the planes of xy and x’y', and the angles
which their line of intersection forms with the axes of x and
,
x ; as described in the following proposition.
93. To pass from the rectangular system of co-ordinates
&, y, % to another rectangular system a’, y’, x’; having given
the angle @ at which the planes ay, «’y’, are inclined to one
another, and the angles @, Wy, which their line of intersection
makes with the axes of w and ’ respectively.
This is effected by passing successively through three
rectangular systems, each having one axis in common with
the preceding, and employing the formule relative to the
transformation of co-ordinates in one plane.
Thus to pass from the system of axes Ox, Oy, Oz,
(fig. 29) to the system Ow,, Oy,, Oz, of which Ov, is the
trace of the plane vw’ Oy’ upon wy, and Oy, is perpendicular to
Ow, in the plane of wy, we must put
@ = @,cos@ — y; sin a (1)
y= a, singd+ y, cosh
without altering x.
Next to pass from the system Ow,, Oy,, Ox, to the
system Ow,, Oy,, Ox’, of which Oy, lies in the plane a’y’,
and Oz’ is perpendicular to Oy, in the plane y,Oz, we must
put
Y,=y, cos 9 — 2’ sin :t @),
% = y, sin@ + 2’ cos@
without altering 2, .
44
Lastly, to pass from the system Ov,, Oy,, Ox’ to the
system Ow’, Oy’, Ox’, of which Oa’, Oy’, are at right angles
to one another in the plane x, Oy,, we must put
@, = # cosy — y' sin)
’ - , apyd te}
Y2=@ sn + y cosy
without altering x’. But we may arrive at the result of
these three successive substitutions by a single substitution,
the formule for which will be formed by eliminating 2, y,,
and y, between the systems of equations (1), (2) and (3); we
shall then obtain for a, y, x in terms of a’, y’, 2’ and the
three angles, the following expressions :
w= sind sin @ + 2’ (cos@ cos yy — sin @ sin yy cos 8)
— y (cos d sin + sin d cos yp cos 6).
y= — x cos sind + 2 (cosy sind + sin yy cos d cos 8)
— y (sin ¢ sin yy — cos @ cos xf cos 8).
=x cos0 +a siny sin@ + y cosy sin 0.
94. If we dispense with the third transformation, we
shall have Ow,, one of the new axes, in the plane of xy.
The system Ow,, Oy,, Ox’, in which the new plane of wy
is inclined at an Z @ to the primitive one, and their inter-
section is the new axis of # inclined at an Z®@ to the pri-
mitive axis of w, is for most purposes sufficiently general.
The formule for it, combining the equations (1) and (2),
and calling the new co-ordinates w’, y’, x’, are !
w= x cos — (y'cos@ — x’ sin @) sing,
y = sind + (y’ cos 0 — x’ sin 8) cos ,
z=y sin@ +2’ cos@;
which evidently agree with the formule of Art. 93, when we
put WW =0.
95. In the preceding transformations we have supposed
the origin to remain unaltered ; if, however, the origin is to be
45
changed, as well as the directions of the axes, we must employ
the formule
e=v' th, yay th, s=2" 4+),
where h, k, ¢ are the co-ordinates of the new origin parallel
to the primitive axes, and a”, y”, =” denote the values of
xv, y, = found in each of the preceding cases.
96. It is important to observe that, in employing any
of the preceding formule to refer an equation F(a, y, x) =0
to new axes, the transformed equation F'(a’, y’, x’) =0 will
always be of the same degree as the primitive equation ;
understanding by the degree of an equation the sum of the
indices of the three variables in that term where it is greatest.
For let this term be Aa? y! z'; then it becomes by substitution,
A (ma’ + ny + 73")? (ma! + n'y! + 9's')1 (ma + ny! + 7" 2’);
now these factors cannot furnish terms whose dimensions
exceed p, gq, t, respectively; therefore if A’a'?’y’Vs'" be
the term of highest dimension, p’ + q + ¢’ cannot exceed
p+q+t; ie. the dimension of the transformed equation
cannot be greater than that of the primitive. Neither can
it be less; for if it could, then, as transformation of co-ordi-
nates can never raise the degree of an equation, we could not
return from F(a’, y’, s')=0 to F(a, y, %)=0, which is
absurd. 5
97. To pass from one system of oblique co-ordinates to
another also oblique.
Let OM=2, MQ=y, QP=2x, (fig. 30) be the co-ordi-
nates of a point P parallel to the axes Ow, Oy, Ox; and
OM'=.2', M'Q’=y, QP=-2' the co-ordinates of the same
point parallel to the axes Ow’, Oy, Oz’. Through the
origin draw a normal ON” to the plane of wy, and on
the same side of it as the positive direction of the axis
of x; and from P, Q', M’ drop perpendiculars Pp, Q’n,
M'm, upon ON";
"6
then PQcos N”Ox = Op = Om+mn+np
= OM’ cos N" 2’ + M’Q’ cos N"y' + Q'P cos N"’2’,
or xcosN” 2 = 2’ cos N”’ a2’ + y’ cos. N"y' + x cos N’’2’.
Similarly, if we draw normals ON, ON’ to the planes of
yz, xx, and on the same side of them as the positive
directions of the axes of w and y, we shall have
«cos Na = a cos Na' + y' cos Ny’ + 2’ cos N2’,
y cos N’y = x cos N’2’ + y' cos Ny’ +’ cos N's
Since cos VW" x = sin (x, wy), &c. these formule may be
expressed without the aid of the auxiliary normals; but they
are more convenient in their present shape, as the employment
of negative angles is thereby avoided.
98. The position of a point P’ in space may be fixed
by the following three variables; (1) the radius vector OP’=r
(fig. 16); (2) the 2 COP’ = @ which the radius vector makes
with the axis of x; (3) the 2 AOA’ = q which the meridian
plane COP’ makes with the fixed plane COA. Of these
angles the former varies from 0 to 180°, and the latter from 0
to 360°, in order that the radius vector may pass through
all points of space. If #, y, x be the co-ordinates of P’, they
can be expressed by means of r, 0, x2 for P’M = r cos 0,
OM =r sin 0;
. w= OMcosh =r sinOcosd, y=rsinOsing, and x =rcos8;
and if these values be substituted for #, y, x in the rectangular
equation to any surface, we shall obtain the Polar equation to
the surface.
Plane sections of surfaces.
99. In the discussion of surfaces, it is useful to know
the nature and magnitude of the curves in which they are
intersected by any planes. To do this, it is not sufficient
to combine the equation to the surface F’ (a, y, ) = 0, with
the equation to the cutting plane z = dv + By +c, so as to
77
eliminate one of the variables, x for instance; for the result
f(a, y,) = 0 would represent only the projection of the curve
required, which is not generally of the same nature with the
section in space, nor sufficient to determine it. But if we
transform the co-ordinates so that the cutting plane may be
that of w’y’, and then put x’ = 0 in the resulting equation, we
shall determine the trace of the surface on a’y’, i.e. the curve
in which it is intersected by the proposed plane.
And it may be here observed that, since the degree of an
equation is never altered by the transformation of co-ordinates,
if the equation to a surface be of the m' degree, the curve in
which it is intersected by a plane cannot be of a higher order
than the m™; but it may be of a lower if, by putting x’ = 0,
we cause all the terms of the highest dimension to disappear
from its equation. If we employ Euler’s formule for trans-
formation of co-ordinates (Art. 93), since x’ is to vanish in
the final result, we are at liberty to make z’ = 0 in the values
of x, y, x before we substitute them, and so we may obtain
the proper substitutions; these however, without being de-
duced from the general case, may be readily obtained by the
following independent method.
100. To determine the nature of the section of a surface
made by any plane passing through the origin.
The most convenient data for fixing the position of the
cutting plane are (1) the 20 at which it is inclined to the
plane of wy, and (2) the 2@ which its trace on that plane
makes with the axis of #; these are readily obtained if we
suppose the equation to the plane given; for-if the equa-
1
tion be s = Aw + By, then cos @ = tA, nad a Be (Art. 29),
A Tht :
and tan @ = — BR? Since Ax+ By =0 is the equation to the
trace.
Let «’ Oy’ (fig. 31) be the given plane, cutting the surface
in the curve 4M, and the plane of vy in the line Ow’, which
take for the axis of a’; and let Oy’, a line perpendicular to
78
Ow’ in the given plane, be the axis of y’, and OR=2’,
RM=y’' the co-ordinates of any point M in the section re-
ferred to the axes Oa’, Oy’; also let OQ=a, QP=y, PM =
be the co-ordinates of the same point referred to the axes Oa,
Oy, Ox. Join RP, then 2 MRP =O the inclination of the
cutting plane to the plane of wy, and £wOza' = @ the angle
formed by its trace on wy with the axis of &.
Then PR =y' cos0, PM =~y'sin@;
OQ = OR cos g + RPsingd, QP= OR sin ® -hPcos@;
.e=y sind, v= cosp+y'cos6@ sin d,
y =w sind — y' cos 0 cos d;
and if these values be substituted in the equation F(a, y, s) =0
to the surface, we shall obtain a relation between a’ and y’
which is the equation to the curve 4M.
101. If the cutting plane pass through one of the co-
ordinate axes, the formulee are simplified, and in many cases
are sufficiently general.
Let «’ Oy (fig. 32) be the cutting plane, passing through
the axis of y; Oa’ in the plane of za the axis of «’,
PM=2', OM =y, the co-ordinates of any point P in the
section, ON=a, NQ=y, QP==s the co-ordinates of the
same point. Join MQ, then 2PMQ= 90, and MQ =~2' cos@,
PQ =. sing;
. v= cos0, y=y, x= a' sind;
which shew that to determine the section made by a plane
through the axis of y and inclined at an 2@ to wy, we have
only to write w cos @ and 2’ sin@ for w and x in the equation
to the surface, without altering y; and the resulting relation
between w’ and y is the equation to the curve. Similarly, if
the cutting plane pass through the axes of s or w.
102. If the cutting plane does not pass through the
origin of the primitive co-ordinates, or if we wish to take a
19
point in the cutting plane different from O for the origin of
the new co-ordinates, we shall only have to add to the second
members of the preceding formule the co-ordinates h, k, 1 of
the new origin reckoned parallel to the primitive axes.
103. To determine the nature of the curve formed by the
intersection of any plane with a surface of the second order.
Let Aa’ + By? + Cx* = 2A'w be the equation, which may
represent all surfaces of the second order; therefore, sub-
stituting for w, y, x the values found in Art. 100,
x= cosp+ y cos @sin @,
y = v' sin pd — y cos 0 cos ,
s=y' sin 0,
the result developed and arranged will be
w (A cos’ p + Bsin’ p) + 2a'y' (A — B) cos Asin dh cos
+ y” (A sin® d + B cos’ P) cos’ 6 + Csin’ 6}
= 24'u' cosh + 24’y' cos Osin d,
the equation to a curve of the second order, which will be
an ellipse, parabola, or hyperbola, according as the quantity
(A — B)’ (cos @ sin d cos p)? — (A cos’ @ + B sin? )
x {(A sin’ + B cos’) cos’ 6 + C sin? 9},
or — AB (cos 0)? — AC (cos ¢ sin 6)” — BC (sin @ sin 6)”
is negative, nothing, or positive.
Hence every section of an ellipsoid is an ellipse, because
all the quantities 4, B, C are positive. For a hyperboloid
of one or two sheets, in which cases one or two of the
quantities 4, B, C are negative, the section may be an ellipse,
parabola, or hyperbola. For paraboloids 4 =0; therefore
for the elliptic, in which case B and C have the same sign,
the section is an ellipse; except when 6=0, or @=0, in
which cases it is a parabola. For the hyperbolic paraboloid,
80
since Band C are of contrary signs, the section is a hyper-
bola; except as before when 6 =0, or @=0, when it is a
parabola.
Cor. Since the section is referred to rectangular axes, it
can never be a circle unless the coefficient of #’y’ vanishes, or
(A — B) cos @ sin f cos h = 0;
Tw Tv ;
hence we must have ees or, pes or ¢=0; which
shew that for a circular section, the cutting plane must be
perpendicular to one of the principal planes of the surface.
In the next article we shall see that this property is confined
to one only of the principal planes of each surface.
104. Every surface of the second order, except the
hyperbolic paraboloid, may be generated in two ways by the
motion of a variable circle parallel to itself, the center of the
circle moving along a diameter of the surface.
Let the surface have a center, and let its equation be
Au’ + By’ + C2’ = D.
Since every circular section must be perpendicular to a
principal plane, let the cutting plane be perpendicular to va,
and inclined to wy at an Z @; and as the center of the circle
must be in the plane of za, let x=h, x= be its co-
ordinates ; then the equation to the section reckoned from that
point as origin will be
A(w cos0 +h)? + By’ + C(a’ sin 8 + 1)° = D,
(Art. 101) which represents a circle if
A cos’@ + Csin’@= B, or A + C tan’@ = B(1 + tan’),
B-A
i GER?
or tan? @
and the relation between h and /, since the origin is the
center of the circle and therefore the coefficient of wv’ = 0, is
Acos@h4+ CsinOl=0;
81
therefore the locus of the centers of the circular sections is
a straight line and a diameter of the surface.
; : D Ah?+Cr
Also the radius of the section = Lot post BE Dh ae
B B
We must now examine, for each of the surfaces, which
axis it is that coincides with the axis of y to which the cutting
plane is parallel.
1 1
1. For the ellipsoid, A=, B==, C=5,
tan @=+-—
an @ 7 v=
-. b lies between a and c, or the axis of the surface to which
the cutting plane is parallel is its mean axis.
2. For the hyperboloid of one sheet, since we cannot
; 1 ]
have B negative, we must put A=-, Bowe Gia aaa
i fe pes
a t Q = ck a CVE 3
an ayy 73
-.b>a, or the cutting plane is parallel to the greater of
the real axes.
3. For the hyperboloid of two sheets, since we cannot
have A and C negative, we must put
1 ]
A=, Bora, Coss
c G4 b*
anes NAS 9?
a 67 — Cc
..b>c, or the cutting plane is parallel to the greater of
the imaginary axes.
Since tan@ has two values, the cutting plane may be
6
82
inclined at an z@ or 180°—@ to the plane of wy, and hence
the surface may be generated by a variable circle in two
different ways; but it will be observed that in every case,
if the surface become one of revolution, the two positions
coincide in one which is parallel to the two equal axes.
105. Next, let the surface not have a center, and let
its equation be
By + Gz? =2A'x.
Then as before, if h, 7, denote the co-ordinates of the center
of the circular section, its equation reckoned from that point
as origin will be
By? + C (a’sin 0 + 1)? = 24'(a' cos + h),
with the conditions
B=Csin?@, Csin@.1= A’ cos@;
‘sing = + JZ. and 1 =~ cot 8.
Hence the locus of the centers of the circular sections is a
straight line parallel to the axis of the surface, 7. e. it is a
diameter of the surface; also B and C have the same sign and
B<C; hence the surface is the elliptic paraboloid, and the
cutting plane is perpendicular to that principal section of the
surface whose latus rectum is the least.
In the case of the hyperbolic paraboloid, since B and C
have different signs, no plane can be drawn so as to intersect
it in a circle.
106. Any two circular sections of a surface of the second
order, provided they be not parallel to one another, are
situated on the same sphere, —
Since all circular sections are perpendicular to the same
principal plane, let (fig. 53) represent that principal plane,
83
and let HA, E’B’ be the projections of any two circular
sections not parallel to one another, and also their diameters.
Bisect LA at right angles by JC, and let C be taken equidis-
tant from E and L’; therefore C is the center of a sphere
intersecting the proposed surface in the circle HA and passing
through £’, and therefore (Art. 78) intersecting the surface in
a plane curve, that is, in a second circle passing through LE’ ;
this circle must be either L’B’, or EA’ parallel to EA; but
it cannot be LA’, because two parallel circles on the same
sphere have their centers in a diameter perpendicular to their
planes, and here JOJ' being conjugate to the chords LA, E’A’
cannot cut them at right angles unless the surface be one
of revolution, in which case 4’E’ and B'E’ become coincident ;
therefore the sphere will contain EB’.
SECTION IV.
ON TANGENT PLANES AND NORMALS TO CURVE SURFACES,
AND THEIR VOLUMES AND AREAS.
107. To find the equation to the plane which touches
a given curve surface at a proposed point.
Let s=f(a#, y) be the equation to the surface, and
x, y, « the co-ordinates of the proposed point P (fig. 35) ;
then the equation to a plane passing through P will be
2s —-s=A(e —-a)+ By’ -y),
x’, y’, denoting the co-ordinates of any point in it.
Through P draw a plane parallel to z#, cutting the
surface in the curve PC, and the plane in the line PT';
then these must touch one another at P, in order that the
plane may be the tangent plane to the surface at that
point. Now, for every point in the line PT’, y’ =y, and
therefore its equation deduced from the equation to the
plane is
x —s=A(a# — 2);
also the equation to the curve PC is deduced from x = f (a, y)
by making y constant in it; and in order that PT may
d :;
touch PC, we must have A=—, y being supposed con-
x
stant in the differentiation. Again, through P draw a plane
parallel to yz, cutting the surface in the curve PD and the
plane in the line PR; then, as before, the equation to PR is
s —-x= By -y);
85
and the equation to PD is obtained from x = f (a, y) by
regarding # as constant ; and in order that PR may touch PD,
dz
we must have Ere being supposed constant in the
Ls
differentiation. Hence, substituting for A and B these values,
the equation to the tangent plane at a point a, y, 2, is
,
sey : ee ral ;
eae SS (VV — — :
i pares! y)
or, as it is usually written,
s—s=p(w™-—a)+q(y-y);
where p and q denote the partial differential coefficients of x
derived from the equation to the surface, and the co-ordinates
may be either rectangular or oblique.
Cor. 1. The plane whose position is thus determined by
the conditions that sections of it and of the surface, made by
planes parallel to two of the co-ordinate planes, touch one
another, contains, as we shall shew in the next Art., the
tangent lines of all curves that can be drawn on the surface
through the point in question. If the given equation to the
surface, instead of having the above explicit form, should be
u = F (x, y, 2) =0, then to determine p and q we have
de au . dus du
iieeoe Nady ads,
the differential coefficients being formed as if w, y, x were
independent ; hence, substituting for p and q their values,
the equation to the tangent plane under its most general
form is
du du du
, ae , ait o _ a O.
(v — a) one (y -y) a + (¢ — 2) FP
Corn, +2... TE ry denote the angle of inclination of the tan-
gent plane to that of wy, then (Art. 29)
1
cos y = ay aT ne 3?
1 ss ee
i | a
86
and similarly, its inclinations to the other co-ordinate planes
may be determined. Also since the perpendicular on any plane
from the origin equals the constant term divided by the
square root of the sum of the squares of the coefficients of the
variables in its equation (Art. 9), the expression for the length
of the perpendicular on the tangent plane to a surface wu = 0
from the origin becomes
da Yay dz
NACE (S-) . ay
da dy ¥ dz
108. To investigate the equation to the tangent plane
at any point of a surface, considered as the locus of the
tangent lines to the surface at that point.
If a straight line be drawn through any point of a surface
and a contiguous point of the same, the limiting position
of this line as the latter point moves up to and ultimately
coincides with the former, is called a tangent line to the
surface at that point. The locus of the tangent lines at any
point of a curved surface is in general a plane, which is then
called a tangent plane at that point of the surface.
Let w=f (a, y, x) = 0, be the equation to the surface,
X-@v@ Y-y Z4-38
n
the equations to a straight line drawn through the point vyz
of the surface. ‘hen if this line meet the surface again in
the point a’y's’,
e=atir, y =y+mr, x =2z4+n7;
nes du , du du Q
Ae = wv & nea ® Tr +— Mr +-— NK + Ce
wp dex dy dx
du du
d
0 =/— + m—
oe da dy t dx
since f(x, y, #) the first term of the expansion of f(a’, y’, <’),
vanishes.
+ terms in 7, (2)
eS
87
ks e / , e e
Now let the point 2’y’s’ approach indefinitely close to
«yx whereby 7 is indefinitely diminished and the straight
line in question becomes a tangent line, then
du du du
Consequently from equations (1) we get
du du du
TAO) eee me) Viens — fe
be at) aig tole i veppity si Trai gs
a relation true for every tangent line to the surface at wysz.
And this is the equation to a plane; consequently it is the
tangent plane at the point wysx.
Cor. 1. In what precedes, we have supposed that
d d
Bo mies are finite, as they will in general be. If such a
dx’ dy’ dz
point be selected for the co-ordinates of which they all vanish,
the above equation gives no result; and the higher terms of the
expansion of f(a’, y’, 2’) must be employed. Taking the
terms of the second order in equation (2), dividing by 7? and
then making r = 0, we get
. at Be du Py Pu ne ad? u ay Pu
dx dy’ ds” dedy dads
+-2mn Pee = 0,
UL du
r i 2
on (4. — y)"= ph Borer
Oa sy (HREM hs ONGicice (Zi a)
9
du
+ rat aia
4
the equation to the locus of the tangent lines, which is a
tangent cone,
88
Points in a surface of this description are comprised
amongst what are called singular points ; the vertex of a cone,
or of a surface of revolution whose generating curve does not
cut the axis at right angles, are obvious examples; and we
observe that at such points the partial differential coefficients
al OS sean the Fortna( Chae 1 Art 107):
de dy 0
Cox. 2. Hence it follows that if a plane pass through a
generating line of a cylinder or cone, and through a tangent to
the base, and consequently through a tangent to every section
parallel to the base, it will be the tangent plane to the surface
at every point in the generating line; for it will contain the
tangents to all the curves traced on the surface through the
various points of the generating line.
109. To find the equations to the normal to a curve
surface at a proposed point.
The normal is a straight line drawn through the point of
contact perpendicular to the tangent plane; and will therefore
have equations of the form
we —-v=m(sx—s), y —y=n(z' -2),
xv, y, x being the co-ordinates of the point of contact; and
since it is perpendicular to the plane whose equation is
x —x=p(s'-a)+q(y'-y),
“. (Art. 25) m+p=0, n+q=0.
Hence, substituting for m and 7 their values, the equations to
the normal at a point wy, are
a’ —-a+p(s'—2)=0, y'-yt+q(s'-2) =0.
Cor. The angles a, 8, y formed by the normal with
the axes of wv, y, x produced in the positive directions, will
be given (Art. 28) by the formule
— Pp -4
cosa = : 9 COs SS
Jit p+ Vitp+¢e
1
cos’ =
89
If the equation to the surface be given under the form
u = F (a, y, ) =0, substituting for p and q their values
(Cor. 1, Art. 107), we shall have
1 du B 1 du : 1 du
=—.—, cospP=—.—, co eas | ae
won he da R dy Y x?
du\* du\* du?
where R= () + ) a (=) 3
and the equations to the normal will assume the symmetrical
forms
dev dy dz
110. To find the length of the portion of the normal
intercepted between the surface and the plane of wy; and
the co-ordinates of the point where it meets that plane.
The co-ordinates of any point in the normal being 2’, y’, 2’,
and wv, y, x those of the point where it meets the surface,
the distance of these points is
V (w= 2) + (y= y) + (8) = (a) V1 peg.
But at the point where the normal meets the plane of zy,
% =0; .. the required length = — a/1 + p+ q’.
Also by making s’=0 in the equations to the normal,
we find
W=U+ ps, Y=yt qs,
the co-ordinates of the point where it meets the plane of wy;
and similarly the points where the normal meets the other
co-ordinate planes may be determined.
111. The equations to the normal may also be found
from the consideration that it is the longest or shortest
line which can be drawn from any point in itself to the
surface.
Let 2’, y’, x be co-ordinates of a fixed point in the
90
normal, and a, y, x of any point in the surface; then if
© = distance of these points,
O° = (a - a)’ + (y'—y)? + (2-8),
a function of two variables w and y, for x is given in terms
of w and y by the equation to the surface. Therefore, in
order that 6? may be a maximum or minimum,
ass, d
a — © + —(s'— 2) =0, yy +7 (2) =0,
These are the equations which determine the position of the
longest or shortest line that can be drawn from the fixed point
w’y's to the surface, and consequently those to the normal.
112. To find the differential coefficient of the volume
of a solid bounded by the co-ordinate planes and_ planes
parallel to them, and by a given curve surface.
Let xs=f(#, y) be the equation to the surface, and
%, y, #% the co-ordinates of any point M (fig. 38) in it.
Through M draw planes parallel to the co-ordinate planes ;
then the volume V of the figure NMEF is a function of
wv and y; and if w and y receive increments h and k, the
point M will be brought to C, and the increment of the
solid contained by the planes ME, SD, SB, FM, will be
dV dV eh ONS ’V GV Fk
Be tay & eda © Gnay higye ys ees
But if we had made @ alone vary, the increment would
have been
dV a .
MBRF =—h-+ e, SLES, Fie
dav da’ 11.2
similarly, if we had made y alone vary, the increment would
have been |
TR hak Pai tl tH
MDEQ = —k + —, —
dy dy 1,2
+ &e.;
therefore, subtracting, we have
2
| 7 hal
l.M =
vol. MCRQ adie
hk + &e.,
91
vol. UCRQ av
hk ~ dady
hence, taking the limit of both sides by making A and k&
vanish, in which case vol. MCRQ, becoming ultimately a
prism whose base is QR and altitude MP, = xhk, and all
the terms in the second member after the first, since they
involve different combinations of positive powers of h and k,
disappear, we have
or
+ &¢.;
av
dady — es
113. Again, to find the differential coefficient of the area
of a curve surface, if S = area of the surface NM, by proceed-
ing as above we may shew that
2
surface MC = liu
dudy
surf. MC @S
a Ce
But surf. MC is ultimately coincident with the portion of the
tangent plane at M intercepted by the prism erected on the
base QR, and therefore = hk sec y, if sy denote the inclination
of the tangent plane at M to the plane of wy;
hk + &c.,
and therefore limit of
~ Ss Ws dz\? (dz\? i kop lie &
ee inp a + (=) a as ° ( rt. O7, or. 2).
114. In applying these results to find volumes and areas
of curve surfaces by double integration, considerable attention
is necessary in taking the integral between the proper limits.
This will be best seen by an example. Suppose it were
required to find the volume contained by two cylindrical
surfaces ferected on given bases ON, AN’ (fig. 37) in the
plane of wy whose equations are w= @(y), v= Wy (y)t, and
the curve surface BPP’ whose equation is z=f(#, y). In-
tegrating first with respect to #, considering y as constant, the
expressl ey
ession
P dudy
=2x=f (a, y), between the limits
w=GN=oly), ©=GN'=\-(y),
92
we obtain a quantity, a function of y only, which when
multiplied by dy is the ultimate value of the slice PP’M
contained between two planes whose distances from za are
y and y+ oy respectively; and if this be now integrated
between the limits y= 6, y=b’, we shall have the volume
contained by two planes parallel to zw at distances therefrom
respectively equal to 6 and 6’.
115. Similarly, for the area of the surface, by integrating
the expression
&S WA i: ae (=) - F (2, y)
dady fk dx dy
with respect to x, considering y as constant, between the
limits v= GN, «= GN’, we obtain a quantity, a function
of y only, which when multiplied by dy is the ultimate
value of the portion of the surface PQ’ contained between
two planes parallel to xa at a distance dy from one another ;
and this integrated between the limits y= b, y =0’, gives us
the area of the surface intercepted by two planes parallel
to 2x.
116. To express the length of the perpendicular dropped
from the origin upon the tangent plane at any point of a sur-
face by polar co-ordinates.
Let ACB (fig. 41, bis) be the tangent plane at a point P
of a curve surface, CPF’ the tangent line to a section made by
a plane through the axis of z, DPE a tangent line to the
intersection with the surface of a right cone described about
CO with semi-vertical angle COP=6; GH the projection
of DE. ‘Then DOP is a tangent plane to the cone, and con-
sequently perpendicular to the meridian section of the cone
COP. Let OA=a, OH=a, OB=b, OG=0', OC H=c.
Then finding the equations to the traces OE, OD, we get the
equation to the plane DOE
93
consequently the equations to a line through O perpendicular
to it are
x (- 7 Fd (; =)
e=-{|--—— = — |- :
GA Gliks seke
which line must lie in the plane COF whose equation is
ycos@ =wxsing, putting angle AOF = ¢;
: 1 1 1 1
sing (7 - 5) ~ 2089 (5-7)
(= @ cos 2) oe (= gm cos )
we ry oy Bi 3
a b a
I cos @ sin ~\ 2
or ts (eee ae)
1 1 cos @ sin 2 2
= beers — > 6 ap wee ae J 1
a” +h b” ( a + b’ ( )
Now let P = perpendicular on ABC from O, OP=r, OK =71',
p = perpendicular on PF’, p’ = perpendicular on GH; then
] 1 1 1 1
=" > ary ea ees hae Is ar ,
ee abide. if a bila
1 1 1 1 cos sin @\ ”
ay oe “2 ae Ww a a “+ (=e - ®) 3
eee OF RY ce a b
1 1 1 1 - @Q)
Now suppose r= f(@, 9) to be the polar equation to the
surface. Then, considering @ as constant, r =f (qd, @) is the
equation to the plane curve PM, and
lst) dine! ( sa)
p” gf °F yt dQ) ~
94
Again, considering @ as constant, 2 = sin Of (@, 0) is the
equation to the plane curve AL, and |
1 1 1 aa 1 1 A 1 eal
peat ee ~ sin? @ |r? ot dd ;
Hence, substituting in equation (2), we finally get
1 tv ft /dF\*> ‘cosec’’ Oo fare
RM avi 68
Peet r iy ao r da
an expression involving the partial differential coefficients of
r = f(@, 0), where @ and @ are independent of one another.
117. The above expression may of course be obtained
from the value of P given in Cor. 2, Art. 107, by changing
the independent variables.
118. In some cases it is convenient to express the dif-
ferential coefficient of the volume of a solid by the polar
co-ordinates explained in Art. 98.
Let ACP (fig. 39) be a curve surface; AP, A’P’, its
intersections with two conical surfaces described about OC
with semivertical angles 9 and 6+60; CP, CP’ its inter-—
sections with two planes through CO inclined to sa at
angles @ and @ + od; then it may be shewn as before,
if V = vol. ACPO, that
pyramid OPP = dV
See Se as
S00 dod
therefore, taking the limit of both sides by making ods O80,
vanish, in which case we may consider the base of the
pyramid as coincident with the surface of a sphere, center
O and radius OP, and therefore the area of the base will
=Pp.P'p=rsin0dd.r00, and the vol. of the pyramid
&e. ;
will =—.r?sin@d60, we have
3
da’ V 1
dOdp 8
which may be also deduced from the result of Art. 112, by
changing the independent variables.
r sin @;
95
119. If in the above expression we substitute for r its
value in terms of @ and @, and integrate with respect to 0,
considering @ as constant, between the limits
9 =COG=F(¢), 0= COG = F,(),
(these values being obtained from the given equations to the
bounding conical surfaces BOD, B’OD’) we shall obtain a
quantity, a function of @ only, which when multiplied by op
is the ultimate value of the volume of the wedge GOH’ ; and
this, integrated between any two values of @ will give the
volume OBD' contained between two planes inclined at those
_ angles to xv, and between the given conical surfaces and the
curve surface. If instead of the vertex O, we suppose the
figure to be bounded by another curve surface whose equation
is x’ = f,(, 9), then the equation to be integrated will be
dv
120. ‘To express the differential coefficient of the area of
a curve surface by polar co-ordinates.
Proceeding as in Art. 118, it may be shewn that if S' be the
area of the surface APC (fig. bes we shall have ultimately
area of surface PP’ =
a
consequently, if p be the length of the perpendicular dropped
from O on the tangent plane at P,
vol. of pyramid OPP’ = 4p (er . ultimately,
d0do
also = 4” sin @. od 60, ultimately ;
iS sind ny
.' : ‘ A 6);
Dod =r A/ rsin'o (5) sint+ (— aa rt.116);
Pp
which may be also deduced from the result of Art. 113, by
changing the independent variables.
SECTION V.
ON TANGENTS, AND NORMAL AND OSCULATING PLANES, TO
CURVES, AND THEIR LENGTHS.
121. ‘To shew how a curve in space may be represented
by equations.
We have seen (Art. 44) that the equation F' (a, y, x) =0
represents a surface, and if to this we join another equation
F’ (@, y, %) = 0 representing a second surface, and suppose
the variables to receive only such values as satisfy both
equations at the same time, we shall determine a series of
points situated in each of the surfaces, that is, in the curve of
their intersection. Conversely, as we have no other means of
determining a curve in space than by assigning two surfaces
each of which contains it, we cannot represent it analytically
except by two simultaneous equations among the variables
Vv, Ys 2. >
122, Among the various surfaces which may pass through
a curve and so determine it, we employ for the sake of sim-
plicity the cylindrical surfaces which are parallel to the co-
ordinate axes, as their equations will contain only two of the
variables. (Art. 43).
Let QPR (fig. 43) be a curve in space, and through all
its points draw perpendiculars Pp, Rr, &c. to the plane of
vy (or, if the axes be oblique, parallel to 4z); the assemblage
of these lines will form a cylindrical surface, called a projecting
cylinder of QR, and meeting the plane of xy in the curve qpr,
which is called the projection of QR on vy. Similarly, if we
drop perpendiculars from all the points of QR on yz and za,
we shall have two other projecting cylinders, and two other
projections ; and the curve will evidently be determined if any
two of its projections are given, for in that case two cylindrical
surfaces will be given of which it is the intersection. Now
97
the projections qpr, Q'P’R’ are determined by equations of
the form
gi (av, y) = 0 ory=@ (2), Vi (8 &) = 0 or ¥ = Wy («),
the complete signification of which, as we know, is the cy-
lindrical surface erected upon each as its base. Therefore
the curve QPR will be determined by the system of simul-
taneous equations
y= (x), x=W(a).
In these, only one of the variables, w for example, is arbitrary ;
and any assumed value of w joined to the corresponding
values of y and x derived from them, will belong to a point
in the curve. The equation to the projection of QPR on the
plane of yz, is deduced from the other two by eliminating 2.
Hence it appears that every curve in space may be con-
sidered as formed by the intersection of two cylindrical surfaces
erected on its projections as their bases, perpendicular to the
co-ordinate planes. |
123. To find the equations to the line which touches a
given curve at a proposed point.
Let PT’ (fig. 43) be a tangent to the curve QPR at a
point P; also let qpr be the projection of the curve on the
plane of wy, and pé a tangent to the projection at p; we
must first prove that p¢ is the projection of PT. Let R be
a point in the curve near to P, r its projection; draw the
lines PR, pr; then pr is the projection of PR, and continues
so, however near & approaches to P, and consequently r
(which is always in a line through # parallel to Pp) to p.
‘Therefore in the ultimate positions of the lines, when R co-
incides with P and r with p, and they become tangents at P
and p, pt is the projection of P7'; that is, the projection of
the tangent at any point coincides with the line touching the
projection of the curve at the corresponding point. The same
is of course true relative to the other co-ordinate planes; if
therefore y = d (vw), s = (a) be the equations to the pro-
98
jections qr, Q’R’, the equations to the lines pt, PT’, or to
the line P7' of which they are the projections, will be
dz
d
s (w’ —#), x -—s= oT v— x);
y -y=
v, y, % being the co-ordinates of the point of contact, and
x, y’, x those of any point in the tangent, and the co-
ordinates being either rectangular or oblique.
124. The two equations to the curve may evidently be
supposed to arise from the elimination of ¢ between equations
of the form « = f(t), y=fi (4), 2 = fo (t); consequently, when
the co-ordinates of a point in the curve are all regarded as
functions of the same variable ¢, the equations to the tangent
line will take the symmetrical forms
dz dy dg
dt dt dt
Or if the equations to the curve, instead of having the
explicit forms assumed in Art, 123, should be
Mie VAG YS) = 0, We, et, (@, Ye) 0,
then considering y and x each as a function of wv, we have
du Hi dudy dwdz _
dw dydzx A dzduv
du, du,dy du,dz
aye Ce a
d ’ :
from which =; pe may be obtained and substituted in the
v dx
equations to the tangent line.
b)
e
9
125. If the tangent to a curve makes angles a, (3, y
with the axes of aw, y, #, since its equations, taking the pro-
jections on the planes of x” and yx, are
99
dy
, 1 , , da i
eae: Fa ©) Vom Ue (Fae ®)s
dx dx
dy
1 dav
cosa LS eee
Vise (Ge) Me) +()
a dx} ' \dwx i dx +(=
dx
COS? =
1+ (2) +(Z)
4 dx dx
126. To find the equation to the normal plane to a
curve at a proposed point.
A curve can have only one tangent at a proposed point,
but it may have an infinite number of normals, that is, of
lines perpendicular to the tangent through the point of con-
tact ; these all lie in one plane called the normal plane. Let
x, y, be the co-ordinates of the proposed point of the curve,
‘then since the normal plane passes through that point, its
equation will be
vs’ —-2=A(e'-a)+ By -y);
and since it is perpendicular to the tangent whose equa-
tions are
dy
1 , , d.
x am (s — 2), Yonsself amar (Ham 8)s
da dx
dy
i dz
A+=—==0 =ume = 0) ;
Pr 4 HER ees
da dx
100
hence, substituting for 4 and B these values, the equation to
the normal plane at a point wy%, is
: , dy ? dz
v—at+y FU) et TAS sn ti Baim
Cor. If a, y, x be each regarded as a function of the
same variable ¢, as in Art. 124, the equation to the normal
plane takes the symmetrical form
dx dy dz
a aki by inks Vee N iseadeeeiye
Dre) fay gl ee ey pinto Pa) tay
127. To find the equation to the osculating plane to a
curve at a proposed point.
The osculating plane at any point is that which has a
closer contact with the curve than any other plane passing
through the same point.
Let «#, y, x be the co-ordinates of the point of the curve ;
then the equation to a plane passing through it will be
ACK Saye BY Syl Ze oF =)
let Oo be the length of a perpendicular let fall on this plane
from a contiguous point in the curve whose co-ordinates are
wth, y+kh, x4+1;
Ah Bk ail
\/ A + B 4 C
dx dy ‘dz ad’ x CY 6a
pelea ty a cS) +}3( 4 Bey cS) ae
( Ai cdi) \ al li, Ge en
tp (Cor. Art. 37).
3
if we suppose the points consecutive, and substitute for h, k,
l their developements in series ascending by powers of 7;
+ being the increment of ¢ the variable of which the three
co-ordinates are assumed to be given functions. Now if we
determine the constants so that the coefficients of + and 7*
101
may vanish in the numerator of 6, the portion of the curve
immediately contiguous to the point wysz will coincide more
nearly with the plane so determined than with any other
plane that can be drawn through that point; and as only the
ratios of A, B, C to one another are required, this can be
effected by establishing the two conditions
dx dy dz dx Ei d’
— a —— =0; 4A — =|
Bri aeat) satin’ ae dae ede eae
whence eliminating A, B, C between these and equation (1),
2)
2
&e., by a’, 2”, &c., we get
di
and denoting a EE
(y/'2"—2!y")(X—a) + (w/a""—a'e”)(Y-y) +(a'y"-y'a")(Z—2) =0,
the equation to the osculating plane at a point wyz.
Cor. If we assume y and g to be functions of a, in the
ordinary way, then
d & d &
Ah+B/( ella ct 1) 40(4= + 4o + Be)
s_ dx dx dx dx” (2)
/ A+ B+ C
dy d’y ad? z
oA — t= Oy ——- — =
eer cS aE ere et
and the equation to the osculating plane becomes
d’y Gydzs dzdy
oa rere ora
which evidently agrees with the above, when & = ¢.
as
— a)+ agit? - Y),
128. In order that the contact may be of the third order,
the coefficient of h* in the numerator of 6 in equation (2)
d? zx d’y
must also vanish, or Cis + Lister =e
ae eT SE A eT
dx’ da da’ da’ :
the condition which must be satisfied at any point of the
curve where the contact is of the third order. When the
contact is only of the second order,
1 We Bs _ dy
bee ty" ily Ro ge
J A? + B+ C {3 (Cas ss ia) ate 4 }
a quantity which changes its sign at the same time that h
does ; consequently the curve generally cuts its osculating
plane. On the contrary, when the contact is of the third
order, the sign of 6 does not change with that of h; and
the curve is said in such cases to have a point of inflexion.
The condition then of there being such a point is
asdy ay ds
eS ea ee SSS
— _—
da’ dx*® da? dx’
129. When a curve in space is a plane curve, if w, y, =
be the co-ordinates of any point in it, they will always satisfy
the equation to a plane
v= Aw + By’ +ce...(1), so thats = d~e+By+e;
therefore the differential coefficients of = and y will be such
as to satisfy the equations
dz iy dy ds d°y
a
dv dx dx dx®’
A and B denoting constant quantities ;
ad’ 2
d [ dx’ Tyds asd'y
ed a\ ai) Esai a Me eae ee
dx*
which must be rendered identically true by the equations
y=@(v), x= \(v), when they represent a plane curve;
103
otherwise, the curve is of double curvature. This condition
being satisfied, if we substitute for 4A, B, ¢ their values in
(1), we find the equation to the plane in which the curve is
situated the same as that to the osculating plane, as might
have been foreseen; for when a curve in space is a plane
curve, the plane in which it is situated is the osculating plane
at every point. Hence when the above condition is satisfied,
the equation to the plane in which the curve is situated may
be obtained by writing for the differential coefficients their
values in the equation to the osculating plane.
If we assume each of the co-ordinates vw, y, x to be a
function of the same variable ¢, a symmetrical expression for
the condition (2) may be easily obtained by eliminating the
constants from the three derived equations of lw + my+nzx=c.
130. The osculating plane at any point of a curve in
space is perpendicular to the line of intersection of con-
secutive normal planes at that point.
The equation to the normal plane at a point wyz is
dy sy ds
ve — Ut CRE mae 28D) Seye & E
: ; dy dz
and at the contiguous point 7+h, y+ he +&c., 2 the + &e.
it 1s
’ dy dy d*y
wa -he(y -y- hg te.) (+ + hot 4 &e.)
d dz d®
+ (¥ ~# ~h— — &e.] (= + a + Be.) = 05
or, combining it with the former and dividing by 4, it is
; d’y dy d’s dz\? s
Bet ET ge (3) + apn (ag) ttm
h, h*, &c. = O.
104
Therefore, making 4 = 0, the equations to the line of inter-
section of consecutive normal planes are
OY dz
w-—at+y a dean ne my eye
2
2 2 2
oF peo Cees) be (=) 5 (=) hee Os
dx” d x” dx dx
(the latter equation being evidently that which results from
differentiating the former with respect to #, considering «’, y’,
%’ as constant) and if we deduce the equations to the pro-
jections of this line on the planes of yz and ga, it will be
seen that the conditions of being perpendicular to the os-
culating plane (Art. 25) are fulfilled.
131. To find the differential coefficient of the length of
the arc of a curve in space.
Let y= (2), * =wW («) be the equations to the curve,
2, y, % the co-ordinates of any point P in it (fig. 43),
at+h,y+hk, 2+, the co-ordinates of a contiguous point R,
s =length of arc QP, Q being a given fixed point in the curve,
and &R = chord PR;
then R?=7? +447
| dy hd’ dz hh dx
ans (A 4 = <7 4 &e V4 + (Aa at &e.]
da rd dx da 2 dx dx’
dy\? dz\*
= h? {1 a : yes
+ (2) + oy fan + &c
DSN He... R a/ dy. * dz\?
, — = | fete — —— ae
ay Imit o h 1 + Gy ae (=
Cor. Hence the formule for the angles a, 3, yy, which |
a tangent at the point wys forms with the axes of a, y, 2,
may be written
dy dz
dax da
cosa=—, cosB=——, cosy =—;
dw dv dn
or, supposing s to be the independent variable,
dxv B dy dz
a oe a eS a) co = ——
COAG = ——s COs rF sty
132. If we employ polar co-ordinates so that x =r cos 0,
# =rsinOcosd@, y =17 sin @sin @, then
d ; dr\* : d d\?
ie / + (=) + r’sin® @ (52) :
ds r
Also, —- = ————, p being the perpendicular on the
dr re p
tangent line at the extremity of r.
SECTION VI.
ON THE DISCUSSION OF THE GENERAL EQUATION OF THE .
SECOND ORDER.
133. To find the position of the center of any surface.
The center of a surface is a point O (fig. 22) such that
any chord of the surface PP’, drawn through it, is bisected in
it. (It must be observed, however, that if PP’ cut the surface
in more points than two, it would be sufficient that these
points combined in a certain order should be, two and two,
equally distant from OQ).
If the surface be referred to any three axes originating
in O, and PM, P’M’ be the ordinates parallel to Ox of the
extremities of a chord, we see from the equal triangles POM,
P’OM’, that these ordinates are equal and of contrary signs ;
the same thing would be true for the other co-ordinates of P
and P’, as well as for every other chord passing through O.
If therefore f (xv, y, x) =0 be the equation to the surface,
and if it be satisfied by 7 =a, y= 6b, x =c, it must also be
satisfied by # = —a, y= —b, s = —¢; that is, it must be such
as not to alter when the signs of the three variables are
changed ; and, conversely, if it have this property, the origin
is the center of the surface. When f(a, y, %) = 0 is algebraic,
it cannot have the above property unless the dimension of
every term be even in an equation of an even degree, and the
dimension of every term be odd in an equation of an odd
degree; for in the former case the equation is not at all
altered by replacing vw, y, s by — a, — y, — #3; and in the latter
case (in which the equation cannot have a constant term) the
sign of every term will be altered, and therefore the whole
equation unaltered.
107
Hence, to find whether a proposed surface admits of a
center, we must refer it to parallel axes, through a new origin
having co-ordinates h, k, 1, by putting
eaa't+th, yay +k, x= +],
and equate to nothing the coefficients of all the terms which
are of a dimension different (as far as regards odd and even)
from the degree of the equation; if these conditions can all be
satisfied by real and finite values of h, k, 1, the surface has
a center, and h, k, / are its co-ordinates; in the contrary case
the surface has no center.
134. To find the co-ordinates of the center of a surface
of the second order represented by the general equation of
the second degree.
The general equation of the second degree is
FS (&, Ys 8) = an? + by? +02? + 2a yx +20 su+ 2c uy
+ 2a" e2+2b"y + 2c"s+d=0.
Here we must make v=a’ +h, y=y +k, s=2'+1, and
equate to nothing the coefficients of terms of odd dimensions,
that is, those involving a’, y’, 3’. The result of these sub-
stitutions is
S(@ +h, y +k, 2 +0,
which when expanded will consist of three parts, (1) terms
of two dimensions which must be the same as those of
FT (2, y’, #’) which are of two dimensions; (2) terms of one
dimension which are
va f(y i). df (h, bl). dfeecan
CMMNM ied aa) as Stas
each of which must disappear; and (3) a constant term
f(h, k, 1); therefore the result will be
aw” + by” +62" 420'y'x' + 2b sv + 2c uy +f (h, kh, D =%
108
df (hs k; !)
dh
with the conditions =0, &c., or
ah+bl+ck+a"=0
bk + al+céh +h’ =0 wl)»
cl+ak+bh+c’ =0
for determining h, k, 7. Multiplying the two latter equations
respectively by indeterminate coefficients ¢, uw, and adding
them to the former, we have
(a+te +ub)h+a"+tb’ + uc’ =0,
provided c +tb+ua'=0, 6 +ta'+uc=0;
these two latter equations give ¢ and w, and then substituting
in the first we find h = = where
D=aa’ + bb” 4+ cc” —abe -2a'l'e,"
N=a’ (bc — a”) +b" (vv — cc) +0" (ac — bb);
aN Wve
similarly, & = —, / = _.
y> D’ D
Hence, provided D is not = 0, these values of h, k, 1, which
are always real, are finite; and the surface will have a
single center of which they are the co-ordinates, and its
equation reckoned from the center as origin will be, suppressing
accents,
ax’ + by? +cex +2a ys + 2b ea4+2ay+ah+b'k
aa cl+d= 0%
for, multiplying equations (1) respectively by h, k, 7, and
adding, we have |
fi, ky Dl) = ah WK +014 d.
135. If the constant term f(h, k,l) disappears, the
surface represented is a cone, since if we combine its equation
with the equation to a plane through the origin s = 4a + By,
109
&
the result will be of the form y = a (p + /q) indicating two
straight lines through the origin; unless the radical be
impossible for all values of 4d and B, in which case the pro-
posed equation represents a point.
If the coefficients of the given equation be such that
D=0, and the three numerators are not all =0 at the
same time, then one at least of the co-ordinates of the
center will be infinite, which signifies that the surface has
no center.
If at the same time that D =0 the three numerators
vanish, then the surface admits of an infinite number of
centers; for in that case the three equations (1) are reduced
to one, or to two realiy distinct equations, as is shewn in most
treatises on Algebra, and may therefore be satisfied by an
infinite number of values of wv, y, x. If they are reduced to
two, that is, if the values of h and k deduced from the two
first for instance, satisfy the third whatever / be, then there
will be an infinite number of centers situated in a straight line
which is the locus of the two independent equations; the
surface will therefore be a cylinder on an elliptic or hyperbolic
base.
If the three equations (1) are reduced to a single equation,
that is, if the value of # deduced from the first, for instance,
satisfies the other two whatever k& and 7 be, there will be an
infinite number of centers situated in a plane which is the
locus of the single independent equation, and the proposed
surface will be a system of two planes parallel and equidistant
from that plane. In this latter case the proposed equation
must be capable of being resolved into two rational factors of
the first degree.
136. The locus of the middle points of a system of
parallel chords of any proposed surface is called its dia-
metral surface. ‘This surface will have several sheets, if
each of the chords has more than two points in common
with the proposed surface; if, for instance, the proposed
surface be of the m™ order, the points of intersection with
110
its chords, real or imaginary, will be in number m, and
their combination on the same indefinite line will form
in(n-—1) different chords, and as many middle points;
and therefore the diametral surface, since it may be met
by an indefinite line in $2 (2-1) points, will have an
equation of the degree $(m—1). For surfaces of the
second order where n= 2, the diametral surfaces can only
be planes.
When any surface admits of a diametral plane, if we
make it the plane of wy, and take the axis of x parallel
to the chords which it bisects, the equation to the surface,
for every pair of values vw =a, y = 6, must furnish for x
values which, taken two and two, are equal and of contrary
signs; and therefore the equation, supposed algebraic, can
only involve even powers of x. And, conversely, whenever
an equation contains only even powers of one of the variables,
x for instance, the plane of wy is a diametral plane, and
is said to be conjugate to the chords parallel to the axis
of zg. Also if a diametral plane be perpendicular to the
chords which it bisects, it is called a principal plane, and the
chords principal chords. Moreover the intersection of any
two diametral planes is called a diameter of the surface.
137. To find the equation to a diametral plane of a
surface of the second order.
Let v=mz, y=ns, be the given equations to a line
through the origin to which the proposed system of chords
is parallel, and f(a, y, 3) =0, the general equation of the
second degree, the equation to the surface. Let h, k, 1, be
the co-ordinates of the middle point of any chord, and let
the surface be referred to axes parallel to the former ones
passing through it; then the equation will become
S(a@ +h, oy +k, x +20) =0,
and the equations to the chord itself will be a = mz’,
y’ =n; therefore the values of x’ belonging to the points
111
of the surface where the chord meets it, are given by the
equation
f(mx' +h, nz’ +k, x +1) =0......(1),
which is of the form
Rz? + S2'+T=0;
and since the values of x’ are equal and of opposite signs,
S=0. But Sis the coefficient of x’ in equation (1) ;
mdf (hy ky 1). df (hy yD af (hy hy 2)
dh eens S/%) Mark Soni lee Whe uae
or mM(ah+Ulock+a")+n(bk+al+ch +b")
tel+avk+h +c" =0...... (2);
or (am+cen+b)h+(bn+em+a)k+(c+0m+an)l
+am+b’n+c" =0,
the relation among the co-ordinates of the middle point of
any chord, or the equation to a diametral plane.
Cor. As the coefficients of h, k, J are possible, there will
be a diametral plane for all values of the constants m and n,
unless the three coefficients should all become nothing at the
same time, when the diametral plane will be situated at an
infinite distance. But the equations
am+cn+b =0, bn+cém+a=0, c+bm+an=0,
containing only two unknown quantities, have an equation
of condition which is the same as D = 0 (Art. 134); in this
case therefore the surface has not a center.
When the surface has a center, every diametral plane
passes through it, or through the locus of the centers; for
equation (2) is visibly satisfied by the co-ordinates of the
center furnished by equations (1), (Art. 134). This also
follows from the definition.
112
138. Any diametral plane of a surface having a center
is parallel to the tangent plane applied at the extremity of
the diameter to which it is conjugate.
Taking the center for origin, the equation to the surface
will be
ax’ + by? + cx? +20 yx 4+ 202742 xy+d=0;
therefore the equation to the tangent plane at a point vyz is
(Cor. 1, Art. 107),
(av + bx +c'y) (# -— 2) + (byt+ax4+ec'x) (y’-y)
+ (cx +a'y4+0'2) (x — 2) =0,
or (av +Uxtcy)a+(bytaxstea)y+(extay+d'x)2
+d=0;
and if it be applied at the extremity of the diameter whose
equations are v = mz, y = nz, the equation becomes
/ ] , / 4 / , d
(am+cen+0')a'+(bnt+em+a')y + (c+ bm +a‘n)z'+-=0,
R
and therefore (Art. 18) represents a plane parallel to the
diametral plane which is conjugate to the diameter x = mz,
y= nz, the equation to which is (putting a” = 6b” =c”’ =0,
in the equation Art. 137)
(am+cn+)a + (bn+emta)y+(ce+ Um+a'n)x'=0.
This result might have been foreseen; because the straight
lines in which a diametral plane and a tangent plane at the
extremity of the conjugate diameter are cut by any plane
through that diameter, must, by the nature of lines of the
second order, be parallel to one another.
139. We shall now proceed to the reduction of the
general equation of the second degree
ax? +by+ex°+2a'yxt2bear2cuvy+2ax42b"y4 2c x+d=0,
f 113
where we suppose the co-ordinates rectangular; for if they
were oblique, by transforming them to rectangular co-ordinates
we should obtain an equation of the same degree as the
above (Art. 96), and which could not therefore be more
general than the one which we have assumed. We shall
prove, as affirmed at Art. 53, that this equation, after being
simplified as much as possible, will always assume one or
other of the forms
Ax’ + By? + Cz’ =D,
By? + C2? =2A'e,
the co-ordinates being rectangular; and therefore the general
equation of the second degree can never represent any other
surface than one of those discussed in Section 2.
140. Every surface of the second order has at least
one diametral plane which is perpendicular to the chords
bisected by it.
Let the equation to the surface be
ax’ + by? +c2°4+2a'ys+2b'za4+2cvyt+2a vt+2b" y+2c'%+d=0,
and v=mz, y= nsx, the equations to the line to which a
system of chords is parallel; then the equation to the plane
which bisects the chords is (Art. 137)
(am+en+b)xut (bnt+em+a)yt+(c+hm+a'n)s
ed Meh +o = 0.
and our object is to shew that real values can be assigned
to mand nm, such that this plane shall be perpendicular to
the chords. The conditions for this are (Art. 25)
am+en+b bn+em+t+a
ee ee OE wat te
, 9
c+bm+an c+bm+an’
from which, by eliminating one of the unknown quantities
m or n, we shall obtain a cubic equation which will always
give a real value for the other; and the direction of the
S~
114
system of principal chords will be determined. But we shall
obtain a more symmetrical result by assuming for the un-
known quantity
c+bm+an=s, ors—-c=bm+an,
then am+cen+b'=ms, or m(s—a)=cn+b e+ (1):
bn+cm+a=ns, orn(s—b)=cm+a
Hence, determining m and 7 from the two latter equations,
m {(s—a)(s—b)-c*} =U (s—b)+ ae
nm {(s —b) (s—a) —c”} =a'(s —a) + yap
And substituting for m and m in the former, we have
(s —a)(s— b)(s —c)—a*(s —a) —b?(s — b) —c?(s—c) -2a'b'c'=0,
or 8 —(a+b+c)s°+ (ab+ ac+be-a’—b’—c")s |
— (abe —aa*— bb”? — cc? + 2a'U'c’) = 0.
This equation, being of an odd degree, will always have
one real root which substituted in (2) will give real values
for m and»; and therefore in every surface of the second
order there is at least one principal plane, or, which is the
saine thing, one system of principal chords. Also there can-
not be more than three, unless the particular form of the
proposed equation of the second order should render any two
of the equations (1) identical, in which case m and nm would
be indeterminate, and the number of principal planes would
be infinite.
Cor. ‘That the cubic has all its roots real, may be shewn
by putting it under the form
(s—c) }(s—a)(s—b)—c} — $a"(s—a)+b°(s—b) +2a'b'c? =0,
and substituting for s, a and ;3 the roots of (s—a)(s—b)-—c?=0.
The results of these substitutions, since (a ~ a) (a — b) = oF.
(a = B)(b- B) =e, are
we fa'\/a —-axt ba — 6}? and + fa'\/a — B + b'\/b — B}*,
115
for upon solving the equation (s — a) (s — b) = c”, it is easily
seen that one of its roots a is greater than both a and 8, and
the other 8 is less, Therefore since +, a, 3, — 0 when
substituted for s give results +, —, +, —, there is one root
greater than a, another between a and #3, and a third less
than p.
141. Every variety of surfaces of the second order referred
to rectangular co-ordinates is comprehended, without exception,
in the equation
Av’? + By + Cx? +24e4+2By4+2Cxe+D=0.
Let the surface be represented by the general equation of
the second degree f(a, y, s) =0, and let it be referred to
three new rectangular axes Ow’, Oy’, Oz’, by the substitutions
of Art. 90, and let the transformed equation be
Ax’ it By”? i Cz” + 2 A'y’s! ate 2 RB’ =x’ ae 2 C'a'y’ 4 2 A" a
+2B"y' +2C"s'+D=0;
also let the equations to a line parallel to a system of principal
chords (the existence of which in every case is certain) referred
to the new axes, be a’ = m3’, y’=nz’, then m and n satisfy
the conditions
Am+Cn+ B=m(C+ Bm+ An)
Bn+Cm+A=n(C+Bm+ An).
Suppose now one of the new axes, that of s' for instance, to
be parallel to the direction of the principal chords; therefore
m=0, n=0; consequently we must have A’=0, B’=0.
Hence whenever one of the rectangular axes is parallel to the
direction of a system of principal chords, the general equation
is freed from two of the rectangles, and takes the form
Aw? + By? + Cx? +2C'a'y'+2 A"a' +2 B’y' +2C 2’4+D=0...(2).
This equation comprehends all surfaces of the second order
without exception, and it may be still further reduced if,
without altering the axis of x’, we turn the axes of a’ and y/
in their own plane through an angle @, so as to make the
term involving ay’ disappear, by the substitutions (Art. 94.)
8—2
116
a’ = x, cosg — y, sin ®,
y =a, sind + y, cos d;
this gives — 2cos@ sin (4 — B) + 2C’ Scos* ph - sin? hb} = 0,
2C"
tan2@ =——_.,
or tan2@ Gap
which value, being real and always admissible even when
A = B, shews that equation (2) may be reduced to the form
(suppressing accents)
Aa’ + By + C+ 2Ae+2By+2C2+ D=0;
which (the co-ordinates being rectangular) comprehends all
surfaces of the second order.
142. It is at this point of the reduction of the general
equation of the second order, that the separation of the sur-
faces represented by it into two classes takes place.
1. If none of the squares of the variables are wanting
in the equation
Aa’? + By +024+2Ae+2By+2C2e+D=0,
i.e. if A, B, C are all different from zero, by writing
eth, y+k, =+1 for x, y, x and determining h, k, 1 by
the conditions
Ah+A=0, Bk+B=0, Cl+C'=0,
(which will give finite values for h, k, 1) the origin will be
changed, but the directions of the axes unaltered, and the
equation will be reduced to the form
Av’ + By? + Cz? =D... (1)
which represents the first class of surfaces of the second order.
2. If one only of the squares is wanting, A for instance,
and the corresponding coefficient A’ is different from zero,
then the term involving w cannot be made to disappear, for
that would give / infinite; but instead of that we may deter-
117
‘mine f# so as to exterminate the constant term, and determiné
k and 7 by the same conditions as before, and the equation
will be reduced to the form
Bye OS = 2A (2)
which represents the second class of surfaces of the second
order, ‘The figures and properties of these two classes of
surfaces have already been discussed, (Art. 53).
143. There are still two particular cases to be examined,
belonging, as we shall see, to the two sorts of cylindrical
surfaces of the second order.
First, suppose that the coefficient of one of the squares,
A for instance, vanishes, and at the same time the correspond-
ing coefficient A’=0, without either of the other squares dis-
appearing; the equation is then
By’ + C2*4+2By4+2C2+D=0,
which, containing only two of the variables, represents a
cylinder perpendicular to the plane of yz on an elliptic or
hyperbolic base; and if the center of this curve be taken for
the origin, the equation will be reduced to the form
By iG ei
which is deducible from equation (1) by making 4 = 0.
Secondly, suppose only one of the squares to remain in
the equation, x? for instance; then writing »+/ and a+h
for s and #, and determining # and 7 so as to exterminate
the term containing x, and the constant term, the equation
becomes
C#+2Aa7+2B y=0;
now, without altering z, turn the axes of w and y in their
own plane through an angle @ such that 4’ sin p = B' cos ,
then the equation will be reduced to the form
118
which represents a cylinder perpendicular to the plane of wy
on a parabolic base, and is deducible from equation (2) by
putting B= 0.
It is unnecessary to examine the case where the three squares
disappear, as no equation of the second order could ever by
transformation of co-ordinates be reduced to that form; the
cases in which it represents a cone, or two parallel planes,
have been noticed.
144. Hence, we conclude, that all surfaces of the second
order with all their varieties, are comprised in the two classes
represented by the equations
Ax’? + By + C2 =D,
By? + C2? =2Aa,
the co-ordinates being rectangular. In the first class, each
of the co-ordinate planes is a principal plane; therefore there
are three systems of principal chords, and consequently the
three roots of the cubic in (Art. 140) are all real. In the
second class, only the planes of s# and wy are principal
planes; therefore there are at least two systems of principal
chords; the third system, determined with the others by the
cubic equation just referred to, must therefore be also real,
but the corresponding principal plane is situated at an infinite
distance. Hence it results from this discussion, as well as
from the form of the equation (as proved in Cor. Art. 140),
that the three roots of the cubic equation which determines
the positions of the principal chords are always real.
145. By means of the preceding results, we are able to
determine the species and form of the surface represented by
any proposed equation of the second order, without recurring
to the laborious process of transformation of co-ordinates.
We must first ascertain whether the surface has a center
by the method of Art. 134, and if it has, find the co-ordinates
of the center h, hk, 1 from the equations which result, by
equating to nothing the three derived equations of the first
———
119
order, w, y, # being replaced by h, k, 1; then taking that
point as origin, the equation is reduced to
aa’ + by? +c3x?4+2ayx4+2bsa+26ay4+ah+b"k+c'l+d=0,
from which the species of the surface and the values of the
real or imaginary axes can be readily determined, as will be
seen in the next Art. In the succeeding Articles, the charac-
teristics of the different sorts of surfaces that have not a center
will be deduced.
146. The equation
au’ + by? + cz? + 2a ys + 2b ea + 2uvy =d,
(d being a positive quantity) belongs to an ellipsoid, hyper-
boloid of one sheet, or hyperboloid of two sheets, according
as the cubic equation
(s —a)(s—b) (s —c) —a?(s — a) —b?(s—b)-c?(s—c) -20'b'c'=0,
gives for s three positive values, two, or one.
Let w=mz, y=nsz, be the equations to a line through
the origin, which will be a diameter of the surface, since
the origin is the center, and let the length of the diameter
be denoted by 27; then the chords parallel to it will be
bisected by a plane, of which the equation is
am+cen+b)a+(bn+emt+ta)y+(c+dm+an)s=0;
Yy
and if this plane be perpendicular to the chords, or if 27
be a principal diameter,
making c+ bm+an=s,
am+en+b=ms
we must have }..0)
bn+cecm+a'=ns,
from which eliminating m and 7, as in Art. 140, we find
(s—a)(s —b)(s—c) — a? (s— a) —b?(s—b)—c? (s—c) -2a'b'c'=0;
120
also multiplying the two latter equations by m and m respec-
tively, and adding them to the former, we find
s(1 +m? + n*) = am? + bn? 404 2a'n + 2b'm + 2Cmn.
But if w, y, x be the co-ordinates of the extremity of the
diameter,
9
~
ra eP+ypPrs= (1+ m +n’) 2;
also since wv, y, % are co-ordinates of a pcint in the surface,
3 (am? + bn? +c0+2an+2bm+2cemn) =d;
d d
2? Pim
Ss
tw)
ae y = or r =
3° x
—.sl+m+n')=
s
ee
Ss
Now d is a positive quantity, therefore according as s has
three positive values, two, or one, r has three real values,
two or one, or the surface has three real principal diameters,
two, or one, and is therefore an ellipsoid, hyperboloid of one
sheet, or hyperboloid of two sheets.
147. Hence, when we have an equation of the second
order reckoned from the center, and the constant term forms
one member of the equation and is positive, if with the
numerical coefficients we form the above cubic, and observe
the signs of its terms, we shall determine the nature of the
surface represented by the proposed equation. For since all
the roots of the cubic are real, there will be three positive
roots, two, or one, according as there are in it supposed com-
plete, three changes of signs, two, or one. If the cubic offers
no change of signs, but only continuations, its three roots are
all negative, and the three axes of the surface are imaginary,
and consequently the surface itself imaginary.
Cor. If we solve the cubic, we may then easily calculate
the lengths of the axes, and fix the position of each by means
of the values of m and n, corresponding to each value of s,
given by equations (1).
Ex. 1. 2a°+5y°+32°42yx%-420-20y+2u+8y-624+8 =0,
121
Forming the three derived equations and replacing a, y, z
by h, k, J the co-ordinates of the center, we get
Diy ra FEA a Te Bo as:
10k +21 —2h+8=0,> which give = —1,
614+2k-—4h—6=0, b= 2.
Then the equation, referred to the center, and the cubic for
determining the axes, become respectively
20° + by? + 32° + Qyz—430 —-Qwy=—-14+446-8=],
s§—10s°+ 78+ 388 =0;
and as this complete equation has all its roots real and has
two changes and one continuation of sign, it has two positive
roots and one negative root, and the surface is a hyperboloid
of one sheet.
Ex. 2. v0 +2y? 4+ 3274+ 2yx4+22u+4+2uyt+ut+y+s=1.
The co-ordinates of the center will be found to be
h= — i, b=0, 1=.0;
and the equation referred to the center becomes
a+ 2y + 32° 4+ 2ye + 2Q2u + 2aHy =F.
Hence the cubic becomes
8 —6s?+8s—2=0,
having three real roots; and as it is.complete and has three
changes of sign, it has three positive roots, and therefore
represents an ellipsoid.
Ex.3. ystxuv+ay=a’.
Here the cubic is s* — 3s — 2 =0, which has one positive
root and cannot have more; therefore the surface is a hyper-
boloid of two sheets. |
Ex. 4. «4+ 2y°+3ys-2ey-6x+7y+624+7=0.
122
The co-ordinates of the center will be found to be
h=1, k=—2, t=,
and the equation referred to that point
uv +2y°? + 3y% —2xy =0,
wanting the constant term; and as it is satisfied by w =0,
y =0, it contains the axis of x, and represents a cone and
not a point.
148. When the proposed equation
axv+by+ex 42a yxet+2b su42Cay+2a vVt+2b y+2c' zx +d=0,
is such that
abe — aa® — bb”? — cc” + 2ab'c = 0,
the surface represented either has not a center, or it has a
central line or plane (Art. 135), and admits of four varieties,
viz., paraboloids, parabolic cylinders, elliptic or hyperbolic
cylinders, and a system of two parallel planes; for each one
of which, certain conditions must be satisfied which are not
all true for the others. Hence we can determine the dis-
tinctive characteristics of each of these surfaces.
1. For paraboloids, since the co-ordinate planes cannot
be all parallel to the axis of the surface, and since it is
only such planes which intersect the surface in a parabola
(Art. 103), one of the quantities
ab, (1)
(which determine the nature of the traces on the co-ordinate
planes) must be different from zero, and negative in the case
of the elliptic, and positive in that of the hyperbolic para-
boloid ; also one at least of the co-ordinates of the center will
be infinite, and therefore one of the equations for determining
those co-ordinates imaginary.
a*—be, b?-—ac, c
2. For parabolic cylinders, since all sections are either
straight lines or parabolas, each of the quantities (1) vanishes,
123
and one of the equations for determining the co-ordinates of
the center will be imaginary.
8. For elliptic or hyperbolic cylinders, the equations for
determining the co-ordinates of the center are reduced to two
distinct equations, and one at least of the quantities (1) is
different from zero, and is negative in the former, and positive
in the latter case. Also, by taking sections parallel to the
co-ordinate planes, it will be necessary further to examine
whether the cylinder is wholly imaginary, or reduced to a line,
or to a system of two planes not parallel.
4. For a system of two parallel planes, the equations for
determining the co-ordinates of the center must be reduced to
a single distinct equation, and the three quantities (1) must
each vanish. It will be further necessary, by taking sections,
to examine whether the two planes are confounded in one, or
are imaginary.
Ex.1. a’ -—2y’?-— 3y2 4+ 32a + ay +42 =0.
The equations for determining the co-ordinates of the
center are
20+3%3+Yy=0,
—4y—-32+4+07=0,
—3y+ 3%+4=0;
and since the last subtracted from the sum of the two former
gives 4=0, this impossible equation shews that the surface
has not a center. Also, c? — ab = (4) + 2, which is different
from zero and positive; therefore the surface is a hyperbolic
paraboloid.
Ex.2. a? +y? +92" + 6y2 — 62a —2ay + 2u—4z% =0,
The three derived equations are ]
2e —-62—-2y+2=0,
2Qy + 6% —2e@ = 0,
18x + 6y — 6% —4=0,
124
the two former of which give 2=0, therefore the surface has
no center; and the traces on the co-ordinate planes cannot be
ellipses or hyperbolas, therefore the surface is a parabolic
cylinder.
Ex.3. #® -y-24+2yx+e+y—-x =a.
The derived equations are
2%+1 =u),
—-2y+22+1=0,
which are reduced to two distinct equations; also the traces
on wy and ws are hyperbolas; therefore the surface is a
hyperbolic cylinder; except when a@=0, when it represents
two planes intersecting in the axis of the cylinder whose
equations are 2v+1=0, 2x -—2y+1=0; for when a=0,
the equation is resolvable into (w + y — 2) (w@-y+2+41)=0.
Similarly, wv + 3y? + 42° -—6yz —2%x” =a represents an
elliptic cylinder; except when a =0, when it represents the
straight line « = %, y = #, forming the axis of the cylinder;
for it may then be written (@ —%)? + 3(y—x)?=0. If a be
negative, the surface is imaginary.
Ex. 4. w+ 4y°+ 2 4+4y2-22a -40y43au—-6y—32=0.
The derived equations are
20-23 —-4y+3=0,
8y +42 —-4v—-6=0,
23+ 4y —-24%-—-3=0,
which are equivalent to only one equation; therefore, the
proposed equation can only represent two parallel planes; and
putting x = 0, the trace on wy has for its equation
w+ 4y? —4vy + 3x —-6y = 0, or (w — 2y)’? +3 (x — 2y) = 0,
representing two real straight lines; consequently the proposed
equation represents two separate parallel planes, and, as we
see, it may be written (v7 — 2y — x) (vw- 2y—%+4 3) =0.
125
149. To find the relations among the coefficients of the
general equation of the second order when it represents a
surface of revolution.
Let av? + by? + c2°+2a yx 4+2b'ax 4+ 2c ary + 2a ex
+2b°y4+2c'x+d=0, (1)
be the general equation to the surface; and let a, 8, y be the
co-ordinates of a point in the axis of revolution. “Then a
sphere whose equation is
(v7 —a)’+(y—-B)y+ @-y)- R=0, (2)
will intersect the surface in two parallel planes
i pircaih. eit
len +my+nz =p.
Now if \ be an arbitrary multiplier,
(1) +A (2) =0
gives the intersections of the surface with any sphere such as
that above mentioned ; hence this equation must be identical
with
O=(la+my+nz—p)(le+my+nzx—p’);
.O0O=PH +m yt ns? +2mnyx4+ 2lnae + 2lmay + &e.
is identical with
O=(a+A)a’?+(b+A)y?+ (C+A) +420 YX 42h az42Cry+&e.;
-P=at+aA, mM=b+r, W=c+”,
mn =a’, In =0’, lm=c
b'e! ; ae’ ih
pel=sat+nr, —-=m=b+)d, —-=Wect+d;
a b C
bo! a’e a’b’
° ee eer b = —— —"¢,
a b
are the required conditions. To this must be added that each
of these equated quantities, being equal to — A, must be finite.
126
Cor. Hence the plane through the origin to which the
circular sections are parallel has for its equation
Veutacy+abz =0,
consequently all chords of the surface parallel to this plane are
principal chords.
Also if we substitute for a, b, c the values furnished by
these equations in the cubic (Art. 140), it will take the form
bic, 20 0 we ean
(+r {srn-“ ~~ hao,
Hence, when the above conditions are satisfied, the cubic
for determining the directions of the principal chords has two
roots equal to — A, so that of the three systems of principal
chords two may be drawn in any manner parallel to the plane
represented by the above equation. Moreover when the sur-
face has a center the equal values of s lead immediately to
the value of its equatoreal axis (Art. 146), and the remaining
value of s to that of the polar axis,
150. To find the equations to the axis of revolution.
By comparing the terms of the first order in the two
identical equations of the preceding article we get
2a"—-2r\a=—-I(pt+p’), 2b"°-2\B =—m(p+p),
2c" —- 2rAy=—n(p+p');
but mn=a’, In=b*, Im=c’, (3)
*. a (a —da) = 5 (b" -AP) =e (c" —Ay);
or, replacing a, 3, y by a, y, s, and X by its values, the
relations amongst the co-ordinates of the center of the sphere
that always intersects the given surface of the second order in
two parallel planes, become
i A a’a”’ u( : bp”! ( ant
TE Ls emerges garry Pi SB re rrr mh eae recs
nae be Leh DN & +o ay)
which represent a straight line forming the axis of revolution
of the surface; and, as appears from equations (3), perpen-
dicular to the parallel planes. Although the results of this
127
and the preceding Article may be so modified as to embrace
all particular cases, yet when a proposed equation wants
several terms, it will generally be better to apply the direct
investigation to discover whether or no it represents a surface
of revolution.
151. Three diametral planes are said to be conjugate
to one another, when each bisects the chords which are
parallel to the intersection of the two others; and the inter-
sections in that case are called conjugate diameters.
When a surface admits of three planes of this sort and
they are taken for the co-ordinate planes, its equation supposed
algebraic can only contain even powers of the three variables
a, y, x Hence the principal planes of a surface of the second
order having a center, are conjugate to one another, since
the equation referred to them is
Av? + By? +C2? =D;
and this, as we have seen, is the only rectangular system of
co-ordinate planes which can give the equation of this form, or
which can be conjugate to one another.
152. We shall now shew that there is an infinite number
of systems of diametral planes oblique to one another which,
taken for the co-ordinate planes, will make the general equation
of the second order to be of the above form, that is, free from
the terms involving za, wy, yx, and which are therefore con-
jugate to one another,
Let w=mz, y=nz be the equations to any diameter
Ox’ (fig. 50), then the equation to the diametral plane «’ Oy’
conjugate to Oz’, is (Art. 137),
df (x, Y, 2) ii AAC Y, &) = df (x, Ys %) as
x
ae O
iy: dy dx ;
or, mAv+nBy+Cz=0.
Since the diameter O2’ is parallel to the chords which 2 Oy’
bisects it must according to the definition be the intersection of
the two planes which are conjugate to wv Oy’; and those planes
will cut the plane #’Oy’ in two lines Oa’, Oy’, such that if
128
together with Ox’ they be taken for a system of oblique axes,
the equation to the surface will only contain even powers of
the three variables, and therefore be of the form
Ae? + Byy? + Ciz" = D,.
But if we make x’ = 0, the result A,a”’ + B,y? = D, is the
equation to the section a Oy’, and by its form shews that
the axes Ow’, Oy’ are.conjugate diameters of that section ;
hence it follows, that any diameter Ox’ being proposed, if
in the section made by the diametral plane which is conjugate
to it we draw any two conjugate diameters Oa’, Oy’, at
pleasure, we shall determine three planes z’Oa’, x’ Oy’, y' Ox’
which are conjugate to one another; the number of such
systems is therefore unlimited ; and the equation to the surface
when referred to a system of conjugate diameters as co-ordinate
axes, may be put under the form
a’, b’, c’, being the distances, real or imaginary, from the center
at which the surface cuts them.
Cor. If Oz’ be in a principal plane, the tangent plane
at the extremity of Ox’ and consequently the diametral plane
conjugate to Ox’ will be perpendicular to that plane, but not
perpendicular to Oz’. Also if Oz’, Oy’, be axes of the section
w’ Oy’, they will be at right angles to one another; or if Oz"
coincide with an axis of the surface, it will be perpendicular
both to Ow’ and Oy which will lie in a principal plane; but
in no case will three conjugate diameters be mutually at right
angles, unless they coincide with the axes of the surface.
153. We have seen (Art. 138) that any diametral plane
is parallel to the tangent plane, applied at the extremity
of the diameter to which it is conjugate. Hence, in a system
of conjugate diameters, the tangent plane at the extremity
of each is parallel to the plane of the two others. Also,
if through the extremities of each of a system of conjugate
diameters we draw planes parallel to the plane of the two
others, we shall form a parallelopiped, which is said to be
129
constructed upon the diameters, and which in the case of the
ellipsoid will be circumscribed about the surface. We shall
now shew that the known properties of conjugate diameters of
curves of the second order may be extended to surfaces of the
second order.
154. Ina surface of the second order, that has a center,
the sum of the squares of any system.of conjugate diameters
is equal to the sum of the squares of the axes; also the
volume, and the sum of the squares of the faces, of the
parallelopiped constructed on any system of conjugate dia-
meters, are respectively equal to the volume, and the sum of
the squares of the faces, of the rectangular parallelopiped
constructed on the axes.
2 y 2 ,
Let wtiatat 1 be the equation to a surface of
the second order, referred to a system of conjugate diameters,
the inclinations of which are Zyz=A, Z2v=p, Ley =r;
and let r=mz, y=nz, be the equations to any diameter ;
then the equation to the diametral plane conjugate to this
diameter is
me ny z
TE DER LES Fae topo de a ae
a b c
and if it be perpendicular to the diameter, we have (Art. 86)
mec*(1 + mcosu + ncosr) = a?(m + CoS v + COS 1),
ne*(1 + mcosu +n cos)) = 6" (nm + m cos v + cos dX).
Let r =c"(1+mcosp + 2COSA) 3;
“mr = a" (m +n Cos vp + COS p),
nr° = b?(m + mcos v + 0s A) 3
Les 10. va Ne
id Sate & +—+ =] =1+m’+n?+2mncosyv+2mMcosu+2N Cosas;
1
a’
*. (as in Art. 146) ris the length of the semidiameter whose
a
130
equations are a= mz, y=nzx. Also if we determine m and
nm from the two latter equations, and substitute them in the
former, the result is
(7° —a’*) (7? —b”) (r? —c’*) —a’*b? (1? —c”) (cosy)? —a’*c?(r* —b”) (cosu)?
— be? (r* — a”) (cos d)* — 2ab”c? cos p cos vy cosA = 0,
or 7° — 7" (a? +b" + 6") 47°f (a'U'siny)’?+(a'c'sinu)?+(b'c'sind)”}
— (a'b’c’)*$1 + 2cosd cos u cosy — (cosd)* — (cos)? — (cos v)*} =0,
But if a, b, c be the three semiaxes of the surface, then
a*, b*, c are the values of r? in the above equation; there-
fore, by the theory of equations, we have
a? +b7° 4+ Pra V+ +0’,
(a’b’ sin v)?+ (ac sinu)?+ (0'e' sind)? = (ab)? + (ac) +(bc)*.... (2);
(a’b’c’)* $1 + 2 cos X Cos pm Cos v
— (cos \)? — (cos nx)” — (cos v)*} = (abc)’.....(3),
which are the three required results, the first member of
the third being the well-known expression for the square of
the volume of a parallelopiped in terms of its three edges,
and their inclinations to one another.
Cor. When one or two of the axes are imaginary,
there will be an equal number of the conjugate diameters
imaginary; and it will be necessary to change the signs of
the squares of these axes and these diameters in the above
equations.
155. The preceding results may also readily be arrived
at by the following method.
Let Ow, Oy, Ox (fig. 29) be the semiaxes of the sur-
face a, b, c; Ow’, Oy’, Oz’ any system of semiconjugate
diameters a’, 6’, c’; let the plane of a’y’ intersect that of
avy in the 4 diameter Ow,=a,; also let Oy,=b, be the
semidiameter of the curve #,x’, which is conjugate to Oa;
a+b, =a? +b”
131
Now the plane aw Oy' is by supposition conjugate to 0,
therefore Ox’, Oy,, Ox,, form a system of conjugate diameters,
or Ow, is conjugate to the plane x’ Oy.
Let the plane x’ Oy, intersect wy in the semidiameter
Oy, = b,, then this plane must contain Ox; for being con-
jugate to Ow, in a principal plane, it must be perpendicular
to that plane (Cor. Art. 152); hence Ox,, Oy,, Ox form a
system of semiconjugate diameters, (for Oz, Oy, are the 4 axes
of the section y,y,%) and any two are semiconjugate diameters
of the plane section in which they are situated ;
U0; Ho =0, ey
a+b =a + bi;
therefore, adding these equations to the former,
a’ “fs b? += q? ib? 6.
Again, employing the same auxiliary systems of diameters
as above, and denoting each parallelopiped by its three edges,
we have
vol. (a’, b’, c’) = vol. (a, be, ¢’),
for these figures have the same altitude, viz. the perpendi-
cular from ’ on the plane of «’y’, and equal bases, viz. the
parallelograms constructed on the two systems of 4 conjugate
diameters a’, 6’, a,, b,, belonging to the same curve in the
plane of ay’. Similarly,
| vol. (a, b,, c’) = vol. (a), 0, ¢),
vol. (a), bj, c) = vol. (a, 6, c) ;
.. vol. (a, b’, c) = vol. (a, 8, c).
Cor. If the conjugate diameters 2a’, 2b’, 2c’ are each
equal to 2R, then 3R*? =a’? +b’?+c’; also since there are
only two equations, viz. (2) and (3) Art. 154, to determine
the angles of inclination A, uw, v of the conjugate diameters,
there may be an infinite number of systems of equal conjugate
diameters, their extremities all lying in the intersection of the
surface and a concentric sphere radius = R.
9—2
132
And of all systems of conjugate diameters of an ellipsoid
the principal diameters have their sum a minimum and the
equal diameters their sum a maximum. For in any proposed
system, if there were two not perpendicular to one another,
by substituting for them the axes of the section of which they
are diameters, we should obtain a system whose sum is less
than that of the proposed system. Again, if there were two
not equal to one another, by substituting for them the equal
diameters of the section to which they belong, we should
determine a system whose sum is greater than that of the
proposed one. And in this manner we could shew that no
oblique system could have their sum a minimum, and no
unequal system their sum a maximum.
156. In surfaces of the second order not having a center,
represented by the equation to rectangular co-ordinates
By + Cz2*°=2/4' a,
the planes of xa and vy only are diametral and principal,
each bisecting perpendicularly the chords parallel to the
intersection of the other with the plane of yz, which is the
tangent plane at the vertex. In this case no three diametral
planes can be conjugate to one another; for, taking a system
of chords parallel to a line whose equations are v = mz, y = 72;
the equation to the diametral plane conjugate to them is
Therefore all the diametral planes are parallel to the axis
of a, and consequently their intersections, which are the
diameters of the surface, are parallel to that axis, and cannot
therefore form a system of co-ordinate axes.
But we can find an infinite number of oblique co-
ordinate planes related to one another in the same manner
as the three rectangular planes mentioned above are, viz. so
that two, ’# and ay’, shall be diametral planes, and each
of them conjugate to the chords parallel to the intersection
of the other with y's’; the latter being a tangent plane to
the surface at the extremity of the diameter in which 3’ a’
133
and a’y’ intersect; and the equation to the surface when
referred to them will therefore preserve the same form.
For let a’ O'y’ (fig. 51) be the diametral plane represented
by equation (1), and which, being parallel to the axis of the
surface, cuts it in a parabola AO’B. In AO'B take any point
O’, and draw O’’ parallel to the chords to which the diametral
plane is conjugate. Then the two co-ordinate planes which
are to go along with this diametral plane will pass through
O's’, and cut the diametral plane in two lines O'v’, O’y’, such
that being taken for the co-ordinate axes they shall give the
equation to the surface under the form
Byy® + Cis- = SA it.
Therefore, making ’ = 0, the equation to AO’B is
By” = 2 Aa’.
which by its form shews that it is referred to a diameter
of the parabola and a tangent to the parabola at the extremity
of that diameter. Hence the position of the second diametral
plane x’ O'x’, and also of the third co-ordinate plane x’ O'y’,
is determined, which latter, since it passes through two tangent
lines to the surface O'y’ and O'’, is a tangent plane to the
surface at O'; and as not only the plane 4O’B is arbitrary,
but also the position of the point O’, the number of systems of
oblique co-ordinate planes similar to the above is unlimited.
157. When the general equation of the second order
represents a paraboloid, to find the position and magnitude of
the surface.
Let aa’ + by? + c2x* + 2a'yxz4 2b ax4+2Cuy+2a' a
+ 2b’y4+2c's+d=0,
be the equation, then
D = aa’ + bb”? + cc? — abe — 2a'b'e' =0;
consequently the cubic which determines the directions of the
principal chords has one root s=0, and the corresponding
values of m and 7 are (Art. 140),
m(ab—c”) =a'c — bb, n(ab—c”*)=)c'- ad’.
134
Since the plane bisecting the chords for which s =0 is at an
infinite distance, these chords are in the direction of the axis
of the paraboloid; and therefore the equations to a line
through the origin parallel to this axis are
A Y Z
MLS PA ea a OL)
ac—-bb be-—ad ab-c®
Also if w,y,% denote the co-ordinates of the vertex,
expressing that the normal at that point must be parallel
to (1) we get the equations
1 du 1 du 1 du
ab—c? dz ac’—bb' dz wc'-—aa’ dy’
which, together with the equation to the surface wu = 0, will
determine a, y, x the co-ordinates of the vertex.
Next to find the values of the parameters of the principal
sections, let
’ , M
aa + by? + cx? + 2ays + 2az + 2Cuy + Dies
where M=a"N+b’N'+c"N"4+dD (Art. 134), be the
equation to a surface of the second order referred to its center ;
then the roots of the equation
D D? 4
s+(a+b+0c) By peat as Lae teat ad a oe) ps _- 0,
PD D? D'
“or gf ln Feit ae - ays 7 0 (Att 146)
are the squares of the reciprocals of the semi-axes a, 3, +.
Now suppose D very small, then one value of s will be very
1 1 1
small, let this be =; and the remaining values of s, ead
a p yy”
will be nearly equal to the roots of
PD QD
s 4+ ——'s + ——_ = 0;
uM** MM?
aye 1 PD\? Pet
eal er ae I
135
D' 1 QD?
(aByy MP’ (Bye MP’
we war tt Q Q’ ak \ tea. C)*
[oe eed ~
Consequently, the squares of the reciprocals of the 4 para-
meters of the ue ag sections of the paraboloid, being the
] 2 Q .
and’ @* = Dr” since
values af poh —
eo 4.9
p= ("= 2Q) =. p+
when D = 0, are the roots of the quadratic,
Q'
im
If 1,7, be those 4 parameters, we have
1 ut (Jay Jakes Cle
pat Cpe ogy! 2S
pt pn ' Var Wr aw
et We Es -
sa ll Q ?
which shews that paraboloids represented by the general
equation of the second order, have the parameters of their
principal sections in the same ratio, or are similar to one
2
another, when has the same value. Also the surface will
be an elliptic or hyperbolic paraboloid according as Q is posi-
tive or negative; for as a is a real semi-axis and therefore a’
positive, Q and M must have the same sign in order that any
surface may exist.
Cor. Similarly we may find the conditions that two
central surfaces of the second order may have their axes
proportional, or be similar to one. another.
As above, the cubic equation
. 1 e e e e
will have roots —, Let this equation in which we
a
1 1
136
may suppose M = 1 without altering the relations of its roots
to one another, be transformed into another whose roots are
1 it ;
(Seen tee) male Be
4 \B Wal ko” Wa pede
the result will be found to be
PQ jie aig 8S es )
Ey) | Slap +3)@-—|——4+1] =0.
ale a) 16\D? D* D* as
. ae ah eee
Now the values of ¢ depend only on the ratios =, — and —;
SRO fi 8
in
consequently these ratios will be the same, provided > and
el 5 hy ae
es =p have the same values; these therefore are the con-
ditions for central surfaces, when represented by the general
equation of the second order, having the ratios of their axes
the same, or being similar to one another.
SECTION VII.
ON CYLINDRICAL, CONICAL, AND CONOIDAL SURFACES, AND
ON SURFACES OF REVOLUTION.
158. Brrore proceeding with the proper subject of this
section it may be useful first to introduce the following
remarks relative to the number of constants in the equations
to surfaces, and to the intersection of surfaces.
If the general equations of the n™, (m — 1)" &c. orders
between two variables x and y be formed, and multiplied
respectively by the quantities 1, z, 2’, &c. %", we shall by
adding all such products together form the general equation of
the 2" order between three variables a, y, x; and the number
of constants which this equation will contain will be
E(m+1)(n+2)+4n(n+1) +h (m—-1)n+...4+$1.2
=1(m +1) (n+ 2) (m + 3).
Consequently, diminishing this by unity we shall have the
number of independent constants in the general equation of
the 7" order between three variables expressed by
N=1n(n?+6n+ 11);
which is also the number of points through which a surface
of the nm order can be made to pass, for the co-ordinates
of every one of these points being substituted in the equation
to the surface would furnish N linear equations for the
determination of the constants.
Let w=0, v =0, be the equations to two surfaces that
intersect; then w+XAv=0, where XA is an indeterminate
constant different from zero, represents a surface passing
through their curve of intersection, since it is satisfied by
all values of wv, y, and x that simultaneously satisfy the
equations w= 0, v = 0.
138
Also if w= 0, v =0, be the equations to two surfaces of
the mn order passing through N—1 given points, then
uw+2Xv=0, since it contains an additional constant ), in-
cludes all the surfaces of the m' order that can pass through
the given points, which shews that all those surfaces will pass
through the intersection of any two of them; therefore all
surfaces of the 2“ order that pass through NV — 1 given points,
have a common curve of intersection.
And if a surface of the m™ order be made to pass through
a number of fixed points two less than the number sufficient to
completely determine it, the surface will also pass through an
additional number of fixed points such that added to the
former it makes up m* the entire number of points in which
three surfaces of the n“ order can intersect one another.
Since an infinite number of surfaces of the m™ order can
be described through the N —2 given points, consider any
three of them whose equations are wu =0, v=0, w =03 then
the equation w~+Av+uw=O0 (1) where A and pw are in-
determinate constants, will include all the surfaces of the
nm order that can pass through the given points, because the
equation of every such surface could involve only two un-
determined constants. But equation (1) will be satisfied by
every system of values of x, y, and x that simultaneously satisfy
u=0,v=0,w=0; that is, all the surfaces will pass through
the points of intersection of any three of them; therefore
all the points of intersection must be fixed points, and the
N —2 given points will determine the n?— N + 2 remaining
fixed points of intersection. |
Again, if the system of equations u,=0, v,=0 taken
together represent the curve of intersection of the surfaces
U, =0, v,=0; and w,=0, v,=0 taken together represent.
the curve of intersection of the surfaces of w,=0, v,=0; then
will 2, + AVv,v; = 0 represent another surface which passes
through both these curves of intersection, and is of the order
indicated by the greater of the quantities ” or r +s.
Hence if v=0, v =0, be the equations to two planes
intersecting a surface w=0; then «+)vv'=0 is the equa-
139
tion to a surface passing through the two plane curves of
intersection; and if the planes become coincident, w+? = 0
represents a surface touching w= 0 along the curve in which
the plane v = 0 cuts it.
A surface of the m‘" order may in general have n (n — ii
tangent planes applied to it passing through a fixed straight
line.
Let V=0 be the equation to the surface, then if its
tangent plane at wysz contain the straight line
A=mZ+a, Y=nZ +0,
we must have
dV dV adaV
— 0
Ce Tae
dV dV dV
— vw) —+(b-y)—-2—=0 (1).
(@- 0) T+ 0-5-8 5-9 O
But if V=w+v=0, so that w is the sum of all the terms of
m dimensions in V, we have by a property of homogeneous
functions
oes, Whe, du pros By
pad pote z—_= = =
da "dy dz
therefore equation (1) is reduced to
Puls Cael dv aay,
———w | Sant av —- ——._- ———_—_- _ a se ad =
de dev dy - dy dz d
which is of the (z — 1)" degree; and the other two equations
from which together with (1) the co-ordinates of the points of
contact av, y, x are to be determined by elimination, are of the
n™ and (n —1)™ degrees; therefore the degree of the final
equation will generally be m(—1)*, which expresses the
number of the points of contact, or of tangent planes that can
be drawn through the fixed line.
159. In the preceding sections we have considered
several instances of surfaces generated by a line, straight or
curved, which so moves and changes its position or form,
as constantly to pass through one or more fixed curves or
directrices. We shall now extend the same considerations
140
to the general case of a generating curve represented by the
equations
SCAR PA EN aah Pi Oy Opec eyed Ra 6
containing two variable parameters, a, $8, and subject to
pass through one fixed curve or directrix. The species
of the generating curve is determined, because the func-
tions f, f,, are supposed to be known, but its position and
dimensions will change corresponding to the different values
of a and 3; and it will generate a solid if the parameters
a and £ vary independently of one another. Thus the circle
represented by
+(y-By=ce-a’, v-a=0,
gives, by the elimination of a, the equation
S++ (y- Blac’,
which represents a sphere, radius c, and center in the axis
of y; and if £6 receive all values from 0 to + o, the equation
will belong to all the points of a solid cylinder, radius c, and
axis coinciding with the axis of y.
But if we suppose the generating line (1) in every position
to have a point in common with a fixed curve or directrix
represented by the equations
y= (#); s=vV(a),
then the equations to the generating line and directrix must
be simultaneously satisfied by the same system of values for
xv, y, * belonging to that common point; if therefore we
eliminate #, y, x between them, we shall have 3 = F(a),
the relation between the parameters, in order that the gene-
rating line may meet the directrix. Hence, the generating
line in any position will be represented by the system of
equations
S(® Ys % a, B)=0, fi(*, y, % a; 6B) =0, B= F(a).
If therefore we eliminate a and 8 between them, that is,
if we determine a and 3 from the two former in terms of
Xv, Y, %, so that B =u, a=v, and substitute in the latter,
141
we shall have w= F'(v) for the equation to the surface; and
we observe that w and v do not change for surfaces of the
same family, that is, for those which admit the same gene-
.rating line, but that F', which depends upon ¢ and y, changes
for each Pcichion! surface of the family.
These considerations we shall now apply to several of the
more common cases, where the generating line is a straight
line, or a circle.
160. ‘To find the general equation to cylindrical sur-
faces.
A cylindrical surface is generated by a straight line
which moves parallel to itself, and always passes through
a given curve.
Let the equations to the generating line in any position
be v=mz+a, y=nzx +f, a and B being variable quantities
depending upon that position, and m and m constant quantities
since the line is always parallel to itself. Also let y= P (a),
#% =) (w), be the equations to the directrix or curve through
which the generating line always passes, and a’, y’, x’ the
co-ordinates of the point in which they meet; then a’, y’, 2
must satisfy both the equations to the directrix, and gene-
rating line ;
.a@ems ta, y=ne +B, y=O(v), x=’);
now by means of these four equations, we may eliminate
a’, y’, s, and there will remain 8 = F(a); but B=y- nz,
a=LX—-Ms;
. y—-nv=F(#- ms),
a relation among the co-ordinates of any point in the gene-
rating line, and therefore the equation to the surface which it
describes. In this case the quantities w and v are y — nz, and
« — mz, which remain the same for all cylinders, whilst the
function F' will alter with the different directrices employed.
Cor. If for the purpose of eliminating the arbitrary
function, we differentiate the equation y—nz = F'\(# — mz)
successively with respect to # and y, we find
—np=F'(4@—mz)(1-—mp), 1—-nq =F" (a — mz) (— mq) +
142
therefore, dividing one result by the other,
np 1— mp dz dz
= > or m—+n— =I,
1—nq mq dx dy
which is the differential equation to cylindrical surfaces; we
may however obtain it more easily by the consideration of the
tangent plane, as follows.
161. To find the differential equation to cylindrical sur-
faces.
One of the distinguishing properties of cylindrical sur-
faces is, that the tangent plane, since it always contains a
generating line, (Cor. 2 Art. 108) is always parallel to a fixed
straight line.
Let 2 = ms’, y’=nz' be the equations to a line through
the origin, to which the generating line is always parallel; then
this line is parallel to the tangent plane whose equation is
s—s=p(e#—x#)+q(y'-y);
dz d
. (Art. 23) mp+nq=1, or mn Ere 1, as before.
Cor. This equation may be employed to discover whether
the surface represented by a proposed equation w = 0, is cylin-
drical or not. For, obtaining the values of p and q, as in Cor.
Art. 107, and substituting, we have
du du du
WP 9 0,
x z
which must be satisfied for all points of the surface, that is,
for all values of w, y, x, if the surface be cylindrical; hence
we must equate to zero the coefficients of the different powers
of the co-ordinates in it, and examine whether the resulting
conditions can be satisfied by real values of m and n.
162. Having given the direction of the generating line,
to find the equation to the cylindrical surface which envelopes
a given curve surface.
143
We have seen that whatever be the nature of the directrix,
the equation mp+nq=1, must subsist between the differential
coefficients p, g, derived from the equation to a cylindrical
surface. But at the points where the cylinder touches the
surface, the values of p, g, are the same for both ; therefore at
those points the above relation subsists between the differential
coefficients derived from the equation to the surface; that is,
the co-ordinates of the points of contact are such that the
equation mp + nq =1 is satisfied. Hence if we differentiate the
given equation to the surface «=0, and write the values
thence obtained of p, g, in the above equation, the result
together with the equation to the surface, will be the equations
to the directrix ; and we know the direction of the generating
line, and therefore can find the equation to the required
surface by Art. 160,
163. To find the general equation to conical surfaces.
A. conical surface is generated by a straight line which
passes through a given point, and always meets a given curve.
Let the co-ordinates of the given point or vertex be
as Cs
@—-a=a(s—-c), y-b=B(s-0),
are the equations to the generating line in any position, a and
f being variable quantities depending upon that position.
Also let y = ¢ (#), % = W (@#), be the equations to the directrix
or curve through which the generating line always passes,
and a’, y’, x, the co-ordinates of the point in which they
meet; then a’, y’, x must satisfy both the equations to the
generating line and directrix,
wv —-a=a(%—-c),y -b=B(e'-0¢), y= (#), X= (e’).
Now by means of these four equations we may eliminate
we, y’, x, and there will remain 8 = F(a);
=U ioe
but 6 =~ sat ease ip
?
yw — 6 w=
14.4
J Y= (2)
Reb EG
a relation among the co-ordinates of any point in the generating
line, and therefore the equation to the surface which it de-
scribes. When the vertex is situated in the origin, the
equation is reduced to Pay (<) , which expresses that the
@ Pe
equation is homogeneous in w, y,%. It is easily seen that if the
directrix be the curve of intersection of two surfaces of the m™
and 2" orders, the equation to the cone will be of the mn" order.
Cor. If from the above equation we eliminate the
arbitrary function by differentiation, as in Cor. Art. 160,
we find
tee d
zen 7 (@—a)t 7 yd),
which is the differential equation to conical surfaces; but
which may be obtained more easily by the consideration of
the tangent plane, as in the following Article.
164. To find the differential equation to conical surfaces,
The distinguishing property of conical surfaces is, that the
tangent plane, since it always contains one of the generating
lines (Cor. 2 Art. 108), always passes through the vertex.
Let a, b, c denote the co-ordinates of the vertex, then
they must satisfy the equation to the tangent plane at any
point wyz; that is, x —s=p(a —«x)+q(y'—y) must be
satisfied by # =a, y =b, x'=c; therefore, whatever be the
nature of the directrix, the differential coefficients p, g, derived
from the equation to the surface, must be such as to satisfy
the equation
x-c=p(w—a)+q(y—4),
which is the differential equation to conical surfaces.
Cor. As in cylindrical surfaces (Art. 161), this equation,
when put under the form
du du du
eae \ eee ny EMRE Cy ee
145
may be employed to discover whether a proposed equation
u = 0 represents a conical surface or not; a, 6, ¢ being the
unknown quantities to which must be applied what was there
said relative to m and n.
165. Having given the position of the vertex, to find
the equation to the conical surface which envelopes a given
curve surface.
Whatever be the nature of the directrix, we have seen
that the differential coefficients derived from the equation to a
conical surface must satisfy the equation
s—-ce=p(w-a)+qty—); (1)
but at the points where the cone touches the surface, the
values of p,q, are the same for both; therefore at those
points the above relation subsists between the differential
coefficients derived from the equation to the surface, that
is, the co-ordinates of the points of contact are such, that
the above equation is satisfied. Hence if we obtain values
of p,q, from the given equation to the surface, and sub-
stitute them in the above equation, the result, together with
the equation to the surface, will be the equations to the
curve of contact of the two surfaces, or to the directrix ;
and we know the co-ordinates of the vertex, and can therefore
find the equation to the required conical surface by Art. 163 ;
and it will be of the m(m —1)™ order, if the given surface be
of the v" order, because (1) will become of the (7 — 1)" order,
in the same way as equation (1), p. 139.
Cor. 1. If the vertex of the cone be considered as a
luminous point, the curve of contact whose equations we
have just found, is that which on the surface separates the
illumined and obscure parts; if it be considered as the place
of the eye, the curve of contact is the line of the apparent
contour of the surface.
Cor. 2. Let w= 0 (1) be the equation to a surface of the
second order; then the curve of contact of the enveloping
cone is easily shewn to be a plane curve determined by an
equation of the first order, v =0 suppose. Hence w +2 .v* =0(2)
is another surface of the second order whose tangent cone
10
146
from vertex (a, b, c) is identical with that to surface wu =0
from the same vertex, whatever be the value of the constant ) 3;
for from Art. 158 the surfaces (1) and (2) touch one another
along the curve in which they are intersected by v = 0, and
therefore each is touched by the same enveloping cone along
that curve. Now no surface of the second order can pass
through the vertex of its enveloping cone; for if it did, that
cone would become the tangent plane; therefore if > be
determined so as to make (2) pass through the vertex of the
cone, the equation w + Av” = 0, will represent the enveloping
cone of the surface wu = 0.
Similarly, if v =0 be the plane of contact of the envelop-
ing cylinder having its axis in a given direction; and 2d be
determined so as to render infinite that radius vector of the
surface % + \v* = 0 which is parallel to the axis of the cylinder,
we shall obtain the equation to the cylinder enveloping a
surface of the second order.
166. To find the general equation to conoidal surfaces.
A conoidal surface is generated by a straight line which
moves parallel to a given plane, and always meets a given
fixed straight line, and ‘a given curve.
Let the given plane, called the directing plane, be taken
for that of wy, and the point in which the straight directrix
OA (fig. 54) meets it, for the origin; then the equations to
OA will be r=mz, y=nz. Also let the equations to the
curvilinear directrix BC be y=@(@), s =W(a); and let
s=B, y=ax+- be the equations to the generating line
DC in any position, since it is always parallel to the plane
of vy; then since the generating line always meets OA, we
must have
nbB=amB+y;
therefore, eliminating y, the equations to the generating line
become
v= B, y-nB=a(w—mB);
we must next express that it meets the curve BC, whose
147
equations are y=@(w#), x= (wv); therefore, eliminating
x,y, %, we have (3 = F(a); hence the generating line in any
position will be represented by the system of equations
e=B, y-nB=a(e-mp), B= F(a);
therefore, eliminating a and , the equation to the surface
generated is
g- (Zo).
C— MZ
Cor. 1. This surface is twisted, that is, no two consecu-
tive generating lines are in the same plane; for the line DC
in passing to the consecutive position D’C’ may be supposed
to glide along the tangent C7’; therefore, in order that DC
and D’C” may be in the same plane, OA and CT' must be in
the same plane, which cannot happen for all the tangents at
least, unless the curve BC is entirely in the same plane with
AO, in which case the surface generated will be a plane,
If the straight directrix be perpendicular to the directing
plane, m=0, m =0, and the equation to the surface, which
is then called a right conoid, becomes z = r(2); at which
we may readily arrive by a direct investigation, as the
equations to the generating line will be s =, y=aa. Also
it is manifest that the equation to the oblique conoid will
a ; :
assume the form z= r(2) if we refer it to a system of
&
oblique co-ordinate axes, of which OA is one, and any two
lines drawn through O in the directing plane are the others.
Cor, 2. If from the equation z = P(t) we elimi-
L—-Ms
nate the arbitrary function by differentiation, as in the
former cases, we find
p (w@—ms) + q(y— nz) =0,
the differential equation to conoidal surfaces; at which we
may arrive more easily as follows.
LO
148
167. To find the differential equation to conoidal
surfaces.
Their characteristic property is that the tangent plane at
any point contains the generating line which passes through
that point, (for it contains the tangent lines to all curves
traced on the surface through that point, and a straight
line is its own tangent,) that is, a line parallel to the plane
of wy, and intersecting another line whose equations are
a =m, y’=nz'. Now the equation to the tangent plane at
a point vyz is
x —s=p(a-x)+q(y'-y),
and if we intersect it by a plane parallel to ry at a dis-
tance x from it, their line of intersection will have for its
equations
s'=s, p(w—x)+q(y-y) =0,
and since this is coincident with the generating line, it will
intersect the rectilinear directrix of which the equations are
wv =m, y=ne';
p(w-—msz)+q(y—nz)=0,
which is the relation that must be satisfied by the co-ordinates
of every point in the surface.
For a right conoid, or for an oblique conoid referred
to the rectilinear directrix as one of the axes, the differential
equation is
p“e+qy=d0.
168. Having given the rectilinear directrix, to find the
equation to the conoidal surface which envelopes a given
curve surface.
As in the former cases, if %=0 be the equation to the
given surface, the curve of contact which forms the curvi-
linear directrix, will be given by the equations
du du
0, Cl atisted eg 88) ED arte
149
which must be employed instead of the equations y = ¢ (2),
x =wW(w#); the remainder of the process will be the same
as that explained in Art 166.
169. To find the general equation to surfaces of re-
volution.
A surface of revolution is generated by a curve revolving
about a fixed axis. If the curve be situated entirely in a
plane passing through the axis, then every section of the
surface through the axis, which is called a meridian, will
reproduce the generating curve; otherwise, the section will
be different from the generating curve. Also every section
perpendicular to the axis, which is called a parallel of the
surface, will be a circle whose center is in the axis and
plane perpendicular to it, and whose perimeter passes through
the generating curve; hence a surface of revolution may be
supposed to be generated by a circle whose center moves
along a fixed straight line to which its plane is perpendicular,
and whose perimeter always passes through a given curve.
The advantages of the second mode of generation are, that
the moveable circle is a generating curve of an invariable
nature and common to all surfaces of the class, whilst the
curve through which it always passes is a variable directrix
which changes for each particular surface, agreeably to the
mode of generation occurring in every instance in this section.
Let v-—-a=m(z-c), y-b=n(s—- Cc), be the equations
to the axis or fixed straight line along which the center of
the generating circle moves, and which we suppose to be
drawn through a given point (a, 6, c) in a known direction.
Then every one of the parallels of the surface may be re-
garded as the intersection of a plane perpendicular to this
axis, with a sphere whose center is any fixed point in the
axis, the point (a, b, c), for instance, and whose radius varies
so as to give the section of the proper magnitude; conse-
quently the equations to the generating circle will be
Neate ()
(vw — a)? + (y — b)? + (ew — cp whis,0 ate ;
ll
7
150
Let y = ¢ (x), = = (2), be the equations to the directrix
or curve which the perimeter of the generating circle always
meets; then these four equations will be satisfied by the same
system of values of #, y, x, which are the co-ordinates of the
point of intersection; hence, eliminating a, y, x from them,
we find B = F(a) the condition to be joined to the equations
(1) in order that the circle which they represent may really
be a parallel of the surface; hence, eliminating a and 3 from
the equations to the parallel, we find
s+me+ny =F }(e—a) + (yb)? + (z- 6),
a relation among the co-ordinates of the generating circle in
any position, and therefore the equation to the surface which
it describes.
Cor. 1. If the axis of revolution coincide with the axis
of zg, and we take the center of the sphere for origin, we
have
m=n=0, a=b=c=0, and z= F(a#+y¥ + 2°),
or x= f(a" +y’),
as we have already seen, Art. 52. In this case it is more
convenient to regard each parallel as the intersection of a
right cylinder with a plane perpendicular to it, that is,
instead of equations (1), to use
= B, a +y’ =a,
which joined to the usual relation 6 = F(a), give imme-
diately
s= F(a’ + y’).
Cor. 2. If we differentiate the above equation with
respect to w and y, and divide one result by the other,
so as to eliminate the arbitrary function, we find
m+p «w-a+(x—c)p
m+q y-b+(s—e)q’
or pjy—b—n(z—c) —q}w-a—m(z-c)} =n(w—a)—m(y-b),
the differential equation to surfaces of revolution, at which
we may also arrive as follows.
151
170. To find the differential equation to surfaces of
revolution.
Their distinguishing property is, that the normal always
meets the axis of revolution. This will be readily seen if
we consider that the tangent plane at any point necessarily
contains the line touching the parallel at that point, and is
therefore perpendicular to the meridian plane passing through
that point; consequently the normal will lie in the meridian
plane, and therefore meet the axis.
The equations to the normal at a point wyxz are
eo —-ve+p(2'-x)=0, y-y+q(e' -2) =0;
let, as above, the equations to the axis be
wo —-a—m(x —c)=0, y —b-—n(s'-c)=0,
then if these two lines intersect, their equations must be
satisfied by the same values of a’, y’, 2’, which will be
the co-ordinates of their point of intersection. Hence if we
eliminate a’, y’, 2’, the result will be the differential equation
to the surface. Subtracting, we find
m (x'—c)+p(2’-2) -@ +a=0, n(x’ —c) +q(%—2)-yt+b=0,
or, (m +p) (x —2) +m(x—c) -v%+a=0,
(n +q)(s —s) +n (e@-c)-yt+b=0;
-, eliminating x’ — x, we find as before
(m+ p) {n (# —0) —y +b} = (m4 q) {m(w—c)-w+a}...(1);
which expresses that a surface is one of revolution about an
axis passing through a point a, 6, c, and having a direction
determined by the quantities m and 7.
If the axis of revolution coincide with the axis of z, the
equation becomes
py—-gqr=d0.
Cor. The above equation will enable us to ascertain
whether a proposed equation w=0, represents a surface
of revolution; fer, deducing the values of p and q, and
substituting them in (1), the result ought to be true for
152
all values of w, y, x. Hence we must equate to zero the
coefficients of the different powers of the co-ordinates, and
we shall obtain among the unknown quantities a, b, c, m, n
a certain number of equations, which must be consistent
with one another, in order that the surface may be one of
revolution. The process applied to the general equation of
the second order will furnish the conditions already obtained
(Art. 149) by a different method.
171. Having given the position of the axis, to find the
equation to the surface of revolution which shall envelope a
given curve surface.
As in the similar cases which have preceded, we must
begin by finding the equations to the curve of contact
which will be the directrix of the moveable circle. Let
wz=0 be the equation to the proposed surface, then at all
points along the curve of contact the tangent planes of the
two surfaces are coincident; and therefore the values of p
and q derived from w = 0, and substituted in (1), must satisfy
it for all points in the curve of contact; the result of this
substitution together with w= 0, will therefore be the equa-
tions to the directrix, and then the equation to the required
surface of revolution may be found as before.
It is manifest that we thus determine a surface which,
if the proposed surface revolved about the given axis to
which it was fixed, would touch and envelope the proposed
surface in every position.
172. The above are the principal propositions relative
to the commoner families of surfaces which admit only one
directrix, or only one of a variable form; for though Conoidal
surfaces have two directrices, yet one of them, namely the
rectilinear one, is constant for all surfaces of the family.
Consequently, for Conoidal, as well as for all the other
families of surfaces considered in this section, the finite
equation involves only one arbitrary function, and the dif-
ferential equation cleared of the arbitrary function, is only
of the first order.
SECTION VIII.
ON SURFACES HAVING MORE THAN ONE ARBITRARY CURVI-
LINEAR DIRECTRIX, AND ON DEVELOPABLE SURFACES, AND
ENVELOPES,
173. In the preceding Section we considered several
surfaces where the equations to the generating curve con-
tained two variable parameters; and the necessary dependence
of these parameters upon one another, in order that a surface
and not a solid might be generated, was established by making
the generating curve in every position have a point in common
with a given fixed directrix. The resulting equation to the
surface consequently involved but one arbitrary function; and
the differential equation, characterizing the nature of the
surface independent of that directrix, was of the first order.
We shall now take a more general view of the subject of
the generation of surfaces, and suppose the equations to
the generating curve
St (@, Yo By As Bs Y &e.) = 0, fi (a, Y, Bs a, B; ie &c.) = 0,
to contain any number 7 of variable parameters a, 9, y, &c.
This curve will not, by passing through all the various
positions and forms corresponding to the various values of
the parameters, generate a determinate surface, unless we
subject it to such conditions as will leave only one of the
parameters, a for instance, arbitrary ; and this may be done
by making it, in every position, pass through m —1 curves
represented by the equations
y=P(%) F=W(*%)s Y= hilt) x= VWn(a); &e.
For in order that it may have a point in common with the
first directrix, its two equations and the equations y = ¢ («)
z= (a) must be satisfied by the same system of values
of w, y, x, which are the co-ordinates of that point; there-
fore, eliminating w, y, * between these four equations, we
154
find F(a, 3, y, &c.)=0, which expresses the necessary
dependence among the parameters in order that the gene-
rating curve may pass through the first directrix. In the
same manner, each directrix will furnish a fresh relation
between a, 3, yy, &c.; so that to represent completely the
generating curve passing through the 2 — 1 directrices, we
must join to its two equations, the 2 —1 equations of con-
dition F' (a, B, y, &c.) = 0, F, (a, B, y, &c.) = 0, &c. And
if from these latter we suppose $B, y, &c. to be all de-
termined in terms of a, and substitute their values in the
equations to the generating curve, those equations will then
involve only one parameter; and if we finally eliminate that
parameter between them, we shall arrive at an equation con-
taining no arbitrary quantity and representing a determinate
surface generated by the moveable curve.
Hence the final equation will contain as many arbitrary
functions as there are directrices, and therefore the differential
equation characterizing the surface independent of the di-
rectrices and consequently not involving the functions, will be
of an elevated order generally exceeding the number — 1 of
functions, and never less than 22—3. ‘These principles we
shall now illustrate, confining ourselves, however, to the cases
where the generating curve is a straight line.
174. Surfaces generated by a straight line are divisible
into two classes, each of which has distinct properties; viz.
into twisted surfaces, where two consecutive positions of the
generating line are never in the same plane; and into de-
velopable surfaces, where two consecutive positions of the
generating line are always in the same plane.
One remarkable difference in the properties of these two
classes has reference to their contact with the tangent plane ;
in both, the tangent plane at any point (since it contains
the tangent lines to all curves traced on the surface through
that point, and a straight line is its own tangent) must
contain the generating line passing through that point; but
in twisted surfaces, the tangent planes along a generating
line are all distinct and each touches the surface only in
155
a point; whereas in developable surfaces, it is one and the
same plane which touches the surface at all points along
a generating line.
For let DC, DC’ (fig. 54) be consecutive positions of
the generating line of a twisted surface; BC, ED sections
of the surface through any points D and C, and TC, AD
tangent lines to those sections. Then the tangent planes
at all points along DC contain DC; and at D and C the
tangent planes are ADC, T'CD, which cannot be coincident
unless AD, T'C are in the same plane; but in that case D’C’
(since between the positions DC, D’'C’, the generating line
may be supposed to glide along 4D, CT’) would be in the
same plane with DC, which is impossible since the surface
is twisted. But if the surface be developable, then since
D'C’ is in the same plane with DC, the tangents 4D, TC
are in the same plane, and therefore the tangent planes at
D and C, as well as at every other point along DC, are
coincident. This property has already been proved for
cylinders and cones, which are evidently particular cases of
developable surfaces.
175. We shall first consider twisted surfaces; and
among the various ways in which the motion of a straight
line may be governed so as to generate a twisted surface,
we shall select the two following: viz. when the moveable
line constantly passes through two fixed curves and remains
parallel to a fixed plane, and when it constantly passes
through three fixed curves.
176. To find the equation to the surface generated by
a straight line which constantly passes through two given
curves, and remains parallel to a fixed plane.
The motion of the generating line is here, manifestly,
completely determined, as we have only to cut the curves
by planes parallel to the fixed plane, and to join the points
of section by straight lines, to obtain as many positions of
the generating line as we please; and as the tangents to the
156
fixed curves at those points of section will not usually be in
the same plane, the surface will generally be twisted.
Taking the directing plane for that of ay, let the equa-
tions to the generating line in any position be
S=a, y=PBat+y,
and let the equations to the two directrices be
y= oie) #=\,(0)3 y= dua) = yA (0);
then successively combining the equations to the moveable
line with those to each of the directrices, so as to eliminate
VY, Y, %, we find
F, (a, B, y) = 0, 1 Gd (rey decry Lal Oe
which we may suppose reduced to the forms
B = p(a), y = W (a).
Hence, eliminating a, 3, y between these and the equations
to the generating line, we have for the equation to the surface
y = vp (zx) +(e).
Cor. ‘To find the differential equation, we have, differen-
tiating successively with respect to w and y, and dividing one
result by the other,
0= (2) + 2p) p+ ¥'()p,
1=0$' (2) 9+ V4
2 oh ea
(=~ 9G)
therefore, differentiating again, and dividing one result by the
other, employing the usual notation,
qr—ps yee qs — pt j
g =- (2) p, ny =— (2) q;
“ Fr—2pqst+ pt =0,
the equation common to all surfaces of this class.
157
177. To find the equation to the surface generated by
a straight line subject to constantly pass through three given
fixed curves. 3
It is easily seen that the motion of the generating line
will be completely determined. For conceive any point in
one of the fixed curves to be the common vertex of two
conical surfaces, having the other two fixed curves for their
directrices; then these surfaces will intersect one another in
a finite number of straight lines, each of which passes through
the three fixed curves and is therefore a position of the gene-
rating line; and as we take fresh points for the vertex, the
successive generating lines belonging to the same sheet will
pass through points contiguous to one another on the three
directrices. Also the surface will generally be twisted, because
the tangents to the three directrices at the points where a
generating line meets them, will not, except in very particular
cases, be in the same plane.
Let the equations to the generating line be
w=axsty, y=Bz +o,
containing four variable parameters, a, (6, y, Oo; and, as
explained in Art. 173, let the three relations among the
parameters necessary for the line passing through the three
given directrices, be obtained and reduced to the form
B=9(2), y=) d=7(a)s
therefore, substituting for 8, y, 0 in the equations to the
generating line, we have
w=azt+W(a), y=xh(a) +7 (a);
and it remains to eliminate a between these equations, in
order to get the equation to the surface ; but this cannot
be done without particularizing the functions, or the direc-
trices on which they depend. ‘Therefore we must retain the
above system of equations to represent this family of surfaces ;
regarding a as an indeterminate quantity to be eliminated,
after the forms of the functions in each individual case have
been determined.
158
It may be observed that the various, other modes of gene-
rating this sort of surface, as for instance when we replace one
or more of the directrices by a surface which the moveable
line is always to touch, are reducible to this method; for in
every case we may take for the three directrices, any three
sections of the generated surface made at pleasure.
Cor. To obtain the differential equation independent
of the directrices, we must, as before, obtain by successive
differentiations of the above system a sufficient number of
equations to eliminate a, and the functions @, vy, mw and their
derived functions; the result is a complicated equation of
the third order.
When in the two preceding cases, the directrices all
become straight lines, the surfaces generated become re-
spectively the hyperbolic paraboloid, and the hyperboloid
of one sheet, as is seen in the Appendix; they are the only
twisted surfaces whose equations do not rise above the second
degree.
Developable Surfaces.
178. We next come to the consideration of the second
class of surfaces which admit of being generated by a straight
line, the characteristic law of the motion being that two
consecutive positions are always in the same plane. Before
proceeding to point out the various modes in which this
condition may be satisfied, we shall shew that surfaces ge-
nerated in this manner are developable; that is, supposing
them flexible but inextensible, they may without rumpling
or tearing be made to coincide with a plane in all their
points.
Let fig. 57 represent a surface of this sort, and let AN,
A'N’, A”N”, &c. be positions of the generating line inde-
finitely near to one another; then from the definition of the
surface, 4N will be intersected by A’N’ in some point m,
A'N’ by A’ N” in m’, and the latter by the next generating
line in m", and so on; so that these successive points of
159
intersection will form a polygon mm’m’”..., or rather a con-
tinuous curve to which all the generating lines are tangents.
Also AN and A’N’, and similarly every pair of consecutive
generating lines, will include a sectorial area Am A’ of inde-
finite length, but infinitely small angle, which may be re-
garded as a plane element of the surface. If now the first
of these elements be turned about its line of intersection with
the second, till they are in the same plane; and then the
system formed by these two be turned about the line of
intersection of the second and third till they are in the same
plane; and if this operation be continued through all the
plane elements, they will all thus be brought into one plane,
and the given surface will be developed without rumpling or
tearing.
179. We have already seen (Art. 174) that the plane
which touches a developable surface in any point M’ is the
tangent plane at every point in the generating line passing
through JZ’; and that to construct the tangent plane to any
point 1’, we have only to draw a tangent line M’T' to any
curve on the surface passing through that point, then TM'N’' is
the tangent plane required.
180. As the generating lines are all tangents to the
curve mm'm’,.. formed by their perpetual intersection, the
surface may be supposed to be generated by a moveable
straight line which is always a tangent to a fixed curve;
the curve must of course be of double curvature, otherwise
the surface generated would be a plane. Hence it is suf-
ficient to assign one fixed directrix (to which the generating
line must be always a tangent) to completely determine a
developable surface.
If the equations to the fixed curve be
v= (2%) y=),
the equation to the surface will result from eliminating a from
the equations to the line touching the curve at a point for
which s =.a, which (Art. 123) are
2 $@)=$@MC-@, ¥-V¥O=V@(e- 45
160
the elimination cannot be effected without assigning the
forms of the functions; but every developable surface may
be represented by the above system of equations, regarding
a in the first as a function of y and x to be determined from
the second.
181. The curve mm'm”... is called the edge of re-
gression of the surface which its tangent generates; the
reason of which is that the two portions of any tangent,
produced both ways from the point of contact, generate two
distinct sheets of the surface which are united in the curve;
so that if we wished to pass from a point in one sheet to
a point in the other without quitting the surface, our path
would have a point of regression at the curve, provided the
two points were not in the same generating line. In conical
surfaces the edge of regression is reduced to a point, in
cylindrical surfaces it 1s removed to an infinite distance.
182. We may also generate a developable surface, by
assigning two fixed directrices through which the moveable
line is always to pass; in that case the points in which
the moveable line meets them must be constantly so chosen
that the tangent lines to the directrices at those points are
in the same plane; then, as the moveable line in passing
into its consecutive position may be supposed to glide along
those tangents, every two consecutive positions will be in
the same plane. Consequently, after having obtained two
relations among the parameters which enter into the equa-
tions to the moveable line by subjecting it to meet the
two directrices, we shall obtain a third by expressing that
the tangents to the directrices at the points where the
moveable line meets them, are in the same plane; hence
only one arbitrary parameter will remain in the equations,
by eliminating which we shall obtain the equation to the
surface.
183." There is yet another way of expressing the gene-
ration of developable surfaces; for since every two consecutive
generating lines include between them a plane sectorial area
161
of infinite length, we may regard these elements as forming
parts of the successive positions of a plane subject to move
according to a given law. This method will usually be found
the most convenient in practical applications; and it leads
easily to the general differential equation to developable
surfaces, and to other results, which we shall now obtain by
means of it.
184, To find the general equation to developable sur-
faces, considered as generated by the consecutive intersections
of the positions assumed by a plane subject to move after
a given law.
The law according to which the plane moves will vary
with each particular surface; but in order that the motion
may be completely determined, and that a single surface
and not an infinite number of surfaces may be generated,
it must leave only one arbitrary parameter in the equation
to the moveable plane. The successive positions assumed
by the plane by virtue of the infinitely small variations of
the parameter, will cut one another consecutively in straight
lines, which taken two and two are in the same plane;
these lines will therefore form a developable surface touched
by all the planes. Hence the law of succession of the planes
which generate a developable surface requires that two of
the three arbitrary constants which enter into the equation
to a plane should be functions of the third, or, which is the
same thing, that they should all be functions of the same
parameter a; let therefore
s=ap(a) + yf(a) + (a)
be the equation to the generating plane in one of its positions
depending upon the parameter a; then the equation to the
plane, which differs insensibly from it in position, will be
ce an) Sat &et + yf f(a) +f (a). da + &c.}
| +W(a) + W (a). Sa + &e.;
and the co-ordinates of the points in which they intersect
LL
162
must satify these two equations; that is, they must satisfy
z= 0(a) +yf (a) + (a),
0O= w$ (a) + &e.t + y sf’ (a) + &e.$ + W' (a) + &e. 5
or if the planes be consecutive, making da = 0, the equations
to their line of intersection are
s=adp(a)+yf(a) + (a)...... Gly.
O0=ad'(a)+yf (a)+ (a)... (2);
and the general equation to developable surfaces will result
from eliminating a between these equations; but as the
elimination cannot be effected without fixing the forms of the
functions ¢, f, .y, that is, without particularizing the surface,
we must retain the above system to represent this family
of surfaces, regarding a in the former as a function of w and y
to be determined by the latter.
Cor. Hence the differential equation to developable
surfaces can be obtained; for, differentiating (1) successively
with respect to w and y, regarding a as a function of # and y
to be determined from (2), we have
p=$(a) + feg'(a) uf) +O} S.
vis prin’ el ibsidetta
q=f(a) + {2p'(@) + of (a) +V(a@)} ar
which by virtue of equation (2) are reduced to
p= (a), g=f(a);
and since p and q are functions of the same quantity, they
are functions of one another,
“p= 7 (q);
which is the differential equation of the first order to de-
velopable surfaces, containing one arbitrary function; this
may be made to disappear by differentiating the equation
163
p = m(q) successively with respect to # and y, and dividing
one result by the other; for we have
=a (q).s, 8=(q)-é;
.rt—s=0,
the differential equation of the second order common to all
developable surfaces.
185. To find the equations to the edge of regression
of a developable surface.
The edge of regression is the locus of the intersections
of the successive generating lines; and to find its equations we
must combine the equations to any generating line (1) and (2),
in which a is an arbitrary constant, with the equations to the
consecutive generating line resulting from (1) and (2), by
changing a into a + 0a; we shall thus obtain four equations
which by reduction are equivalent to the three
z= ad(a)+ yf (a) + W (a),
0=ag (a) + yf (a) + ¥ (a),
0= ag" (a) +f (a) + W' (a);
these, for any value of a, will furnish the three co-ordinates
x,y, % of the point where the generating line corresponding
to that value of a, is intersected by the consecutive generating
line; so that in eliminating a from the above system, we
shall have the two equations to the locus of all such points of
intersection; that is, to the edge of regression of the surface,
by the tangent line to which the surface may be considered
as generated.
186. ‘To find the equation to the developable surface
which touches at the same time two given curve surfaces.
Let the equations to the two surfaces be
F, (a, Yis %1) ee Ag ee, (#5, Y25 Bo) = 0,
dz dz
fie
dase tt dy
11—2
and let p,4i, P2q2, denote the values of for the
164
first and second surface respectively. Then the moveable plane,
since it touches the first surface, will have for its equation
B—-R= pi (©-X)+UnY-Y%);
and in order that it may touch the second surface, that is,
in order that it may coincide with the tangent plane at
the unknown point «,y,%,, we must join the conditions
Pi = Por Nh = Joo F1 — PiG — NY = Fe — P22 — G2Yr2-
If therefore from these three equations, and the two equations
to the surfaces, we determine the five unknown quantities
Yis Zip Voy Yoo Vy in terms of &,, the equation to the moveable
plane will take the form
x =ad(a) + yf(a) + (a);
and by eliminating the arbitrary quantity #, between this
equation and the derived equation with respect to v,, we shall
obtain the equation to the required developable surface.
If we suppose an opaque body to be illuminated by a
luminous one, the surfaces which circumscribe the umbra and
penumbra occasioned by the interposition of the opaque body,
are two sheets of the developable surface which touches
both the bodies; and its lines of contact with the surfaces
of the bodies are, one, the curve on the opaque body which
separates the illumined and obscure parts; and the other,
the curve on the luminous body which separates the illumi-
nating part from that which can send no rays to the other
body. Consequently this problem is the general problem of
umbras and penum bras.
Similarly, we may find the developable surface generated
by the intersections of the tangent planes to a surface along an
assigned curve traced on the surface.
187. We may arrive at the differential equation to de-
velopable surfaces, and at the equation to the generating line
passing through any point, by the following method.
Since a tangent plane applied to the surface at any point,
touches it in a series of points the projection of which on any
165
of the co-ordinate planes is a straight line, and since the
equation to the tangent plane at a point wysx is
= pa +qy +8 —pxr—qy,
if y’ =ma' +n be the equation to the projection of the ge-
nerating line which passes through the point wy, and in
the values of p, g, and s — paw —qy, we change y into ma + n,
the above equation must remain the same for all values of
xv whatever, and therefore the quantities p, q, 7 — px —qy,
which become functions of # only, must have their differential
coefficients with respect to w equal to nothing,
dp dy
_— = —~=0
dw 4 u ane: ;
d d
q =s+f y = 0,
dav dav
d 1733 4 BS ( +
FMS aiN a WP Gar HP Oe
te sg :
Hence, eliminating as rt —s* =0, the equation required ;
x
: . s ;
and y -—y=— 5 (# —uv)y=-— - (x — x) is the equation to the
projection of the generating line which passes through the
point xy%.
188. When a developable surface is made plane, the
absolute lengths of any determinate portions of the gene-
rating lines, as well as of any curve traced on the surface,
are not altered; and the angle which the tangent line to the
curve at any point forms with the generating line through
that point also remains unaltered ; but the curvature at any
point is changed.
For let AN, A’N’, &c. (fig. 57) be consecutive positions
of the generating line of a developable surface, and let PR,
P’R, &c. be the chords produced of a curve traced on this
surface, and meeting the generating lines in the points P, P’,
&c.; and for the surface let us substitute the series of plane
sectorial elements Am JA’, A’m’ A”, &c., and for the curve the
166
polygon PP’P” &c. Then in turning the first plane element
about A’m to bring it into the same plane with the second,
and in turning the system formed by the two about A”m’ to
bring it into the same plane with the third, and so on, it is
evident that neither the lengths of PN, P’N’, &c., nor the
angles which they form with PP’, P’P’, &c., nor the lengths
of the latter, are altered, but that the angles RP’R’, RP’ R",
&c. will be changed; and as this continues true, however
small we take the plane elements, the properties announced
above for the surface and the curve traced upon it, are
established.
189. The shortest line on a developable surface has the
property that its osculating plane at every point is perpen-
dicular to the plane touching the surface at that point, or
contains the normal to the surface at that point.
If the polygon be such that, upon bringing all the plane
elements into the same plane, it becomes a straight line, then
two consecutive sides must always make the same angle with
the intermediate generating line; that is, for every point we
must have
L4mPR=em PR;
for when this condition is fulfilled, in the developed surface
every two consecutive sides will be the prolongment of one
another. Also since RP’, R’P’, make equal angles with
P'N', they may be regarded as two generating lines of a
conical surface whose axis is P’N’, and therefore the plane
RP'R’ will ultimately be the tangent plane to this surface
along P’R, and therefore perpendicular to RP’N’, which is a
meridian plane of the cone containing P’R; hence, passing
to the curve, the plane RP’R’ is its osculating plane at P’,
and RP'N’ is the tangent plane to the surface at the same
point, and these planes are perpendicular to one another;
also since, when the surface is developed, the curve becomes
a straight line, it is the shortest line which can connect two
points on the surface through which it passes, and it has the
property announced at every point.
167
190. That the same property is true for the shortest
line traced on any surface whatever, appears thus.
If on any proposed surface we conceive the shortest line
between any two points to be traced, we may describe a
developable surface which shall touch the surface according
to that curve; consequently the curve of contact will be the
shortest line between the same points on the developable
surface, and therefore its osculating plane at every point must
contain the normal to the developable surface, i. e. the
normal to the proposed surface at that point. Hence if
y = d (x), x = yy (x) be the unknown equations to the shortest
; d
line on a surface for which = 7, - =q; expressing that
the osculating plane at any point contains the normal to the
surface at that point, we find
dy (1 =) d’sf dy );
dx? beers = Sag -4
d d
which together with e =p+t+ 7 will determine the curve.
v c
If we take s the length of the arc for independent variable, the
equations to the curve will assume the symmetrical forms
ax d’z d’y d’z
= : a
ds” Leryn ?- ds? os Tae
0.
191. From the property that the lengths of the ge-
nerating lines of a developable surface, and of any curve
traced upon it, as also the angles which the generating lines
form with the curve, are not altered by the development of
the surface, we can find the nature of the curve when the
surface is made plane; or, conversely, a curve being traced
upon a plane, we can find its nature when the plane is applied
to a given developable surface. This is shewn in the case of
cylindrical and conical surfaces, in the appendix.
Envelopes.
192. Developable surfaces, considered as generated by
the successive intersections of planes drawn after a given law,
168
are a particular case of a general family of surfaces to which
we shall in the next place direct our attention.
193. To find the equation to a surface which envelopes
a series of surfaces described after a given law.
Let w=f(x, y, %, a)=0 be the equation to a surface
containing, besides other constants, a parameter a; then if
we give a particular value to a, the form and position of
the surface in space will be completely determined; and if
we give to it all possible consecutive values, we shall obtain
an infinite number of corresponding surfaces, usually inter-
secting one another two and two. The surface formed by
these successive intersections, and having with each one of
the surfaces the infinitely narrow zone in common which is
contained between the curves in which that individual surface
is intersected by the preceding and succeeding ones, (and
which consequently envelopes each one of the first series of
surfaces, supposing them all to exist together, and touches
it according to a curve of intersection) has been named by
Monge the Envelope. Also the curve in which any two
consecutive surfaces intersect, he has named the Characteristic
of the Envelope, because it indicates the mode of its genera-
tion; thus the characteristic of developable surfaces which
are generated by the intersections of planes, is a straight line ;
and that of surfaces generated by the intersections of spheres,
a circle, :
To find the equation to the envelope, we observe that if
after having given the parameter a determined value a in
u = 0, we give it a new value a + oa differing insensibly from
the former, we obtain the equation to a second surface differing
in form and position insensibly from the first, and intersecting
it in a series of potnts the co-ordinates of which satisfy the
equations
du . @u (da)®
= 0, U-+ ——.0a + ——.
da d a’ 24
du
or u=0, apes ee
a
169
or if the surfaces be consecutive, w= 0, — =0;3
it follows therefore that the equations to the characteristic or
curve of intersection of two consecutive surfaces, corresponding
to the value a of the parameter, are
d
S(@, Y, % a) =9, Spake Y,.% a) =0;3
and since the envelope is formed by the assemblage of the
characteristics, its equation will result from eliminating a
between the same equations. As this elimination, however,
cannot be effected without assigning the form of f; we must
retain the above system of equations to represent the envelope,
regarding a in the former as a function of w, y and z to
be determined from the latter.
194, Again, if in the equations to the characteristic,
after having given the parameter a particular value a, which
determines the position of the characteristic in space, we give
it a new and insensibly increased value a+ da, the two
resulting equations will be those to a second characteristic,
differing insensibly in form and position from the first, and
intersecting it in general in a finite number of points, the
co-ordinates of which satisfy the two sets of equations
du
u = 0, ory
du du du
—. ae U) pa ——— , eee
Uti da + &c. = 0, FEEL da + &e :
which latter two equations, by virtue of the former, are
equivalent to
du & " ad’ u
rere Cs =
da i ‘ da’
Consequently the co-ordinates of the points in which two
consecutive characteristics intersect, satisfy the three equations
du d?u
u =,0, wa () - =0;
+ &c. = 0.
170
and from these equations we may either determine a, y,
and x in terms of a, that is, the co-ordinates of the point
in which a given characteristic is intersected by the con-
secutive; or eliminate a between them, and so find the two
equations in w, y, and x, to the curve formed by all the
successive points of intersection. This curve will be touched
by all the characteristics, and will be the edge of regression
of the envelope; for, as explained in the case of develop-
able surfaces, the portions of the characteristics which are
on contrary sides of their points of contact with the curve
under consideration, will form two distinct sheets of the
envelope, and these sheets will touch one another in that
curve and have it for their common limit.
195. The propriety of the term Characteristic will be
more apparent, if we generalise the equation w= 0, and
suppose it to contain together with a a function of a,
@ (a), so as to have the form
f 2, Y, ®, ay @ (a)} =n)
Then whatever form we give to @, the intersection of two
consecutive surfaces represented by this equation will always
be a curve of the same sort, and will therefore offer a
character common to the whole family of surfaces. For this
curve of intersection will be represented by the equations
Smaking ¢@ (a) = B}
Bes df dB
Tid, ys. a, (3) = 0; at, da Saas
and we see that by changing A we shall only alter the
constant parts of these equations, without at all affecting
the way in which a, y, # enter into them; so that for all
values of B; the curve represented by them will be of the
same species.
196. ‘To determine the equation to the envelope when
the equation to the series of surfaces contains two independent
parameters.
Suppose the equation #«=0 to contain two parameters
171
a and £ independent of one another; then the equation to
the surface which differs insensibly from that represented by
u = 0, will be
du
1 ——~ fa Sh. OB + &e. me =
da
and the co-ordinates of the cy in which these surfaces
intersect must satisfy their two equations, that is, they must
satisfy
a
wu = 0, ot om 0B + &c. =0;
or if the surfaces be consecutive
- du x du
Radia aay.
where A = limit of d8+da; but since A may have any value,
the latter equation resolves itself into
du du
da ” dB
these then together with w=0, are the three equations for
determining the curve of intersection of two consecutive sur-
faces; from whence we may obtain its two equations involving
only one of the parameters; and finally by eliminating that
parameter we may obtain the equation to the surface generated
by the perpetual intersection of the first series of surfaces.
= 0,
SECTION IX.
ON THE CURVATURES OF CURVES IN SPACE.
197. Preparatory to finding the radius of curvature,
and evolutes of a curve in space, consider figure (59), where
for the curve is substituted an equilateral polygon mm'‘m”...,
and through the middle points of its sides are drawn planes
respectively perpendicular to them, which intersect, two and
two, in the lines-kh, k’h’, k”h”, &c. Then the plane which
contains the two consecutive sides mm’, m’m”, is perpendicular
to each of the planes gh, g’h’, and therefore to their common
intersection kh; let kh meet this plane in the point q, then
q is the center of a circle passing through the three angles
m, m’,m”; and every point in the line kh is likewise equi-
distant from the same three angles.
The lines kh, kh’, &c. will be parallel only when the
sides of the polygon mm’m”...are in the same plane; in other
cases, if they be produced till each meets its consecutive, they
will form a polygon hop..., the angular points of which are
equidistant from four consecutive angles of the first polygon
mm 'm'’...The point o for instance, since it is situated in kA, is
equidistant from m, m’, m”; and again, being situated in k'h’,
it is equidistant from m’, m”, m’”; that is, it is the center of a
wr
sphere passing through four consecutive angles, m, m’, m”, m’”.
198. The preceding results being true when the number
of sides of the polygon is indefinitely increased, it follows that
the normal planes of a curve generate, by their perpetual
intersection, a curve surface; also, since of the lines of
intersection kh, kh’, &c. every two consecutive ones are in the
same plane, the surface which they generate is developable.
173
199. On the same supposition, the polygon hop...be-
comes a curve, to which all the lines ko, k’p, &c. are tangents ;
that is, hop...is constantly touched by the straight line
which generates the developable surface, and is therefore the
limit to the surface, for no part of the surface can fall within
the space towards which the curve is concave. Moreover its
tangent, being produced both ways from the point of contact,
generates two sheets of the surface which are united and
terminated in the curve; consequently, as was explained in
Art. (181), the curve hop...is called the edge of regression
of the developable surface, from the analogy it bears to a
point of regression in a plane curve. This curve is also the
locus of the points of intersection of three consecutive normal
planes, or of the centers of spheres passing through four
consecutive points of the first curve mm’m”..., that is, it is
the locus of the center of spherical curvature.
200. The plane ggg’ passing through two consecutive
tangents to the curve, and which is perpendicular to the
intersection of two consecutive normal planes, is the osculating
plane, whose equation was found in Art. 127; and the point
qg, in which that intersection meets it, is called the center of
absolute curvature.
201. To find the equation to the surface generated by
the perpetual intersections of the normal planes to a curve
in space.
Let y= («), = = y (2), be ‘the equations to the curve,
dy dy
and let qn? da? &c. be denoted by 9, y”, &c.; then we have
seen Art. (130), that the equation to the normal plane
X—a2+(¥-y)y +(Z-%)% =0, (1)
and the derived equation with respect to a, viz.
(Y—-y)y + (Z—-s) a" -1-y*-x°=0, (2);
are the equations to the line of intersection of two consecu-
tive normal planes; XY, Y, Z, denoting the co-ordinates
174
of any point in that intersection. If therefore we eliminate
w between them, we shall have the equation to the surface
generated by the line of intersection in all its successive
positions. On this surface lie all the evolutes of the curve,
as will be shewn; it will be a cylinder if the given curve lie
all in one plane, and if the given curve be traced on the
surface of a sphere it will be a cone with its vertex in the
center of the sphere.
202. ‘To find the radius, and co-ordinates of the center,
of spherical curvature of a curve of double curvature at any
point.
The center of spherical curvature, or of the sphere which
passes through four consecutive points of a curve, is the point
of intersection of two consecutive generating lines of the sur-
face considered in the last article; hence we must join to
equations (1) and (2) of last article, the derived equations
with respect to v, as explained Art. (185).
If therefore , Y, Z be the co-ordinates of the center
of spherical curvature, we have in order to determine them,
the equations,
X—-a+(Y-y)y + (Z—2)2 =0,
(Y-y)y" + (Z—-s)2"- A +y? + 2%) =0,
(P-y)y" + (Z—#)2" — 3 (y'y" + 2'2") = 0,
and the radius of the sphere
- a V/(X— a) + (Fy) + (Z- 2)".
Moreover if from these three equations we eliminate a,
there will result two equations between X, Y, Z, which will
be those to the locus of the center of spherical curvature, or
to the edge of regression of the developable surface generated
by the intersections of the normal planes.
203. To find the radius, and co-ordinates of the center
of the circle of absolute curvature.
175
The center of absolute curvature is the point where the
line of intersection of two consecutive normal planes meets the
osculating plane.
Let s be the length of a curve from a given point to a
point 7, y, x; and instead of taking two equations between
&, y, x to represent the curve, let us suppose each of them to
be a function of another variable ¢; and let a’, wv’, &c. denote
their differential coefficients with respect to ¢; then the equation
to the osculating plane at wyz is
(y's — 2'y") (X - a)
+ (2 a” —a's")(Y —y) + (e'y" -y'2")(Z-z)=0. (1)
The equation to the normal plane at the same point is
we (X-—wv) +y (Y-y) +2 (Z-2)=0, (2)
and this, in conjunction with the equation obtained by dif-
ferentiating it .
a (X-a2)+y" (F-y) +2" (Z-2)=8", (3)
(since a? + y? +2" =s8", and». a’ a” +y'y" + 32/2" =8's") (4),
defines the line of ultimate intersection of two consecutive
normal planes (Art. 130). Hence if .Y, Y, Z, satisfy at once
these three equations, they are the co-ordinates of the center
of absolute curvature at wyz; and the radius R will equal
V(X — «)? + (¥ -y)? + (Z - 8)?
Now from (1) and (2) taking account of equations (4), we
get by ordinary algebraical operations,
AX —& Y-y Z-—
savas’ sy —ys" sls" — 2! 8"
5 : from (3)
rom (;
s a wy + y”? 4 9/2 _ 32 oe! 4. yf” 4. 9/2 — gl ?
which equations give the radius and co-ordinates of the center
of the circle of absolute curvature.
Cor. If we make w the independent variable by putting
176
, Ad ”? d a
t=, so that 7 =1, 7 =0, and y'sy &c., become et “ y
die da’
we get by reduction
(1+ y? +s”)
= ja eh Gisele yy
Sy + 8? + (2"y! — xy")
204. Also if we make s, the length of the arc, the in-
dependent variable instead of ¢, we get immediately
1
R ad J x se y” “ 2
i” dx
where wv denotes ds?” &c.; and the expressions for the co-
ordinates of the center assume the following simple forms,
a ”
A-a“7= a” 4/8? 43 1129 Y-y= a? 4 of 4 agit?
y gl!
-Z%=
WN
wey + 3/2
205. A curve of double curvature may either have three
consecutive elements in the same plane, or two consecutive
elements in the same straight line; the first is called a simple
inflexion, that is, when the curve at that point becomes plane,
and therefore the radius of spherical curvature infinite; and
the second a double inflexion, as necessarily including the
former, that is, when the curve at that point becomes a
straight line, and therefore the radius of absolute curvature
infinite.
If in Art. 202, we actually determine the values of XY — a,
Y-y, Z-—z, we find for their common denominator the
expression
vIn OP she Md 2
wy -YyY ® 3
hence the condition
177
expressing that the center of spherical curvature is at an
infinite distance, will determine the points of simple inflexion,
and agrees with Art. 128.
At points of double inflexion, the radius of absolute curva-
ture changes its sign, and the curve, after being concave in one
direction, crosses its tangent line and becomes concave in the
opposite direction; hence also its projections on the planes of
sv and wy cross their tangents, and have corresponding points
of contrary flexure; at such points therefore, and at cusps,
d’ z | d’y
——=0, Or ©, — =0, or ©;
da® div®
which are the equations for determining the abscissee; and
which we observe make the expression for the radius of absolute
curvature infinite or evanescent, as ought to be the case.
206. The tangent lines of the given curve mm’,..(fig. 59)
will generate a developable surface, whose edge of regression is
the curve mm’...; and since the tangent line is the intersection
of two consecutive osculating planes, this surface would also be
generated by the perpetual intersection of the osculating planes.
But the lines of intersection of the normal planes are perpen-
dicular to the osculating planes; therefore the angle between
two consecutive lines of intersection, that is, two consecutive
tangents of op..., is equal to the angle between two consecutive
osculating planes of mm’... Again, the angle between two
consecutive tangents of mm’... is equal to the angle between
its corresponding normal planes, that is, to the angle between
two consecutive osculating planes of op... If therefore we call
the angle between two consecutive tangents of a curve of
double curvature its first flewion, and that between two con-
secutive osculating planes its second flewion, we may enunciate
the above properties as follows. The first flexion of a curve
of double curvature is equal to the second flexion cf the edge
of regression of the developable surface generated by its
normal planes; and the first flexion of the latter, is equal to
the second flexion of the former.
12
178
Evolutes of a curve in space.
207. Every curve, whether plane or of double curvature,
has an infinite number of evolutes, all of which lie on the
developable surface generated by the perpetual intersection of
the normal planes.
Recurring to fig. 59, since kA is perpendicular to the plane
through two consecutive sides mm’, m'm”, of the polygon, any
point f in it is equidistant from the middle points of the sides
g and g’. If therefore we draw gf in the first normal plane,
and g’ff’ in the second, the lines gf, g’f are inclined at the
same angle to kh; and if with center f and radius fg a circle
be described, it will touch the two consecutive sides mm’, m'm”
in their middle points g and g’, and when the describing radius
comes to g’ it will be confounded with g’f’. In like manner if
we draw 2 f'f” in the third normal plane, the lines g’f’, gf’
are equally inclined to k’h’; and a circle described from f’ with
radius f’g’ will touch the consecutive sides m’m”, mm” in
their middle points, and the describing radius will be con-
founded with g”f” at the point g”. By continuing this con-
struction the polygon fff’... will be formed, by unwinding
a thread gf from which, circular arcs, touching every two
consecutive sides in their middle points, will be described.
Therefore when the number of sides is indefinitely increased,
the polygon fff’... will become a curve traced on the develop-
able surface formed by the intersections of the normal planes ;
and by unwinding a string from it, the proposed curve of
double curvature will be traced out.
Also since the direction of the initial radius gf is arbitrary,
any other direction would have produced a curve endowed
with the same properties as ff/f”...; it follows therefore that
every curve has an infinite number of evolutes all contained
on the surface generated by the perpetual intersections of its
normal planes; in the case of a plane curve all the evolutes,
except that in its own plane, are curves of double curvature,
traced on the right cylinder whose base is the evolute in its
own plane.
179
208. Since ze fk =e" f'k' = h'f'f’, if the plane hk’
were brought into the same plane with h’k” by being turned
about kh’, ff’ would be brought into the same straight line
with ff’; and since the same may be proved successively of
all the other sides of the polygon fff’..., it follows that if
the surface formed by the intersections of the normal planes
were developed, the evolute ff'f’... would become a straight
line, and therefore would be formed by stretching a thread
in the direction gf and applying it freely to that surface.
Hence if from any point in a curve of double curvature, any
line be drawn touching the surface formed by the normal
planes, and be applied freely to that surface, it will form an
evolute of the proposed curve; and will be the shortest line
that can be drawn between any two points of the surface,
through which it passes.
209. It is to be observed that the centers of absolute
curvature are not situated on an evolute of the curve.
For if g’q’ be perpendicular to kh’, then it cannot pass
through the point g; because it would then coincide with g"q,
and therefore cut kh at right angles, which would require that
kh and kh’ should be parallel; and this can never happen as
long as the proposed curve is of double curvature. Hence
the consecutive radii of absolute curvature gq, gq’, since they
are in different planes, and do not meet one another in the
common intersection of those planes, therefore they do not
meet one another at all; and therefore cannot be consecutive
tangents of the same curve; that is, the locus of the center
of absolute curvature is not an evolute.
210. ‘To find the equations to any proposed evolute of
a curve of double curvature.
As we know the equation to the surface on which all the
evolutes lie (Art. 201), it only remains to find the equation
which particularizes a given evolute. Let X, Y, be the co-
ordinates of a point in the projection of an evolute, wv, y, the
co-ordinates of the corresponding point in the projection of
12—2
180
the curve; then a tangent to the projection of the evolute,
must pass through the corresponding point in the projection
of the curve;
Sed Oey
Ad Xs
This joined to the two equations
oe +4 Yo y) y+ (Z—2)x'=0,
(VY —y)y"+(Z —2)2"- (14+ y? +2”) =0,
which belong to the generating line of the developable surface
containing the evolutes, will give by eliminating w the two
equations to the evolute, one of them being a differential of
the first order; and its integral will introduce an arbitrary
constant, by means of which the evolute may be made to
satisfy that condition which particularizes it; as, for instance,
to pass through a given point of the developable surface on
which it is traced.
SECTION X.
ON THE CURVATURE OF SURFACES.
211. ‘To find the requisite conditions for a contact of
the first, second, &c. order, between two surfaces.
If two surfaces, referred to the same origin and axes,
pass through the same point, the co-ordinates of which are
xv, y, %; and if we change w into w+h, and y into y +k,
the equation to the first surface will give for the value of
the new ordinate,
st+ph+qk+4 (rh? + 2shk + tk’) + &e.
and the equation to the second surface
s+ Ph+Qk+h(RW4+2ShK 4 Th) + &e.;
the distance of the surfaces, measured in the direction of their
ordinates, will therefore be expressed by
(P-p)h+(Q-gQk+3}(R-r)h?+2(S-s)hk+(T-d kh} + &e.
If we suppose the equation to the second surface to
contain a certain number of arbitrary constants, we may
determine them so as to make the first terms of this dis-
tance vanish; and it will follow that any other surface, for
which these terms do not disappear, cannot be situated
between the two former with reference to the points which
are contiguous to their common point; at least so long as
we take h and & so small, that the sum of the terms of
the first order may be more considerable than that of all
the terms of succeeding orders. When we have P—p=0,
Q - q=0, the surfaces will have a contact of the first order ;
if besides these, we have R-r=0, S—s=0, T'—-?#=0,
the contact -will be of the second order, and so on.
182
212. Let therefore V=0 be an equation between three
variables a’, y’, x and a certain number of arbitrary constants,
and according to the usual notation, let P, Q, R, &c. denote
the partial differential coefficients of ’ with respect to a’ and
y 3 by the determination of the constants we may make this
surface have with one completely given (the co-ordinates of
which we shall represent by a, y, x, and the partial differential
coefficients of z, by p, qg, &c.) a contact of an order which will
depend on their number. ‘The first condition to be satisfied is
that the surfaces may have a common point, that is, that by
changing #’ into a, and y’ into y, in V= 0, we may find x’ = x.
Besides this, for a contact of the first order, upon making the
same substitutions for # and y’, we must have P= p, Q=@q;3
and for one of the second, besides all the foregoing conditions,
we must have R=r7r, S=s, J'=
Hence it appears that a contact of the first order requires
three arbitrary constants, and one of the second, six; and
in order to have a contact of the » order, since the terms
of three, four, &c. dimensions in # and & in the develop-
ment of the difference of the ordinates in the preceding
Article are, in number, 4, 5, &c., the number of disposable .
constants must be |
14+424+3+4&c.4+ (n+ 1)=4$(n +1) (7 + 2).
213. But if we confine our attention to the sections
of the surfaces made by a plane containing their common
ordinate #, and inclined to the plane of za at an angle
whose tangent is m, then, making k=mh, the difference
of the consecutive ordinates in these sections will be
$P—p+(Q-9)mth+4{R-r+2(S—s)m+(T-t)m t{th’+&e.
Hence if the surfaces have a common point and a common
tangent plane at that point, that is, if upon making a’=a,
and y= y, we have x=, P=p, Q=gq, then the first term
of the above difference will disappear, and the second will
also vanish if we add a fourth condition, viz.
R-7r+2(S-s)m+(T'-t)m =0.—
183
Hence four disposable constants in the equation to a surface
are sufficient in order to establish between it and one com-
pletely given a contact of the first order at a proposed point,
and also to establish a contact of the second order between
the sections of the surfaces made by a See: plane containing
their common ordinate.
214. Previously to applying these considerations to es-
timate the curvature of any surface at a proposed point, we
may observe that we cannot attempt to assimilate the curvature
of a surface in general to that of a sphere; because in the
latter the curvature is uniform about the same normal, whereas
for surfaces in general that is far from being the case. The
mode of proceeding must therefore be to imagine several planes
drawn through the normal at the point under consideration,
to calculate the radius of curvature of each of these sections
at that point, and by comparing them to judge of the greater
or smaller curvature of the surface in those directions about
the point, as well as towards what parts the curvature is
turned; for, as we know, certain surfaces contiguous to any
point are situated partly above and partly below the tangent
plane at that point.
In making different planes pass through the same point
of a surface, so that some contain the normal to the surface
at that point, and some do not, we shall find the radii of cur-
vature relative to that point both of the normal and oblique
sections, bearing remarkable relations to one another, inde-
pendent of the particular form of the surface. We shall
begin by demonstrating the simple relation which exists be-
tween the radii of curvature of a normal and an oblique
section, made by planes passing through the same tangent
line to a surface, first noticed by Meunier. We shall then,
by the doctrine of contacts, deduce the radius of curvature
of any normal section of a surface; and afterwards prove
Euler’s theorems relative to the curvatures of the principal
normal sections of a surface. ‘The circumstances under which
the normal at any point may be intersected by the normals
at immediately adjacent points will next fall under considera-
184
tion, as throwing great light on the subject and leading to
many results of interest.
215. If a normal and an oblique section of a surface
be made by planes passing through the same tangent line
to the surface, the radius of curvature of the oblique section
is equal to the projection on its plane, of the radius of
curvature of the normal: section.
Let the tangent plane to the surface be the plane of
wy, the point of contact the origin, and the tangent line the
axis of w; then the normal to the surface will be the axis
of =; let OP (fig. 62) be the normal section in the plane of
sv, OP the oblique section made by a plane s’Ow through
Ow, and inclined to the normal section at an 22Oz=6@;
NP, NF’, ordinates to the two curves corresponding to the
common abscissa ON =h; also let NP=2, and let h, &k, 2’
be co-ordinates of P’, Then, (assuming that in any plane
curve when the axis of the abscisse is a normal at the origin,
the radius of curvature at the origin is equal to 4 limit of
oe eee)
———— | we.have
abscissa
milidevorMbureature of Onests0 oe ion: cee
FA=radius of curvature o at O= 5 Imit o WP?
ON?
Casita ; ; a ( ee ise ye sa ee...
R= radius of curvature of OP at O=4% limit of NP”
~
NP’ x’ sec
— = limit of ——= limit of - -— =sec@. limit of
NP x
a | &
But since the plane of wy is the tangent plane at O,
p=0, q=C, and s=trh?+shk+4tkh + &e.
and making k = 0,
ee
pas trh? +t+—.h' + &e.3
185
tk k : ;
because limit of re since Ow is a tangent to the pro-
)
jection of OP’ on the plane of wy;
. R= Rcoos8.
Cor. Hence it follows that the osculating circles of all
sections of a curve surface that have a common tangent
line, are situated on the surface of a sphere the radius of
which is the radius of curvature of the normal section pass-
ing through the same tangent line; for OA’= OA cos AOA
(and consequently AdA’O is a right angle) if OA be the
diameter of the circle of curvature of the normal, and OA’
that of the circle of curvature of the oblique section; the
latter circle is therefore situated on a sphere whose diameter
is AO.
216. To find the radius of curvature of any normal
section of a surface at a given point, in terms of the co-
ordinates of that point.
Let w, y, x be the co-ordinates of the given point P
(fig. 63) of the surface, p, q, 7, &c. the partial differential
coefficients of x expressed in terms of those co-ordinates ;
PT the tangent line through which the normal section is
to pass, which, since it lies in the tangent plane at P,
will be determined by the equation to its projection on the
plane of vy, viz. y—y=m(a'—«). Then if we determine
a sphere having a contact of the first order with the surface
at P, and whose section by the vertical plane PQT' has a
contact of the second order with the section of the surface
by the same plane, all planes passing through P7' will cut
the sphere in circles, which are the circles of curvature to
the corresponding sections of the surface, and therefore the
radius of the sphere will be the radius of curvature of the
normal section; and this we are enabled to do, because the
four disposable constants in the equation to a sphere will
enable us to satisfy the four requisite conditions, which are,
186
that upon making 2’=«a and y'=y in the equation to the
sphere, we must have
s=2, P=p, Q=q, R-7r+2(S-s)m4+(T-t)m’ =0.
Now the equation to the sphere gives, by differentiation,
(2 — a)? + (y'- By + @'- y= 8
a’—-at P(s-y)=0, y'-B+ Q(2’-y) =9,
14+-P°+R(s'-y)=0, PQ+S8(%-y)=0, 14Q°?+T7(s'-y)=0;
hence, there result between the constants a, B, yy, 0, and
the co-ordinates vw, y, x, the following relations
(w—a)’+(y- 8)? + @-yyP =e,
w-atp(s-y)=0, y-B+q(?-y) =9,
1 2 2
Mica +rgo( Pt 4) m + (= +t) m= ;
e—%¥ z-¥ z
The latter gives
1+p°+2pqm+(14+q)m
y+2sm+tm
e-y=—
and then a, 2, 6 are known from the equations
w-a=—p(z—-y), y-B=-9g(e-y), P=(e-y)P(+p+e);
Vi1l+ pit’ {1 +m? + (p+qm)°}
. the radius =
ry +2sm +tm’
Hence we have determined the radius and co-ordinates
of the center of the circle of curvature of the normal
section at a point wy, whose intersection with the tangent
plane at that point is projected into a line represented by
the equation
y-—y=m(a'— 2).
Cor. Suppose the tangent line to make angles dA, pu, v
with the axes of a, y, x, and let o be the length of the
arc of the normal section, then (Art. 28 and 131),
187
COS ph
m =——, and
cosr
do HAG (54) +m + ( + mg = (2 2
(=) poe: da +(= m4 P iy = | ,
hence, by substitution, we get for the radius of curvature
Y1l+p+¢
~ # (cosA)* + 28 cos dA Cos p + t (cos nu)?”
217. But if the equation to the surface be proposed
under the form F'(a, y, #) = 0, the radius of curvature of a
normal section may be more readily obtained by considering
(Art. 200) that the center of curvature of the section is the
point in which the normal meets the line of intersection of two
consecutive normal planes.
dk d dF
lor ee eae eal — = W, so that
dx dy dz
Ude+ Vdy+ Wdz=0;
then the equations to the normal at wy are
AX-« Y- Z—s
a ays a ees Q, suppose (1).
Now if ds be an element of the arc of the normal section,
and da, dy, dz its projections on the co-ordinate axes, the
equations to the normal plane to the section will be (Cor.
Puric. 120. }
(X -x)dx+(Y-y)dy+(Z-—sz)dz=0; (2)
and the line of intersection of this with its consecutive is given
by (2), together with its differential, viz.
(X — 2) @a+(Y-y) dy +(Z—2)a@x=ds*. (3)
Hence if XY, Y, Z be determined so as to satisfy the above
three equations, they will be the co-ordinates of the center of
curvature of the normal section; and if R be the radius of
curvature, we shall have from (1)
CMS OL yy + (2-3 x)
v= \/U? + VV? + W?.Q
188
but from (3) we get
(Uda + Vay + Wd’s) Q= ds’;
ds?.4/U? 4+ V2.4 W
Oy are ae
Udu+ Vay + Wax
But since Ude + Vdy + Wdz =0, if we differentiate
and call
dF d’F dF
Teak —-=UV, ——~=W;
dye? dx"
aon pains aay ihe fatdtin tie
dydzs’ dads’ dudy
we get
UPx+ Va@y + Wd’: + uda’ + vdy + wdz + 2uwdydz
+2vdedz+2wdady =0;
consequently, substituting, and calling
dx } dy dz
Bei —=m, —=N,
ds ds
ds
et
we have
\/ U? + V24 W
~— at mv + niw + 2imw' + 2lnv'+ 2mnw
3
which coincides with the former expression for R, if we
suppose (wv, y, 7) =2—f(a, y) = 0.
218. Hence we can find the conditions, in order that
the curvatures of the normal sections of a surface at a pro-
posed point may be all in the same direction, or in opposite
directions; that is, in order that the surface may be convex
or non-convex about a proposed point.
If in the value of # (Art. 216) we always take the radical
positive, as the numerator 1 + m* + (p+mq)’ is incapable of
changing its sign, the sign of J will depend upon that of
its denominator
]
r+ 28m RE in S(mt +s)? +rt— 8°};
189
if therefore xv, y, x the co-ordinates of the proposed point be
such that rf — s*>0, then # cannot change its sign for any
value of m, and the surface contiguous to that point will
be situated entirely on thé same side of the tangent plane,
or will be convex at that point; and if this be true for
every point, as in the ellipsoid, the surface will be entirely
convex. If, on the contrary, r¢—s*<0, there are two
values of m for which the denominator will vanish, and R
will be infinite and will change its sign as m passes through
each of these values; therefore the surface will have opposite
curvatures about the point in question, or will be non-convex.
Thus in the surfaces considered (Art. 176), of which the
equation is
gr—2pqs+pt=0, or (qr—ps)’ +p? (rt —s’) =0,
rt —s° is necessarily negative, and therefore surfaces of this
class are non-convex at every point.
For developable surfaces, where r¢— s* = 0, the denomi-
nator of # is a perfect square, and consequently retains the
same sign; therefore the radius of curvature will haye the
same sign for every point of the surface; only it will at every
point be infinite when mt +s = 0, the direction so determined
evidently coinciding with that of the generating line passing
through the point.
219. 'To determine the normal sections of greatest and
least curvature at a given point of a curve surface.
For the same point of the surface, the expression for R
varies with m, and we may find what value of m will make it
a maximum or minimum by putting the differential coefficient
of & with respect to m (for the quantities p, q, r, &c. are
in general independent of the position of the tangent line
which fixes the normal section) or rather of x —y on which R
depends, equal to 0; hence putting equation (1) (Art. 216),
under the form
(x -—y) (v7 +28m + tm’) +14 p> + 2pqm + (1 + q’) m® = 0,
we have (x -—-y) (s+ tm) + pat (1+q’)m=0;
190
therefore eliminating successively x —-y and m from these
equations, we get
S(14q°)s—pqt}m?+ }(1+q°)r-(1+p*)t} m— { (14+ p*)s—paqr} =0,
(rt-s°)(x-y)?+{(1+p’)t-2pqs+(1 +7 )r}(s-y)+14+p°+q=0.
These equations being of the second degree, it follows
that in general the radius of curvature of the normal section,
as the cutting plane revolves about the normal, will have only
one maximum and one minimum value; which, in absolute
magnitude, may be two maxima or two minima, when the
curvature of the surface changes its sign about the given
point. These two normal sections, one corresponding to the
least, and the other to the greatest radius of curvature, are
called the principal sections of the surface relative to the
point at which the normal is drawn, and the corresponding
radii the principal radii of curvature.
Cor. Hence the former of the above equations enables
us, for each point of the surface, to determine the directions
of the principal sections, and we might also shew from it
that these directions are at right angles to one another.
To determine however the angle between the principal sec-
tions in a simple manner, let us suppose the plane of vy to
be coincident with the tangent plane at P, or only parallel
to it, (this of course we may do without at all altering the
form of the surface or the positions of the tangents to the
principal sections) then we must put p=0, g=0, and the
equation becomes
Brg ae PGR TTD 28,
the two roots of which are always possible and satisfy the
condition
therefore the principal sections always exist and are at right
191
angles to one another. Also the same supposition gives for
the radius of curvature of any normal section (Art. 216),
1 +m?
e+ 25m4+tm-
Having however made the foregoing process answer its pur-
pose of giving the radius of curvature of any normal section
of a surface in terms of the co-ordinates of the point, we shall
not deduce from it all the important results which it is capable
of furnishing, but obtain them by a direct and much simpler
method.
220. ‘The sum of the curvatures of any two normal sec-
tions at right angles to one another, is a constant quantity at
the same point of a surface.
Let AO, the normal at any point A of a curve surface,
be the axis of zs, and wdAy, the tangent plane at the same
point, the plane of wy (fig. 64). Let AB be a section of the
surface made by any plane passing through AO, AC the
corresponding section of the tangent plane, which therefore
touches the plane curve 4B at 4; and let y = tan@.« be the
equation to AC, so that 0@=ZNAC, Then if 40 =R be the
radius of curvature of the plane curve 4B at A, and BC be
parallel to 40, we have by a well-known theorem
A 2
R= 3 limit of = -
Let dN=h, NC =k, be the co-ordinates of C;
“ BCa=trh?+shk+ttkh + &e,
for p=0, gq =0, since the plane of wy is the tangent plane ;
also kA = tan@0.h;
+i
* R= limit of —— —_ —__—
2 Lrht-+shk+ 4th + &e.
a): Petal Oe
= limit of ;
dr
r+2stan@+étan?@ + met htan?@ + &e,
aw
192
1 + tan?@
or 2 = —
r+ 2stan0+¢tan’@’
where 7, s, £ are the values of
d’z d’x d’s
du?’ dady’ dy’
at the point A derived from the equation to the surface.
Hence, if R’ be the radius of curvature of another section
inclined at an angle = 90°+ @ to AN, we have
1 : Ke
rea 7 cos’ @ + 2s cos @ sin @ + ¢ sin’ 0,
] ; : .
—=rsin?@ — 2ssin 0 cos@ + ¢ cos’ @;
R'
which (taking, as usual, the inverse of the radius of curva-
ture to measure the curvature) expresses that the sum of
the curvatures of any two normal sections at right angles
to one another, is a constant quantity at the same point of
the surface.
221. Of all sections of a curve surface made by planes
drawn through the normal at any point, to determine those of
greatest and least curvature at that point; and to shew that
they are at right angles to one another.
We have
fay dart ;
R= 7 cos’ O + 28 sinO cos O + ¢ sin’ A,
he r (1 + cos 20) + 28 sin 20 + ¢(1 — cos 20)
=r+i#+4 (r —- ¢t)cos20 + 28 sin 20,
which is to be a maximum or minimum by the variation of 0;
193
therefore putting its differential coefficient with respect to @
equal to zero, we get
— (r — f) sin20 + 2s cos20 =0;
28
*, tan20 = ari tan (180° +20); (1)
, 28 yr—t
sin 20 = £ ——______—. , cos 20 = ok Sey
J (r — ty + 48° VJ (r —t)? + 48
2 (r — ty 48°
We —_ = 7 + Zt + yg de a ;
a JS (r—t +48? SA (r—f) + 48
: 1 17 Sete ot
or Fy es (r — t)* + 48°,
which are the values of the reciprocals of the greatest and
least radius of curvature; and the positions of the sections
to which they belong are fixed by the angles 0 and 90° + @
obtained from equation (1); consequently the sections of
greatest and least curvature at any point of a surface, called
the principal sections at that point, are at right angles to
one another.
222. The curvature of any normal section is equal to
the sum of the curvatures of the principal sections, multiplied
respectively by the squares of the cosines of the angles which
its plane forms with their planes.
Suppose the axes of w and y to be drawn in the principal
28
planes, then since 0=0 and tan26 = , we must have
yr —t
s = 0; hence
1 :
2 cet
—=rcos'@ + ¢sin’@;
R
consequently, if we denote by p and p’ the radii of curva-
ture of the principal sections corresponding to @=0 and
8 = 90°, we have
1
wi ye
p p
194
therefore substituting, we have for any normal section,
hp roe D daisy
a p
This theorem, due to Euler, joined to that of Meunier,
contains the whole theory of the curvature of surfaces; for
it is hence sufficient to know at any point of a surface the
directions and curvatures of the principal sections, to deduce
the curvature at that point of every other section, normal or
oblique.
223. When the two principal radii have the same sign,
the formula
ha —cos’@ + oa sin® 0
La p
proves that every normal section will have a radius of that
sign, and therefore the surface will be entirely on the same
side of the tangent plane at the proposed point; also the
principal radii will be the maximum and minimum values
of the radius of curvature at that point, and the form of
the surface will be that represented in fig. 65, where AO,
AO’ are the principal radii, and AQ is the radius of the
intermediate section AP.
When the principal radii are equal, as well as of the
same sign, the formula gives R =p, and therefore all the
normal sections have the same curvature, and may be
regarded as principal sections; of this we have examples
at the vertex of a paraboloid of revolution, and the poles
of a spheroid.
224. When the principal radii are of contrary signs,
p positive and p’ negative for instance,
1
1 e €
cos® 9 — — sin’ 0,
it
ft aep p
which vanishes, or R is infinite, when @ has such a value w
195
that tanw = + Je. hence for values of 6 between 0 and +
P
R is positive, or the section situated above the tangent
plane; and for all other values, R is negative, or the section
situated below the tangent plane; also p will be the minimum
of the positive radii, and p’, numerically, the minimum
of the negative radii. Hence at the proposed point, the
form of the surface will be that represented in fig. 66; DBE
being the intersection of the surface with a sphere center 4;
AE, AF sections having respectively a positive and negative
radius, and AB the section made by the limiting normal plane
throu gh AO.
Thus in the hyperboloid of one sheet, we know that
the tangent plane at any point cuts the surface in two straight
lines, limiting the normal planes which give positive and
negative radii of curvature, or which separate the convex and
concave parts of the surface. In surfaces of a higher order,
the limiting normal planes will cut them in curves having
with the corresponding sections of the tangent planes a contact
at least of the second order, since the curvatures of those
sections are infinite. In developable surfaces, if we take a
principal plane at any point for that of sa”, then s =0,
and since rf —s°=0, r=0 or ¢=0; therefore one of the
principal radii is infinite. In fact, we know that the tangent
plane to a developable surface does not cut it, but touches
it along a generating line; and that this generating line is
a principal section whose radius of curvature is a maximum
and infinite, whilst the minimum radius belongs to the section
made perpendicular to the generating line. Hence, the
curvature of any surface at every one of its points may be
assimilated to that of a surface of the second order at one
of its vertices, as will be seen in the following Article,
where we shall take a paraboloid, although an ellipsoid would
do equally well.
225. To determine a paraboloid of the second order,
which shall have at its vertex, a complete contact of the
second order with a given surface at a proposed point.
13—2
196
Let A (figs. 65 and 66) be the given point of the sur-
face, Az the normal, and let the principal sections meet
the tangent plane in Aw, Ay, which lines take for the axes.
Also, let
z= os A se
2p 2°
be the equation to a paraboloid, p, p being the principal
radii of curvature of the proposed surface at 4. Then if AP
be a section of this paraboloid through its axis inclined at
an Z2@to ga, and AN=r, PN =2, be co-ordinates of P,
the equation to AP, putting rcos@ for # and rsin@ for y,
(Art. 101) is
1 Lhe
sat ef cos’ @ + — sin? af :
2 if
which represents a parabola, the reciprocal of whose semi-
latus rectum, and therefore of the radius of curvature at
its vertex,
Hence the curvature of every section of the paraboloid is
the same as that of the corresponding section of the surface,
and therefore the paraboloid has a complete contact of the
second order with the surface.
Cor. What we here assume, viz. that if two surfaces,
having a common point and common normal at that point,
have the curvatures of all normal sections equal, (or, which
comes to the same thing, have their principal sections coin-
cident and equally curved,) they have a complete contact of
the second order at that point, agrees with Art. (211); for
this, from the general expression for the radius of curvature
of any normal section, requires that x and its differential co-
efficients p, q, 7, s, #, should have equal values in the
equations to the two surfaces, when in them we substitute
the co-ordinates w and y of the point under consideration.
197
226. In the equation to the paraboloid, suppose x
constant, and =e, therefore
x y
rel oge
2 pec 2p ec
1;
this is the equation to a section of the paraboloid, or to a
section of the surface if ¢ be indefinitely small, Hence it
appears that if the surface be cut by a plane parallel to
the tangent plane at any point, and indefinitely near to it,
the section is ultimately a curve of the second order whose
center is in the normal, and axes in the planes of greatest
and least curvature; also the square of the diameter in which
the curve is intersected by any plane drawn through the normal,
is proportional to the radius of curvature of the corresponding
section of the surface.
This curve has been called by Dupin, the Jndicatrix
of the surface, because it indicates the directions of the
curvatures; if at any point it be an ellipse, p and p’ must
have the same sign, that is, the curvatures are in the same
direction; if a hyperbola, the curvatures. are in opposite
directions; and if a circle, p= p> and the curvature of
every normal section is the same. For instance, it has been
shewn (Art. 104) that an ellipsoid may be generated in two
ways, by a circle of variable radius moving parallel to itself;
consequently there are four points on its surface at which
a plane, parallel and indefinitely near to the tangent plane,
cuts it in a circle; that is, the indicatrix at those points
is a circle, and therefore the two radii of curvature equal.
These points are called wmbilicit, and are symmetrically
placed in the four angles of the principal section containing
the greatest and least axes. It is manifest that at these
points, a sphere can have a complete contact of the second
order with the surface.
Intersection of Consecutive Normals of a Surface.
227. As two lines in space will not intersect unless
the constants which enter into their equations satisfy a
198
certain equation of condition, so if from any point in a
curve surface at which we have drawn a normal we _ pass
to a contiguous point, the normal at the latter point will
not intersect that at the former, however near the points
are to one another, unless the second point be taken in such
a direction as to satisfy the equation of condition for the
intersection of the normals. ‘The consideration of consecu-
tive normals throws great light on the subject of the
curvature of surfaces, as will appear from the following
propositions.
228. Having given a point on a curve surface, to find
the directions in which we must pass to consecutive points,
in order that the corresponding normals may intersect.
Take the normal at the given point for the axis of x,
and the tangent plane for that of wy, and let h, k, Jd be
co-ordinates of a contiguous point situated in a_ section
through the normal made by a plane inclined at an angle
0 to that of zw, so that K=hAtan@; then the equations to
the normal at that point are
wv —h+p,(2' -l) =0, y —kign(s-J) =0,
where p,, g, denote certain functions of h and k (p and q
d d:
denoting the values of 7. and a
v y
fore each = 0). Then in order that this line may meet the
axis of x, its two equations must agree in giving the same
value for x’ when wv’ and y’ = 0, that is,
at the origin, and there-
/
2 aay
s =—+land vs = —+],
Pi Nh
must be the same. But if the points be consecutive, then
the ultimate values of x’ must be the same when h = 0, that is,
(expanding p, and q,),
ie A h 1
SY scilimit of pee cae 3
rh+sk+&c. r+stan@
199
k tan @
sh+tk+&c. 8s +#tan9
d’°s dz d? x
(7,8, ¢ being the values of qa®? bray ; ia the origin) must
and x = limit of
be identical ;
..s+é#tan@ =rtan@ +s tan’ 0,
2 tan @ 28
or tan 29 = ——————. = ——_
1—tan®@ r-—f#
r!
an equation which being the same as that for finding the
positions of the principal sections through the origin, shews
that there are two, and only two, directions at right angles
to one another in which we may pass from a proposed point
in a curve surface to a consecutive point, so that the normals
may intersect; and that these directions coincide with the
principal sections through the proposed point. Also the
value of
1 | ee poe oe eee
sar+stanO =} frt te (1-2) + 48°}
shews (Art. 221) that the points where the normal is in-
tersected by consecutive normals, are the centers of curvature
of the principal sections.
229. This property of consecutive normals enables us
in some cases to determine at once the principal sections of a
surface at a proposed point; for instance, in surfaces of
revolution the plane of the generating cwrve, supposed to be a
plane curve, is necessarily one of those sections, because in
it consecutive normals intersect, and therefore a plane through
the normal perpendicular to it is the other. The first series
of consecutive normals intersect in the evolute of the generating
curve, the second in the axis; hence, in a surface of revolution,
the loci of the intersections of consecutive normals, or of the
centers of curvature of the principal sections, are the axis of
revolution and the surface formed by the revolution about
that axis of the evolute of the generating curve. Hence
200
we can find immediately the radius of curvature of any
section of a surface of revolution, the principal radii being
the radius of curvature of the meridian, and the portion of
the normal intercepted between the proposed point and the
axis of revolution.
Lines of Curvature.
230. A series of points on a surface determined by
the condition that the normals at any two consecutive ones
intersect is called a line of curvature of the surface. Every
point of a surface, as appears from the above investigation,
is situated on two curves of this kind, whose dircctions
coincide with those of the principal sections through that
point, and cut one another at right angles. Thus in start-
ing from a point P of a surface (fig. 68), there are two
contiguous points P’ and Q the normals at which will inter-
sect the normal at P; and if of these we take only that
which is in the same direction with KP, namely P’, and
then advance in the same direction to the contiguous point
whose normal intersects the normal at P’, and so on con-
tinually, we shall obtain KPP’n a first line of curvature
of the surface; the second line of curvature which passes
through P will be obtained in the same manner, and will
be RPQE; and as these constructions may be repeated for
every point of the surface, we shall thus form two series
of lines of curvature dividing the surface into curvilinear
quadrilaterals whose sides in space cut one another at right
angles.
It must be observed that the lines of curvature which
pass through any point and the principal sections through
the same point, although they have common tangent lines
at that point, will not usually coincide. Thus in a surface
of revolution the meridian PB (fig. 67) and the section
through the normal perpendicular to the meridian QPG are
the principal sections at P; but only one of them is a line
of curvature through P, viz. the meridian; the other line
of curvature being manifestly the parallel PD, which has a
common tangent with PQ at P.
201
231. To find the differential equation to the projection
of the lines of curvature which pass through a given point of
a curve surface.
The equations to the normal at a point wysx of a curve
surface, are
we —ax@t+ p(s -x)=0, y —-y+q(% — 2) =0;
and at a contiguous point v+h,y+hkh, x +1, the equations
to the normal, are
wv —xvx—-h+p,(2’-x-l=0, y-y-kig(s'-x-l=0;
Pir Qs» denoting the same functions of «+h, y +k, that
p and q are of w and y, as derived from the given equation
to the surface x = f(v, y). If these two normals intersect,
their equations must be satisfied by the same values of a’, y’, 2’,
which will be the co-ordinates of their point of intersection ;
eliminating therefore wv’, y’, x’, between the four preceding
equations, we shall have an equation of condition expressing
that the normals intersect, and which will establish a relation
between A and k& or fix the direction in which we must pass
from the proposed point to a consecutive point in order that
the normals may intersect. Subtracting the first equations
from the second, we have
—h+(p,— p) (# - 2)-pil=0, -—k + (4-9) (# - 2) -Ql=9,
and these equations must agree in giving the same value
for 2,
oS h+pl_ pi-P
k+ ql ara
Now let the points be consecutive; then expanding p,, q,, and
/, and retaining in their developments only those terms which
involve the simple powers of h and k, we must have
h + (; *h +sk)(ph k ie A h+sk
limit of ec ei) (2 oe ) = limit of sath :
k+(q + sh+tkh) (ph + qk) sh + tk
But if 6 be the angle which the tangent to the projection of a
line of curvature on the plane of wy makes with the axis of a,
202
we have & = Atan@ ultimately ; hence substituting this in the
above equation, and making / vanish, we have
1+ p* + pq tand r+stan@
tan0d+pqt+q'tanO s+é#tand@
But if we now consider wv and y to be the co-ordinates of the
ae : d
projection of the line of curvature, we have tan @ = one hence,
Lv
substituting and reducing,
{(1 + 9°) s — pqtt (5%)
airy ’
+f +g) r= (+p) = — {14 ps - par} = 0.
It remains to substitute for p, q, 7, s, ¢ their values in terms
of w and y derived from the equation to the surface and to
integrate; the result will be the equation to the projection of
the lines of curvature.
e © . . e ° d °
Since the above equation is of two dimensions in — its
x
integral when completed by the arbitrary constant C’, will be
of the form C’ + Cd (a, y) + (w, y) = 0. Suppose the line
of curvature is to pass through a point for which vw =a, y = 6;
therefore C? + Cq@ (a, 6) + (a, b) = 0, which will give two
values of C; and by substituting them successively in the
complete integral, we shall have the equations to the two lines
of curvature passing through the given point.
Cor. We may find the differential equation to the lines
of curvature of a surface whose equation is given under the
form F' (a, y, ) = 0, as follows.
Let U, V, W have the same meanings as in Art. 217, then
the equations to the normal at a point wyz and at a con-
tiguous point are
X-w Y-y Z-2z
_U Ley eye + W)
Ape da eae y — dy ee Gass tds
Ted Wade? ewe awe
203
the second of which by expanding and neglecting small quan-
tities of the second order becomes
dv dU # nie =
Sly be ya ee ms ss
v + ( v) i +(Y- py wt (Z Aye ye Lares
Eliminating 1, Y, Z between (1) and - we get for the
condition that consecutive normals may intersect,
U(dzdV —dydW)+V(dedW —dzdU)+W(dydU-dedV)=0,
the integral of which joined to the equation F(a, y, x) =
will determine the lines of curvature of the surface. If we
suppose (x, y, 3) =f(a, y)-#x=0 so that V=p, V=gq,
W=-1,dU=rdx+sdy,dV=sdax +tdy, dW =0, we fall
upon the equation given above.
232. The points in which the normal at any point is
intersected by the consecutive normals, coincide, as we have
seen, with the centers of curvature of the principal sections
at that point; and the portions of the normal intercepted
between these points and the point of the surface at which
it is drawn, (which, as we have seen, are the radii of curvature
of the principal sections of the surface at that point,) have been
called by Monge the two radii of curvature of the surface at
that point, as being the radii of two spheres which alone can
touch the surface in two consecutive points. [or if with
the points O, O’ (fig. 65) as centers, and radii OA, O'A, we
describe spheres, the former will touch the surface along 4G
and the latter along AD, because consecutive normals to the
surface in both these sections meet 4O. But if with Q, the
center of curvature of any section AP, as center, and radius
AQ we describe a sphere, it will not touch the surface along
AP, because QP cannot be a normal to the surface, since it
intersects AO; only, this sphére will have the same tangent
plane as the surface at 4, and its section made by the plane
AQP will have with the corresponding section of the surface a
contact of the second order.
From the consideration of consecutive normals we can now
find the expressions for the lengths of the principal radii of a
surface at any point much more easily than by the former
method.
204
233. To determine the radii of curvature at any point
of a surface in terms of the co-ordinates of that point, from
the consideration of consecutive normals.
The equations to the normal at a point wyz of a curve
surface are
we —vt+p(s%-xs)=0, y -y+q(s'—-2)=0,
and at a contiguous point vw +h, y +k, x+/, the equations
to the normal are
wv —v-h+p,(2'-x-l=0, y -y-k+9q,(% —z-/) =09.
The values of w’, y’, 2 which simultaneously satisfy these
four equations are the co-ordinates of the point of inter-
section of the normals; subtracting the two former from the
latter, we have
—h+(pi-p)(® -2)-pl=0, -k+(u-9@ -*)-ul=0,
both of which the co-ordinate x’ must satisfy. Now let the
points be consecutive, and let the second point be situated on
a line of curvature the projection of whose tangent on the
plane of wy makes an 20 with the axis of w, then k = h tan 0
ultimately ; hence, expanding p,, q,, and J in the above
equations, substituting for k, and then making h = 0, we find
the equations
(r + s tan 0) (x’ — 3)
(s + ¢tan @) (x — z) = tan0 + pq + ¢’ tan 0,
which must be satisfied by the same values of x’ and tan@ ;
hence, eliminating tan @, we must have
1+ p’ + pq tan 0,
Le Pe ee ae ee ne
“@aas—pqg 9 14+ ¢- (= a)t?
or (rt — 8°) (x! — x)? - $1 + p*)t- 2pqs+( + qr} (e — 2)
+1+p?4+q°=0,.....(1),;
the equation which determines z’— x, the projection of a
principal radius of curvature on the axis of z, so that
p=-(® -s)V1l +p ?+q.
Hence if we find the two values of x’ — x from the above
equation, and substitute them in the value of p, we shall
205
obtain the values of the two principal radii of the surface at
the proposed point; also knowing z’ — z, the other co-ordinates
w and y of the centers of curvature are known from the
equations to the normal
we —xv+p(s —x)=0, y —y+ q(x —2) =0.
Cor. The roots of equation (1) have the same or different
signs according as r¢—s° is positive or negative; therefore
the curvatures are in the same or different directions (as we
have before seen) according as rf — s’ is positive or negative.
Also in order that the two radii of curvature may be equal
but of different signs, we must have
(1+ p*?)t-2pqs+ 1+q)r=0.
234. In order that the two radii of curvature may be
equal but of the same signs, we must have (solving equation
(1) and putting the part under the radical.equal to zero)
,(1+ p*)t-2pqs+(1+q")r}?-4(1 4+ p?+q’) (rt — s*)=0...... (1)
or {(l+p)t-(1+q)r+2(1 + 9@°)r -2pqs}?
—43(1 +p) (1+ 9°) -— p’g’t (rt —8*) = 0;
and expanding the first term and reducing, we find
S(i+p*)t-(1+q@)r}?4+43 +p’)s—pqr $(1+q°)s—pqt} =0...(2).
Let (1+ p*)s —par=pqu, (1+ 9°)s— pat= pau;
o Sl +p*)v— (1+ q)uh?+4uv = 0;
Qu )*
b l+p)v-(l+¢
Vice eae tes
‘ 4(1+4q°) , 4u
=jJ(l+p)v-(C+q)ul? + 4uv— — —~ u? + ——__;
ee a eae
a {tp )o- (144 _ —) a + Canty 4u°=0...(3),
an equation which can only be satisfied by putting
u= OQ, v= 0,
or (1 + p')s— pgr = 0, (1+q')s — pqt =0,
206
the conditions that the two radii of curvature may be equal
and of the same sign.
From this process it appears generally that the roots of
the equations (Arts. 231, 233) which give the principal radii,
and the directions of the principal sections, at any point of a
surface are always real; for these roots involve the square
roots of the polynomials (1) and (2), which are here shewn to
be equivalent to one another, and to (3) which is a form that
never can become negative.
235. The points of a surface at which the two radii of
curvature are the same both in magnitude and sign, are called,
as has been stated, umbilici. The co-ordinates of such points
must satisfy the double equation
l+p? pq 14+¢
5
ry s t
at which we may arrive immediately by observing that, since
at such points all normal sections have the same curvature,
the expression for the radius of curvature of any normal
section,
V1+pit+g fi +p +2pqm + (1 + g’)m"?
y+ 2sm + tm
R=
must be independent of the quantity m which particularizes
the normal sections; and for this it is necessary and sufficient
that the coefficients of like powers of m in the numerator and
denominator should be equal ;
Pep) pgs A
Also at these points the equation which determines the
directions of the principal sections at any point, since it may
be put under the form
1 Yo —- (1 K
m + {utero m—-uU
PY
becomes identical, and therefore the directions of the principal
sections indeterminate; in fact, all the normal sections having
= 0,
207
the same curvature, each of them may be considered as a
principal section.
And the shortest distance between the normal at the um-
bilicus and the normal at a consecutive point taken in any
direction, will vanish as far as it depends upon small terms
of the first order; and for certain directions it will vanish as
far as it depends upon small terms of the second or higher
orders likewise. For the first member of the equation
(Art. 231)
(h+pl)(1-9)-4&+a90(p-p)=9 (1)
is the numerator of the shortest distance between the normals
at wyx and any adjacent point ~+h, y+k, x47; and ex-
panding p,, q,, / in powers of # and k as far as the second,
we get
P=p+dp+tdp, n=qt+dq+ hdq, l=dz+hd's
where dp=rh+sk, @p=uh’ + 2vhk + wk’,
and similarly for the others, so that the first member of (1)
becomes
sh+(p+dp+ 3 d’p) (dx + 4d’z)} (dq+4d’q)
—jk+(q+dq+4d'q) (dz + £d’z)t (dp + 4d’p).
But if wyz be an umbilicus, all the terms of less than three
dimensions in h and k are identically zero; therefore writing
down only the terms of three dimensions we get
(h + pdx) d’q + pdzdq —(k + qdz)d'p — qdizdp,
which put equal to zero gives a cubic for determining
tand=k—h;
so that in general there will be either one or three directions
in which we may pass from an umbilicus to an adjacent point,
so that the normals shall have a coincidence of a higher order
than for all other adjacent points. If the coefficients of the
cubic should be identically zero, we must have recourse to the
terms of a higher order in the expansion of (1), and we shall
generally determine a finite number of directions relative to
which the normals coincide more closely with the normal at
the umbilicus, than do the normals at other adjacent points ;
208
and those directions more especially deserve the name of lines
of curvature through the umbilicus.
236. To find whether a given surface admits of umbilici,
we must substitute the values of p, g, 7, 8, ¢, derived from its
equation in the double equation
= i
and examine whether these together with the equation to the
surface can be satisfied by real values of a, y, x. Consequently
the number of umbilici of a given surface will in general be
limited ; only if the double equation should be reducible to a
single distinct equation, then that equation joined to the
equation to the surface will determine a curve on the surface
every point of which will be an umbilicus, and which is called
a line of spherical curvature, because about each of its points
the surface is uniformly curved like a sphere.
237. If through all the points of a line of curvature
KPP (fig. 68) we draw normals to the surface, since every
two consecutive oues intersect, they will form a developable
surface whose edge of regression will be the locus of the
centers of the first curvature relative to KPP’. Proceeding
in a similar manner for each line of the same curvature LQQ’,
&e., we shall obtain a series of developable surfaces of which
the edges of regression will form by their assemblage a surface
which is the locus of all the centers of the first curvature,
and to which all the normals will be tangents. Also this
surface will have a second sheet which will be the locus of the
centers of the second curvature, resulting from the edges of
regression of the surfaces generated by the normals along the
lines of the second curvature RPQ, MP’Q’, &c.; and which
will be touched by the same normals as the other.
To obtain the equation to this surface, we must eliminate
w, y, = between the equation which gives x’ — x, the projection
of a principal radius upon the axis of x (Art. 233), and the
equations to the normal. When the two sheets intersect,
their intersection will be the locus of the centers relative to the
line of spherical curvature of the surface under consideration.
APPENDIX.
Tur following Problems, distributed into Sections cor-
responding to those into which the Treatise is divided, will
furnish occasion for applying the results in the Text as they
are successively obtained.
ProspiteEMs on SeEctTion I.
1. To find the equation to a plane considered as generated
by a straight line always passing through a fixed point and a
fixed straight line.
Let , y, x be co-ordinates of P any point in the generating
line CQ (fig. 7) passing through the fixed point C in the axis
of zx, and the fixed straight line AB in the plane of wy. Let
x’, y’ be co-ordinates of Q, and a, b, c the lines OA, OB, OC;
then
oe yf ev ON -c—x# *y- ON. ¢-—z
=+>1;, bout.—=—— = De es ee
Fh A, a OR Cc y OQ G
~+0=1--, the equation required.
2. The equation to the plane generated by a straight
line that always meets a line whose equations are v7 = mz +a,
y=nx-+b, and that remains parallel to a line through the
origin whose equations are w = m’z, y =n’, is
e-mx—-a y—n'zs—b
m — m' n—n'
3. To find geometrically the distance of a point from a
plane.
E4%,
210
Let a’, y’, x’ be the co-ordinates of the given point P
(fig.9), s=Av+ By+c the equation to the given plane ABC.
Through P draw a plane GAH perpendicular to the trace
AR; then this plane is perpendicular both to the given plane
ABC, and to AOB, and contains the ordinate PN. Let GH
be the intersection of the two planes meeting PN in Q; draw
PR perpendicular to GH. Then PR is perpendicular to the
plane ABC; and PR = PQsin PQR = PQcos GHK.
But PQ=PN- QN =2'- (Aa'+ By' +0),
since Q is a point in the plane ABC, for which w=a’, y=y ;
1
and cosGHK = €Art. 29);
J/i+ 4+ B
e— Aa'— By -c_
2/14 A+ Be
the radical being taken with that sign which makes the whole
expression positive.
.. PR, the required distance, =
4. To find the equations to a straight line which cuts
perpendicularly each of two straight lines not in the same
plane, whose equations are given.
Let
/ ,
L=MN2FZ+Aa VC=mese+a :
\, : | be the equations to the
y=nst+b y=ns+b
given lines; then if x= da + By+e be the equation to a
plane parallel to them both, dm+Bn=1, Am'+ Bn'=1,
(Art. 23)
, ,
n—-n m— mm
= Saree O B=————_, 7°
mi —- mn Mm — M7
Let x = A’a + By +c’ be the equation to a plane which
contains the former of the given line and also the required
line, and which therefore (Art. 41) cuts the plane parallel to
both the given lines at right angles; therefore (Art. 23, and 35)
AA'+ BB +1=0, Am+Bn-1=0, A’'a+Bb+c=0;
hence c= — d'a— B’b,
211
and A'(dn - Bm)=-(B+n), B'(Bm-An) = -(4 4m);
“. =A’ (w@—a) +B (y - bd),
or = (dn - Bm) = - (e@- a) (B+) + (y—-b) (44m),
or, substituting the values of 4 and B,
x Ym (m'— m) +n (n'— n)} = (w — a) §m'—-m+n(m'n - mn')}
+ (y —b) j{n’-n+m(mn'—m'n)},
is the equation to the plane containing the former of the given
lines and the required line.
Similarly, the equation to the plane containing the latter
of the given lines and the required line, by interchanging
m,n, a, b, and m’, n’, a’, b,, is
= }m'(m —m') +n (n-n')h =(w-a’) {m-m'+n'(mn'- m'n)}
+ (y-0)}n-n'+m'(m'n—-mn')t.
These two equations determine the position of the line re-
quired, since they are the equations to two planes each of
which contains it; the equations to its projections may be
found by Art. 20.
5. To find the length of the shortest distance between
two straight lines whose equations are given.
If a plane be drawn through each of the lines parallel
to the other, the perpendicular distance of these planes which
will of course be parallel to one another, will be the shortest
distance of the lines (Art. 41).
8
I|
Let
meta) wvemsz+a’
) : py be the equations to the
y= no +b y=nszt+b
lines, x = da + By +e the equation to the plane which is
parallel to the second and contains the first,
- Am'+ Bn'=1, Am+Bn=1, da+Bb+c=0,
which give for A, B, c, the values
n'—n m’ — m
A =————— , B= - ——,, c=- Aa-— Bb.
, a > , , b)
mn—-m nr mn — mn?
14—2
212
Similarly, the equation to the plane which is parallel to the
first and contains the second, is = dva+ By+c’, where
A, B have the above values, and c’ = — Aa’ — Bb’; and the
lengths of the perpendiculars dropped upon these planes from
the origin are respectively (Art. 37. Cor.)
— Cc —C
V1 + ANE an lee
— (c'-c) A (a’— a) + B(b'-b)
nat at aera /1 + A+ B
therefore, substituting for 4 and B their values, the shortest
distance between the two lines
(n’— n) (a’— a) — (m’'—™m) (b’— 5)
V (m'— m) + (n'=n)? + (mn'— m'ny? ;
Cor. When c in the equation to the first plane, and
c the corresponding quantity for the second plane, have
different signs, the origin is situated between the two planes ;
and therefore the distance of the planes will be found by
taking the sum of the perpendiculars. When the shortest
distance vanishes,
(n’— n) (a’— a) = (m’- m) (b'— b),
which is the condition in order that the lines may intersect,
found in Art. 22.
whose difference =
6. Having given the lengths, the least distance, and the
inclination of two opposite edges of a tetrahedron, to find its
volume.
Let 4 (fig. 11) be the vertex, and BOC the base of any
tetrahedron; complete the parallelogram Cy, and join Ay;
let Nw cut each of the opposite edges 4B, OC, at right
angles; then OC is parallel, and Nw is perpendicular to the
plane ABy. Hence, since they are on the same base, and
their vertices are in a line parallel to the plane of the base,
tetrah. dy BO = tetrah. dyBu = 4 AB. By.sin ABy. 4 Ne;
but tetrahedron Ay BO = tretrahedron AOBC,
213
since they have a common vertex, and equal bases in the
same plane ;
‘, tetrahedron AOBC = 1 4B. OC.Nvw.sin 8,
where @ denotes the angle between AB and OC.
7. A sphere may be described touching each of the six
edges of a tetrahedron, provided the sum of every two
opposite edges be the same.
8. A cube may be cut by a plane so that the section
shall be a square whose area is to that of the face of the cube
as 9 to 8.
9. Ifa plane be drawn so that the sum of the perpen-
diculars let fall upon it from m given points a’y’s’, vy’, &c.
shall be always equal to a given line A, then it will always
touch a sphere the co-ordinates of whose center, and whose
1
radius, are respectively F th of the quantities =(#’), = (y’),
>= (z’), and R.
10. The shortest distance between a diagonal of a cube
and an edge which it does not meet is a+»/2, @ being an
edge.
11. If @ be the inclination of two planes /a+my+nz=a,
lv + m’y + nx =, the distance of their line of intersection
wns 1 > ik ea = Aes He
from the origin = arp wo 2a(3 cos 0 + 2°.
ProspLEMs ON Section II.
The following Problems furnish examples of finding the
equations to surfaces from the given geometrical mode of
describing them. As the simplicity of the result will in
every case greatly depend upon properly selecting the posi-
tions of the co-ordinate axes, some instances are here intro-
duced with that special object.
214
1. If a straight line have always three given points in
three fixed planes at right angles to one another, to find the
locus of any other point in it.
Let the planes be taken for the co-ordinate planes, and
let the line meet them in the points 4, B, C, (fig. 23);
and let P be the describing point the co-ordinates of which
are On=a, NQ=y, PQ=z2. Alsolet PA=a, PB=b,
PC=c, ZACa=i, £Cbx=0, Ca being the projection of
CA on wy. Then
z=csini, w=acosicos#, y=bcosisin®@;
2
ne y Pe i aie
—+—+— = 005 @+sin*2z=15
C
a* b*
hence the locus of P is an ellipsoid,
2. To find the locus of the intersection of two planes
drawn through two given lines, so as always to be perpendi-
cular to one another.
Let 44’, BB’, (fig. 24) be the two given lines, AB
their shortest distance which make the axis of x; also take
the plane bisecting AB at right angles for that of «y, and
the line bisecting the angle between the projections Oa, Ob
of the lines, for the axis of « Then if O04 = OB=ce, and
tanaOw=m, the equations to the lines 4d’, BB’, are
Y=oMe, ®=C}; y= —Me, £=—C.
Let zs = Av + By + € be the equation to the plane passing
through 44’; .. d+ Bm=0;
w-en A (v2) Bi PAS BAe ‘hit Se
m me—y me—-—y
Similarly, if * = A’w + B’y -c be the equation to the plane
passing through BB’, d’- B'm=0;
A eae: ppp TOS.
me+y Lina ty)
215
But, because the planes are at right angles to one another,
AA + BB+1=0, ©: ples aaa SES ee Tt eI),
(mv) -—y (mx)? —y
or (m* —1)2°+4+ ma’ — y? = (m? — 1) ce’,
which, since only one of the coefficients is negative whether
m> or <1, represents a hyperboloid of one sheet.
Cor. It may be easily shewn that each of the lines
AA’, BB’, lies in the surface, and that a plane perpendicular
to either of them will cut it in a circle. If the lines are
parallel, m=0, and vy’ + 3’ =c*, which represents a cylinder
as it ought.
3. To find the locus of a point which is equidistant
from two lines given in space.
Retaining the same axes and notation as in Prob. 2, it
will be found that the surface is a hyperbolic paraboloid
having for equation
may + (14+ m’)cz=0.
4. Two points move in straight lines with uniform
velocities, to find the equation to the surface generated by
the straight line which joins them.
Retaining the same axes and notation as in Prob, 2,
and besides denoting by a, a’, the initial values of w for
the points moving in 44’, BB’, respectively, and by 7 : 1
the ratio of their velocities, it will be found that the equa-
tion to the surface is
(1 +1)c(cy—mazx)—(n—1)c(mea—zy)=m(a-na)(c? —32").
5. ‘To find the equation to the surface generated by a
straight line which constantly passes through three fixed
straight lines.
Let a parallelopiped be constructed, having its three
edges (of which no two either intersect or are parallel to
216
one another) in the three given lines; take its center for
origin and lines parallel to its edges for the axes; then the
equations to the given lines will be
Let the equations to the generating line in any posi-
tion be
a—h —k
v=mzt+h, y=nzr+k, a“ atl ;
m
then since it meets each of the given lines, we have three
equations of condition, viz.
a—h b+k
—-a@=mc+h, b=-ner+hk, = ;
m n
and it remains to eliminate m, n, h, k, between these and
the equations to the generating line. The result is
ayz+bzx+cxy+abe=0.
Cor. If the signs of the quantities a, b, c be changed
this equation is not altered; therefore the same surface would
be generated by a straight line constantly passing through
three different straight lines whose equations are
v= ‘t hs Mat
3 3 *
s=-€ Zi ,uc y= 6
The surface, as we know, is a hyperboloid of one sheet; the
latter therefore is the only surface that can be generated in
the manner described above; the origin is its center, and the
axes of the co-ordinates are situated in the surface of the
conical asymptote.
6. To find the surface generated by a line which always
intersects two given lines, and is parallel to a fixed plane.
Let the fixed plane be taken for that of yx, and let 4B
(fig. 24), the line joining the traces of the given lines on that
217
plane, be taken for the axis of x, and its middle point for the
origin. Also, let the plane of wy be parallel to the two given
lines, and the axis of w bisect the angle between their pro-
jections made parallel to Oz. Then referred to this system of
oblique axes the equations to the given lines are
=Me#, Z=C3 Y= —-MX, F=—=—C.
>
Let v=a, y=nz+b, be the equations to the generating
line since it is parallel to yz; then since it intersects each
of the given lines
ma=ne+b, —-ma=—nc+b;
“. b=0, and ma=ne, or eliminating a and n, mzx=cy
is the required equation, representing, as we know, a hyper-
bolic paraboloid.
7. To find the surface generated by a straight line con-
stantly meeting three given lines that are parallel to the same
plane.
Take one of the given lines for the axis of wv, and a line
intersecting it and parallel to another of the given lines for
the axis of y, so that the plane of wy is that to which the
three given straight lines are parallel; and for the axis of x
take a line joining the origin with the point in which the third
given line meets the plane containing the axis of y and the
line to which it is parallel. Then the equations to the three
given lines are
y = 0 v= 0 Y= Ma
ane set =k |
and the equation to the surface will be found to be
kyz+m(h—-k)ex=hky,
representing a hyperbolic paraboloid since the surface has not
a center, and is generated by a straight line that does not
continue parallel to itself. No two of the fixed lines must
be in the same plane; for if so, two planes will be generated,
viz. that containing the two lines and meeting the third
at an infinite distance, and that passing through the point of
intersection of the two lines and the third line.
218
8. The locus of a point, whose distance from a fixed
point is always equal to times its distance from a fixed line,
; y ne NC
is a spheroid with semi-axes ——_,, —=—., c being the
1-n V/1 — n°
perpendicular distance of the fixed point from the fixed line.
Let A (fig. 6) be the given point, HA-=c the perpen-
dicular from it on the given line 47’; produce 4H to O so
that 40 = ——., and consequently HO = — ;- Draw Ox
—n
parallel to 47’ for axis of x, and take OA for axis of wv, and
let P with co-ordinates wv, y, x be a point in the locus so that
Ei ee), aE
ne 2 2
( - 2) +y +n’, MA =an’ ( _- 2) +n'y’;
lL —-n
: be Set ee
or CU itp) Sia ste Stag (I sartt) rag oe
—n
which represents a spheroid with the semi-axes above stated.
9. To find the locus of a point whose distance from a
given point always equals m times its distance (measured
parallel to a fixed plane) from a given line.
Take a plane through the given point parallel to the
fixed plane for the plane of wy, and a plane through the given
line perpendicular to the fixed plane for that of vg, and sup-
pose the axis of y to contain the given point, then
a+ (y—b)? +2? = n'y? +n? (mz+a- 2)?
is the required equation, v= mz +a being the equation to
the given line in the plane of vz.
10. The locus of a point whose distance from a fixed
plane (that of wy) always equals its distance from a line
inclined at an angle a to that plane, has for its equation
y? — vy sin2a + (a — 2°) sin? a = 0.
219
11. To find the equation to the surface of a cable-ring,
having given a and ¢ the radii of its inner and outer boundaries.
Also to shew that the section made by a plane touching the
inner boundary is a Lemniscata when ¢ = 3a.
Let ACB a diameter of a circle whose center is C meet
Oz, a line in the plane of the circle, at right angles in O;
to find the equation to the surface generated by the revolution
of the circle about Oz. Take ON, NP co-ordinates of P a
point in the circle, then
PN’ = AN.NB=(OA-ON).(ON — OB),
or 3° = (w—a)(c - @)
is the equation to the generating curve in the plane of za;
therefore (Art. 52) the equation to the surface generated is
= (/ a + y —a)(e- J v? + y).
To determine the section made by a plane touching the
inner boundary let v = a, then
v= (fat y—a)(c- Vaty’),
or (2° + 9’)? = (c + a) $(c — a) y’ — 242°},
which becomes the equation to the common Lemniscata if
Cras SG:
12. If the axes of a hyperboloid become evanescent,
always preserving a constant ratio to one another, the surface
is changed into its asymptotic cone,
Let the equation to the hyperboloid be
2 2 2
a, b, ec being certain given values of the semi-axes; and sup-
pose the semi-axes to diminish continually, but always to be
proportional to a, b, c; then they may be represented by na,
nb, me, where m is a common multiplier continually growing
220
less; therefore the equation to the surface in its successive
changes will be
a y° x? x
oe ie ie OTe
na nb ae CD
3 a
Cc
therefore, making 2 = 0, or the semi-axes na, nb, ne, evan-
escent, the equation to the surface becomes the same as that
to its conical asymptote, viz.
ei it
13. If two fixed lines pass through a point, the locus of a
third line passing through the same point and making angles
a, a’ with the former, such that tanda.tan4q’ is invariable,
is a cone of the fourth order.
14. To find the locus of a circle whose center is the
origin, and which always passes through the axis of z, and
through an ellipse whose equation is a’y’ + b’a* = a®b’.
15. If three lines mutually at right angles and passing
through a fixed point C intersect the surface of a sphere,
then the center of gravity both of the triangle formed by
joining any three points of intersection, and of the pyramid
having C for its vertex and that triangle for base, will lie in
spherical surfaces described about C as center with radii
vas ae 2 ay 22 _ oO?
=3V 3r 2d and 44/8r 2d
respectively, » being the radius of the sphere and d the
distance of its center from C.
16. If the paraboloid Ja + Uy? =Jlz be cut by planes
through its axis, the directrices of the sections will lie in a
surface whose equation is 4% (Ja + I'y”) + Ul’ (a + y’) =0.
17. If one of the co-ordinates of an ellipsoid be produced
so that the part produced may equal the sum of the other two,
its extremity will trace out an ellipsoid of the same volume as
the original one. =
221
18. The locus of a circle that always touches the axis
of x at the origin, and also passes through a fixed straight
line ay + bw = ab in the plane of wy, has for its equation
x (ay tbe) +(e + y’) (ay + bw - ab) = 0.
Prospitems on Section III.
We shall here give some applications of the principal
results obtained in this section, relative to the projections
of plane surfaces, the transformation of co-ordinates, and the
intersections of surfaces by planes.
1, The pyramid whose vertex is the origin, and base any
plane surface, is equal to the three pyramids whose common
vertex is any point in the plane of the surface, and bases its
three projections on the co-ordinate planes.
Let d, a perpendicular from the origin upon the plane in
which the surface lies, make angles a, 3, y with the axes of
X,Y, %; also let A denote the area of the surface, and a, y, z
the co-ordinates of any point in its plane,
wcosat+ycosf + cosy =d.
Hence, multiplying by 4.4,
Acosa.tx+AcosB.4y+ Acosy.4%=A
ve
or, dy. + Ay. + Ay.
Go | ee
1}
ae
which (since the volume of any pyramid is equal to the
area of its base multiplied by one third of its height) ex-
presses the property announced. The quantities d,, 4,, 4,
will be positive or negative according as a, 3, y are acute
or obtuse.
2. To find the sum of the projections of any number of
areas upon a given plane.
222
Let D’, D", &c. denote the areas; a’, By’; a’, By’;
&e. the angles which the perpendiculars upon their planes
from the origin make with the rectangular axes of a, y, %
produced in the positive directions. Also, let a, B, y be
the angles which a perpendicular to the given plane makes
with the same axes. ‘Then the areas must be multiplied
respectively by the expressions
f , ,
cos a cos a’ + cos (3 cos [3 + cos vy cos +y
cos a cos «+ cos ( cos 3” + cosy cosy”, &e.,
which are the values of the cosines of the angles between the
plane of projection, and the planes in which the areas lie.
The sum of these products will be
S' = cos a (D' cos a’ + D" cos a” + &e.)
+ cos 3 (D' cos 3’ + D” cos 3" + &c.)
+ cosy (D' cosy’ + D" cosy" + &e.) ;
or, if we denote the sums of the projections on the co-ordinate
planes of yz, za, vy by S,, S',, S, respectively,
S' = S, cosa + S,, cos B + S, cos vy;
which enables us, from knowing the projections of any areas
upon three planes at right angles to one another, to find
the sum of their projections upon any plane whose position
is given relative to the same planes. In forming the values
of S,, S,, S, regard must of course be had to the signs
of cosa, cosa’, &c.
3. To find the position of the plane on which the sum
of the projections of any number of areas is a maximum.
Let S,, S,, S, denote the sums of the projections of
the areas on the co-ordinate planes of yx z#, wy respec-
tively; S,, S',, S, the sums of the projections on three
other planes y'2’, s'v’, a'y’ at right angles to one another
having the same origin,
223
. Sy = S, cos wae + S, cos v’y + S, cos v’x,
t ’ , ,
S, = S,cosyxv+S,cosyy + S, cosy x,
S,, = S, cos z'v + S', cosx’y + S, cos 2%.
Hence, squaring and adding and taking account of the equa-
tions of condition to which the cosines are subject,
y2 2 2 2 2 2
Sy + 8, + S, = S, +S, + 8;.
Hence, if the planes of y's’, 2’a’, ay’, be so situated that
the sums of the projections on two of them vanish, the sum
of the projections on the third will be the greatest possible ;
let this be the plane of y’z’, then S,=0, S,=0, and
the greatest sum S'= Sy = V/s? + 5) + S%.
But if we had begun by supposing the projections on
the planes of y's’, 2a’, ay’ given, we should have had
the equation
S, = Si cos va’ + S,,cos xy’ + S, cos wz’.
In this case therefore, S$, = S, cos va’; similarl
>) @ oH >) 3
S, = S,cosya’, S,= Sy cosza';
which three equations determine the position of the plane (or
rather of its normal, viz. the axis of 2’) on which the sum
of the projections is a maximum, since S‘y = JS? as SY + 82.
Cor. Let S” denote the sum of the projections on a
plane whose normal makes angles a’, 6’, y’ with the axes
of w, y, x, and an Z=8 with the normal to the plane of
greatest projection, and let the latter normal make angles
a, 2, y with the axes;
.. cos @ = cos a cos a’ + cos 3 cos 3’ + cosy cosy’
= S', cosa’ + S,, cos 3’ + S', cos ry’ ~~ S
S SS”
.*. S" = S'cos 0.
224
Hence, the sum of the projections vanishes on all planes per-
pendicular to that on which the sum is a maximum.
4. To find the area of the surface of any portion of a
right cone on a circular base.
Let ACM (fig. 33) be a section through the axis of the
cone perpendicular to the cutting plane CPB;
AC=c, AB=c, LCAB=B;
also let Cpb be the projection of CPB on a plane perpen-
dicular to the axis, then
Cb = (c +c’) sind B = major axis of projection ;
and 4/ce' sin 1 =4 minor axis both of CPB and Cpb, since
it Is a mean ioe between the perpendiculars dropped
from B and C upon the axis of the cone;
es (c+c) Vee
. the area of projection Cpb = z sin® : :
But Cpb is the projection of the conical BUMAEE CABP,
every portion of which is inclined at an Z = 90°- +B to the
plane CM,
B(ce+ (c-+¢) See / ce
‘, area of surface CABP = r sine TC Ba
5. Three straight lines mutually at right angles and
meeting in a point, constantly pass through a plane curve
of the second order; to find the locus of their point of
intersection.
Let Aa’?+ BY =C
= 0
h, k, & the co-ordinates of the point of intersection of the
three lines; and let the curve be referred to that point as
origin, and to the three straight lines as axes; let the co-
sines of the angles formed by the new axes of a’, y’, 2’, with
the axis of w, be denoted by m, m, 7; and similarly for the
be equations to the plane curve,
225
axes of y and zs; therefore by the formule of Art. 90, the
transformed equations to the curve are
A(ma'+ ny + rx’ +h)? + B(m'a' + n'y + 7'2'4+ kP=C,
mae + n'y +r es +1=0.
In order that the axis of a may meet the curve, these
two equations must agree in giving the same value for 2’,
when y’ and x’ =0;
“ y
“. A(ma’ +h)?+ B(m'e' + ky =C, ma’ +1= 0, or a’ = —-—;
m
hence, substituting this value of a in the former equation,
and proceeding in a similar manner with respect to the axes
of y’ and 2’, we have
A(m”h— ml)? + B(m’k —- m'l)? = Cm’”,
A (n"h — nl)? + B (nk - n'l)? = Cn",
A (rh — rl)? + B(r"k - 7'l? = Cr”.
Therefore, adding these equations together and taking account
of the equations of condition to which the quantities m,n, &c.
are subject,
A(h?+l?)+B(K+P)=C,
the equation to the locus; which is therefore a surface of
the second order concentric with the curve.
Cor. If the given curve be an ellipse, this equation
becomes, replacing h, k, 1 by «#, y, 2,
Ba” + a’y’ + (a? + b*)2? = a’? b’,"
that to an ellipsoid; if a hyperbola, the equation to the sur-
face is
ba? — a’y? — (a — b*) 2? = a’d’,
that to a hyperboloid of one or two sheets, according as
the transverse axis of the hyperbola <or > the conjugate axis.
In the case of the ellipse, if we remove the origin to
15
226
the extremity of the major axis, by writing w — a for a, the
equations become
we | 02a : ‘ :
——-— + = = 0, Ba +a’y’ + (a? +0°)2 = 2ab'a;
a a b-
now let b°>=2ap —p’, p being the distance of the focus of
the ellipse from its vertex, and make a infinite ;
~YyY=4pa, y+" =4pH;
which shew that when the curve is a parabola, the locus
required is the paraboloid which it would generate by re-
volving about its axis.
6. Three straight lines mutually at right angles and
meeting in a point are applied to a surface of the second
order; to find the locus of their point of intersection.
Let the equation to the surface be da’ + By? + Cx? =D;
then referring it to the three straight lines as axes and
to their point of intersection as origin, the transformed
equation is
A(ma’+ ny +rz' +h) + B(m'a! + n'y +7°x' +k)
+C(m'e' + n'y +7's' +l)’ =D.
Now making successively y’, 2’, =0, a’, 2’, =0, a’, y', =0,
in order to determine the points where the axes of a’, y’, 2’
meet the surface, and calling Ah?+ Bh? + Cl’—- D=G, we have
v?(m?A + m?B + m'?C)+ 220 (mAh +m Bk +m’Cl) + G=0,
y? (nA +n? B+ n'?C) + 2y'(nAh+n' Be +n" Cl) + G=0,
8°? At rP B+ r?C)4+2ek(7Ah +r Bk + 7° Cl) + G=0.
But since the axes of a’, y’, x’ touch the surface, the two
points in which any axis meets it coincide in one; therefore
the roots of each of the above equations are equal, and the
first members are perfect squares ;
“(mA + mB + m'C)G = (mAh + m'Bk + mC),
(A+ n?B+n?C)G =(nAh +n'Bk +n’ Cl)’,
Ate Bir * C)Ge (Ahk +r Bk +4 Cly:
2947
Hence, adding together these equations, and taking account
of the equations of condition to which the quantities m, n, &c.
are subject,
(44 B+ C)G= Ah? + Bi’ + CP,
or, restoring the value of G, and reducing,
A(B+C)h?+ B(A+ OC) 4+C(44 BYP =(A4+B+C)D,
the equation to the locus, which is therefore a surface of the
second order concentric with the proposed one.
Cor. For the ellipsoid, replacing h, k, 1 by a, y, 2,
this equation becomes
(P+e)a+(P+ec)y + (2 +0) = 0B? + ae? + b’c’,
x’ y” 3?
or Se aetee et
& s-
if g, g’, g” denote the three altitudes of the triangle formed
by joining the vertices of the ellipsoid (Cor. Art. 82).
=)15
ae)
If we remove the origin to the extremity of the semi-
axis a, and then make a infinite, the equations become
2 Pad
7 4 en =0, Yrs a4(pt pat app’;
2p ° 2p
if therefore the given surface be a paraboloid, the locus
required is a paraboloid of revolution.
If the given surface be a hyperboloid of one sheet, making
c’ negative, the equation to the locus is
(P-—c)e + (@—e)y + (C84) =e -e(a +8);
the locus is therefore an ellipsoid when c, the imaginary semi-
‘nat ie ab ,
axis, is less than —,———— and of course less than either
J/ a +b?
of the real semiaxes; a point, viz. the center, when
ab F
c¢ = ————; imaginary, till c = b the less of the real semi-
a? + b° :
15—2
228
axes; a hyperboloid of two sheets till ce =@, when it becomes
a hyperbolic cylinder; and a hyperboloid of one sheet for
all future values of c, i. e. whenever the imaginary axis is
greater than either of the real axes.
If the given surface be a hyperboloid of two sheets,
making 6b? and c’ both negative, the equation to the locus is
(PP +c%)a7 4+ (P-a) + (CP -—a)P’=CWC (C+) - Be’,
which is imaginary, as long as a the real semiaxis is less
be :
than ir: af ‘and of course less than either 6 or ec; a
ge Oe, Cc
: be ees 4
point, when a = ————-; afterwards an ellipsoid, till a =e
\/ b+ &
the less of the imaginary semiaxes, when it becomes an
elliptic cylinder ; a hyperboloid of one sheet, till a = 6 when
it becomes a hyperbolic cylinder; and for all future values
of a, a hyperboloid of two sheets, i.e. whenever the real
axis exceeds each of the imaginary axes.
e a e e e e ]
Also, if in the original equation we put D=0, 4 =—,
=
I 1 ; :
B= Be? C= ai we have the equation to a conical sur-
face of the second order, and the equation to the locus be-
comes
(0? + c*) a’ + (a? + c*)y? + (a’ + b’)2* = 0,
which is also the equation to a conical surface of the second
order; one, or two of the quantities a*, b’, c® being negative.
7. To find the equation to the section of an oblique
cone on a circular base.
Let the diameter of the base on which a perpendicular
from the vertex falls, be taken for the axis of « Then
(Art. 49) making 6 = 0, the equation to the surface is, putting
a-2r=a,
cy” = (ex — az) (er+ a's - ex);
229
therefore, the equation to the section made by a plane through
the axis of y inclined at an 26 to the plane of the base, is
cy” = (ecos 8 — asin@) }2cra’ + (a’ sin@ — ecos@)a"?.
Cc
a!
As long as tan@ lies between - and , or 8 between SAB
a
and SEB (fig. 34), 7. e. as long as the plane cuts both sheets
of the cone, the section is a hyperbola.
When tan @ = an or the plane is parallel to ES‘, the
a
section is a parabola. In other cases it is an ellipse, which
becomes a circle when
c’ = (c cos 8 — asin 8) (c cos 8 — a’ sin @),
or, c’?+c’ tan’@ =c’—c(a+a) tan@ + aa tan’@;
c(a +a’)
. tan@=0, or tan@ = —— ie
aa —c
The first value of 0 gives the base of the cone ; the
second gives
tan 9 = tan (SAB + SEB),
or if DAy be the plane of the circular section,
DAE = SAB + SEB, «. DAS = BES;
that is, the cutting plane makes the same angle with one
side of the principal section of the cone, that the base does
with the other. ‘This circular section is called the subcon-
trary section.
8. The sum of the squares of the reciprocals of any
three semidiameters of an ellipsoid mutually at right angles,
is equal to the sum of the squares of the reciprocals of the
semlaxes.
Let a, B, y be the angles which any semidiameter r
makes with the axes, then the co-ordinates of its extremity
arev=rcosa, y=rcosP, =7 cosy;
1 cosa cos" — cos"
a = ae eae ++ :
r a b? Cc ?
230
and forming similar equations for 7’ and 7’, and adding, since
the semidiameters are mutually at right angles, we find
1 vk 1 1 1 sity
APN Ota Fe
9. Ifa sphere be placed in a eee of revolution,
the section of the paraboloid made by any plane touching the
sphere is a conic section having the point of contact for a focus.
]
eT
ph inte
10. The eccentricity of any section of a paraboloid of
revolution is equal to the cosine of the inclination of the
cutting plane to the axis,
11. To shew that among the constants m, n, r &c.
employed in passing from one rectangular system of co-
ordinates to another, the following relations hold ;
man — nr, m =n"r—nr"', m’=nr —n'r;
and similarly for n,n’, n”; andr, 7’, 7’.
Eliminating y’ and 2’ from equations (3) Art. 90 we get
a’ = (n'r" — n"7') a + (nr —nr")y + (nr -—n'r)z
=rAma«+rm'y +rm’sz from (5).
Aman’ —-n"r, rm =n"r —nr”’, Am" =n —n'7;
N= (nr — n"7')? + (nr — nr’)? + (n7! — n'ry?
= (n? 47? +n’) (r+ 7? 4 7) sin? 6 (Cor. Art. 30)
= 1,0 being the angle between the axes of y’ and 3’; and
“, sin @=1. HenceA= = 1, and the proposed results are esta-
blished.
12. ‘To shew that
(mm'm’)? + (nn'n"”)? + (rer)? = (mar)? + (m'n’s’)? + (Mm nr’)?
we have (mm’m”)?
= mm’ (nn + rr")(nn'+ rr’) from (7)
Lf Se AO
=m m'n'n”’.n? + mm” rr? mn’. mm’ ny + m'nr’.m’n’r,
231
and forming similar expressions for (mm‘n”)* and (rrr?) and
adding them together, observing that
m?>m'm” (n'n” + 9’) = — (mm'm")?,
we get 2(mm'm")? + 2(nn'n")? + 2(rr’r’)?
pales ” , ” , ry Ay, ray ou , ” Woe
=m n'r.m' nr +m’ nro mn'r’ + mn'r m'n"r + mn’. mn’
+m'n'y.mn'r + mn’r.m'nr’ ;
and as the second member does not alter when we simul-
taneously interchange m’ with », m” with r, n” with 7’, the
proposed result is established.
13. To pass from the rectangular system of co-ordinates
«, y, x, to another rectangular system a’, y’, 2°; having given
the 2@ at which the planes of wy, ay’, are inclined to one
another, and the angles @, vy, which their line of intersection
makes with the axes of w and w’ respectively.
Let the new plane of a’y’ intersect the plane of vy in
the line Ov, (fig. 29), and be situated above it; and let
Ox’ perpendicular to the plane a’y’ be the axis of x’, and
Ox’, Oy’, the axes of w and y’; and let a sphere be
described about O as a center with radius 1, cutting the
axes in the points a, y, z, &c.; and suppose these points
to be joined by arcs of great circles. Then if va, = ¢,
and Za'a,y=0, or xz’ = 0, the plane of a’y’ and the axis
Oz’ will be completely determined; also if wa, =, then
the positions of Ow’, Oy’, in the plane w,Oy’ will be fixed.
Also we have
, , , , , ,
C= cosxxty cosyx+2 cosza, &C.;
and our object is to express the nine cosines in terms of
0, p, , which may be done by means of the fundamental
theorem in Spherical Trigonometry for finding one side of
a triangle in terms of the other two and the included angle.
First, from the triangle a’a,2, in which Zwv’a,x = 180° ~ 0,
COS v'&@ = COs @ cos yy — sin @ sin W cos 0 ;
232
and changing W into 90° + w,
cos ya” = — cos d sin Wy — sin d cos yy cos 8 ;
also from the triangle z’v,a, in which 2’a, = 90°, za, = 90°— 0,
cos *# = sin @ sin 0.
Secondly, from the triangle w’x,y, in which a,y = 90° — @,
cos wy = cos yy sin @ + sin Wy cos d cos 8 ;
and changing y into 90° + Wp,
cos y'y = — sin sin @ + cos ycos @ cos 0 ;
also from the triangle x’a,y, in which z’a, = 90°, x’ay = 90° + 0,
cos ¥y = — cos @ sin 0.
Lastly, from the triangle w’v,z, in which za, = 90°, 2’v,z =90° - 0,
cos a's = sin Wy sin 6;
and changing vy into 90° + W,
cos y's = cos vy sin @; also cos xs = cos @.
PROBLEMS ON SeEcrTion I[Y.
The following Examples will illustrate the method of
drawing tangent planes and normals to curve surfaces, and
finding their volumes and areas.
1. To find the equation to the tangent plane to an
ellipsoid at a proposed point.
2
® e @ y* Pod
The equation to the surface is — + — + — =1,
a Ds eae
ve sdz y sds
we — a tae sey ee O ae 8 oe coe
a’ ede °° Bt ce’ dy
Webicd dz dz ’ }
Hence, substituting for — , —, their values in
dx” dy
dz ave,
os 4 Tiere ae O — ¥)s
233
the equation to the plane which touches the ellipsoid at the
point vys, Is
fe + 5% (a0) 45% yy) =o,
2 , 2
so 8 we
Cheer apr Raith: Tae Tia eset Lie v= 0,
c c fi b? b?
7
YY 88
or — + ——+—=1.
= b? Cc
Cor. To find where the tangent plane cuts the axes,
; a
making y’ = 0, s’=0, we get w =—, the distance from the
@
center at which it cuts the axis of #; similarly the distances
from the center at which it cuts the axes of y and x, are
5? Cc
— and —
y ra
2. To find the equations to the normal to an ellipsoid at
a proposed point.
dz ca das cy
Since — = —- —-, —= ——%, the equations are
dx a’ xs dy b? 3”
; Cw
a 2 =[2(@'—2), f y= G2 Ge - 2).
Cor. By making »’ = 0, we find the co-ordinates of the
point where the normal meets the plane of wy,
, c\ , Gi
x“ =x(1-4), Yy -y(1-<)s
also, the length of the normal, intercepted between the sur-
face and the plane of wy, is
ct a ca? ‘ x 2
eI Re eR Ber ea Kae
&
3. To find the locus of the extremity of the perpen-
dicular dropped from the center upon the tangent plane to an
ellipsoid.
The equations to a perpendicular from the center on the
23 4
Cw e AREY) 5 J
tangent plane are wv = —-—2 = —~ 3", which, combined
8 P os ee J Bs ’ ‘
with the equation to the tangent plane, give by eliminating #
and y,
(a? + y”? +2”) = cs";
ce? 3 aa b?y’
i ye. f eo aengnat 9 a= : —
in which expressions a’, y’, 2° are the co-ordinates of the
point of intersection of the tangent plane and perpendicular ;
and w, y, * the co-ordinates of the point in the surface to
: au y gt
which the tangent plane is applied so that = ae ia 4 mis 1;
therefore the required equation is
(aa’)? + (by’)? + (c2x')? = (w? + y? + 2)?
Hence if R be the central distance of a point in the ellipsoid
the normal N at which makes angles a, 3, y, with the axes,
and P be the perpendicular on the tangent plane at that point,
we get from equations (1), considering that P? = v? + y? + 2”,
Pcosa=2, a’ cosa = Px, &e.
fh) J im TLL me die
5, 7 =e i eS
P as bt t a
2 Q 2
Gey
P? =P? |—+ = +—)=a@ cosa + Bb’ cos’B +c’ cos’ y,
ie en I pil hl oe
R?P? = P’ (a? + y? +2") = a' cos’a + b' cos’ B + c* cos’.
4. If three tangent planes to an ellipsoid are mutually at
right angles, their point of intersection will trace out a sphere
concentric with the ellipsoid.
Let P the perpendicular from the center on a tangent
plane make angles a, 3, yy with the axes of a, y, x; then the
equation to that plane is a’ cosa + y’cosB + 2’ cosy = P, which
must be identical with the equation to the tangent plane
expressed by the co-ordinates of the point of contact, viz.
ve yy 82
Pnerre ean e o
2
a” b
235
xv P yP xP wana. «38
*. cosa = cos 3 =——, cosy = —, but —~+—4-—=1
a a. > B i? 2) Py Cc ? a b° Cc )
a P\* y P\? 3P\? P
. Pia ( oo (=) 4s (==) =a’cos’a + b* cos’ 3 + ¢ cos? +y
a Cc
as before, which gives the length of the perpendicular in terms
of its inclinations to the axes. Now let there be two other
perpendiculars P’, P’, which make with the axes the angles a’,
, ,
, °
B's a”, B’, ys respectively,
“. P®? = a’ cos’a’ + b° cos’ 3’ + & cos’y’,
P’”? = a? cosa’ + b’ cos? 3” + ¢* cos’ry’” ;
and suppose the three perpendiculars to be mutually at right
angles; then since a, a’, a’ are the angles which a line (viz.
the axis of v) makes with three rectangular axes, viz. the
three perpendiculars, if we take account of the condition to
which they, as well as B, 8’; B’, y, y'5 y's are subject (Art.
5), and add, we find
(Pete Pcs Pie Gay +e
but the first member is equal to the square of the distance
from the origin of the point of intersection of the three
planes; this point therefore describes a spherical surface con-
centric with the ellipsoid whose radius = J a + b? +c’.
In the case of hyperboloids, one at least of the quantities
a’, b’, ce’ is negative, and hence their sum may be negative or
nothing ; in the former case there is no point in space through
which three rectangular planes touching the hyperboloid can
be drawn, and in the latter, the center is the only point which
has that property.
Cor. If we remove the origin to the extremity of the
semiaxis a, and then make a infinite as in Art. 72, the equation
to the ellipsoid will become
236
and the equation to the sphere will become
oe —2an+y +2" =2ap — p> +2ap — p”,
or, making a infinite, - # = p +p’; therefore the locus, in the
case of paraboloids, is a plane perpendicular to the axis of the
surface, and at a distance = — (p +p’) from the vertex.
5. Three planes mutually at right angles constantly
touch the perimeter of a plane curve of the second order ; to
find the locus of their point of intersection.
Let the equation to the curve, and to one of the rect-
angular planes, be respectively
Li beet of; : : ;
atiani, acosaty cos3 + % cosy =d;
then making x’ = 0, the equation to the trace of the plane on
wy is x cosa + y' cos3 = d, which must be identical with the
equation to the tangent of the curve
U ,
CR yy vd yd
Sy yer Boel that cosa = GP? cos} = ra
2
way Ned ate
= =") an (4) = a° cos’a + 6’ cos?3; and similarly
, © id
d® =a’ cos*a + 6 cos*', d’? = a’ cos’a” + b’ cos’.
Hence, adding, d?+ d? +d”? = a? + b*, which shews that
the locus is a sphere, concentric with the curve, whose ra-
dius = \/a? + b°.
Cor. If the curve be a hyperbola, the radius = J a —
therefore we must have the transverse greater than the con-
jugate axis, otherwise the problem is impossible; if a = 6,
asin the rectangular hyperbola, the locus is a point, viz. the
center. If the curve be a parabola, it may be shewn, as in
Prob. 4, that the locus is a plane perpendicular to the plane
of the parabola passing through its directrix.
6. If two concentric surfaces of the second order have
237 :
the same foci for their principal sections, they will cut one
another every where at right angles.
x’ y 3? £ y 3
Behr ag chs bing sails oe y Ga = 1,
be the equations to the surfaces; then the equations to the
tangent planes are
Po
nx ie! y cena? © YY Se
‘ b) Sha. F Tiare 3
a? vege a’? b? ce? ‘
and in order that these may be at right angles, we must have
Ga?” BBP ee?
(hence one of the surfaces must be a hyperboloid, for some of
the quantities a’, b*, &c. must be negative;) and this equation
gives the relation among the co-ordinates of the points in
which the surfaces may intersect at right angles. But by
subtracting the two equations, we find
1 1 3 1 1 4 1 1 ,
aba as) ge OND Abbi F 2 Netacok eames
for the relation among the co-ordinates of the actual points
of intersection, which must be identical with the former if
the surfaces intersect every where at right angles. Hence,
equating the ratios of corresponding coefficients,
a—a’?=? —-b? = — ec”,
6 , , iD , © a
or @ —- P=a’*-b*, @-eC=aa*?—c*, Yb -—e’=b? —c?,
which three equations express that the principal sections of the
surfaces have the same foci, or that the surfaces are homofocal.
Cor. In like manner, if the surfaces have not a center,
and we represent their equations by
2
Y
Ir la ars
b*agiia
i Za 3°
(Sat aa ihe
by eliminating w between these equations, and comparing the
238
result with the equation expressing that the tangent planes are
at right angles, we find that the surfaces will intersect every
where at right angles, provided
BF 0 ae ee — 2p”
12
CC — aes
2a Pig at 2a 2a)
that is, the foci of the principal sections of the surfaces must
be coincident.
7. If three surfaces of the second order have the same
foci for their principal sections, and if each of three planes
mutually at right angles touch one of them, the point of inter-
section of the planes will trace out a sphere.
8. If the tangent planes to two homofocal surfaces of
the second order be parallel, the difference of the squares of
the distances of those planes from the center of the surfaces
is invariable.
9. The locus of the projection of the origin of co-
ordinates upon the tangent planes to wyx = a*, has for its
equation 27a°wyz = (a + y’ + 2°)’.
10. If the equation to a surface be \/v+\/y+\/ 2=\/a,
the sum of the portions of the co-ordinate axes, intercepted
between the origin and tangent plane at any point, is constant,
11. Ifwyszbe the point in a curve surface the tangent
plane at which forms with the co-ordinate planes a pyramid
of the least volume, the equation to that plane will be
12. The number of normals to a surface of the n‘" order
that may pass through a given point cannot in general exceed
V—-n +n.
Take the given point for the origin, and suppose the
equation to the surface to be
w= 3" + Vis"! 4 Vis"? + ...4+ 0,124 V, =0,
239
where V, denotes an integral function of w and y of r
dimensions; then
du du R+@Q ie
RS a5 ee © =0, or +Qp=0;
similarly, S + Qq = 0;
consequently the points of the surface through which the
normal drawn from the origin may pass, will be determined
by the values of w, y and x that simultaneously satisfy the
three equations
u= 0, Qe -Rz=0, Qy-—Szx=0; (1)
and as each of these equations is of m dimensions, they will
determine * points. But if x =0, w and Q are reduced to
V, and V,_,; and equations (1) are replaced by
%=0, y= Very 0s
which determine m (m — 1) points in the plane of vy that may
be foreign to the question, and such (not that the normal at
them passes through the origin but) that at them the tangent
plane is parallel to the axis of x; consequently the number
of points determined by equations (1) that satisfy the proposed
condition cannot in general exceed n’?— ”? +n.
13. If p, 7 be the perpendicular on the tangent plane
and the radius vector at any point of a surface, then p* =r
will equal the length of the perpendicular on the tangent plane
at the corresponding point of the surface which is the locus of
the extremity of p.
Let wv, y, be the co-ordinates of a point in the first
surface, a, 3, yy those of the corresponding point in the second
surface; then we have the relation
av+Bytys=p=ae+h?+y’, (1)
since a, 3, y are the co-ordinates of the extremity of p. But
this being the equation to the tangent plane at (wyz), the co-
ordinates w, y, may receive indefinitely small variations
240
without the magnitude or position of p being altered; there-
fore we may differentiate considering a, 3, -y as constant,
. adw+ Bdyt+ydzx=0. (2).
Now if V = f(a, B, y) = 0 be the equation to the surface
traced out by the extremity of p, and P the perpendicular on
the tangent pas to that surface,
poate Bagt va, +N (a) + Ga) + Ge):
and from the oats to fe surface
ond +o 5 dB + dy =0 (3)
But differentiating (1) pa it and taking account of
(2) we get
(w - 2a)da+ (y-—2B) dB + (z - 2y) dy =0;
hence comparing the two latter equations we have
d
Therefore substituting for ae &c. in the value of P, and
‘A
reducing by means of equation (1) we get
_a(w7—2a)+BYy-2B)+y(8@-2y) P+ PhP +y Sa
J (@—2a)* + (y— 23)? + (s-2y)? "a +y°42? 7
Application of the Formule for the Volumes and Areas of
Surfaces to Examples.
1. To find the volume and area of the surface of a
sphere, (fig. 41). Here
av
— — SL ae talers
dudy od / a v Y's
241
+ C,
dV é
. —-=taVYa-y-wv#+4(ae-y’) sin
a” — y°
and integrating between the limits 7 =0, r= MQ= /a — ¥°,
*, integrating between the limits y = 0, y= CM,
volume AMBQ = (a’y —4y’),
3
and, making y = a, vol. ABCD = = = = ae
ae Andra
*. vol. of whole sphere = 8 a= sos 2
as as\4 dz\?
Ranger a, (<* ca
oe dady / i e %3 i
ds
oe aot ib +C;
dy / a? Le y?
and taking the integral between the limits #7 = 0, v=\/ a?-y’,
dS 1wa_
dy 2”
.. integrating between the limits y = 0, y= CM,
area of surface PBAQ = 7,
; Ta
and making y = a, area of surf. DAB = ai
2
7 a
‘, area of surface of whole sphere = 8 ers 47a’.
16
242
2. The volume of the solid which is common to two
equal spheres that have their centers in one another's surfaces,
5ara
12
1S
8. To find the area of the spherical surface intercepted
by a right cone whose vertex is in the center of the sphere,
and whose vertical angle is 2a.
ds > @
As in Ex. 1, — = asin7=1———— +0,
dy
fay
and integrating from # =0 to w= / a? sina — y’,
ds Vie asin’ a — y”
—— = asin~ ag
y
a —y?
and again integrating from y=0 to y=asina (Integ. Cal.
Art. 89),
9
4
a
S= a (1 — cos a).
4. To find the area of the spherical surface intercepted
between two meridians inclined at an angle @, and a plane
perpendicular to one of them at a distance c from the center,
and parallel to their line of intersection. |
dS :
= Pte See Bt Cc
dx J a? — x
: wv tan @
=a sin“} , from y=0 to y=a@£ tan@;
\/ a — 2
x tan @ v
= - @ tan @ f ——______
Va wv # (a? — x) \/a? — x sec? 0
* §'=awsin—}
xv tan@ een es
f SIN Ge eee OO,
Va — a / a? — x?
=aasin=!
243
and integrating from #=c to w=a, we get the surface
required,
Pie we OSI 7 : c tan @
2S = 2a" sin~* ————— _— 2acsin™
en
J/ ae aA ile?
5. To find the volume intercepted between the surfaces
whose equations are
Party, e=ar+y’
dxdy
dV ——__— yt /e+ ye
Ten dy Vara + hat log (VEVE EY)
and taking the integral from y=0 to y =\/a? — a’,
9
dV = 3 a a — a
Lan/@ — a + yeaa
av
therefore integrating between the limits vy = 0, w =a, we get
(Integ. Calc. Ex. 5, p. 97)
Te ra ra
le ail See la ae
ey rye
Aa a
*. volume required = 8V =
6. The axes of two equal cylinders intersect at right
angles, to find the volume and superficial area of the portion
which is common to both.
Let OC (fig. 46) be the axis of one cylinder, O.4 that
of the other; then the equation to AB or to the cylinder
whose base it forms is s = 4/ a? — y’,
av dV
=<\/a’>—y’, and [ma Bais nk sal 8
ia dady —
16—2
24:4
integrating between the limits 7=0, w=ON= J a -y’;
hence, integrating again between the limits y =0, y=a
ae 16a3
vol. ADMB = oe and whole vol. intercepted =
dz -y dz ad’ S a
A 1 ——— ee | ee “gee 0, one = —_}~ ——_ §
Bane? dy \/ a? a y : daw dady (yg y
- ax
+ C=a from #=0 to va/fa-y <n
at é
hence area of ADPB = a’ = area of CDPB, and area of whole
surf. = 16a’.
7. If the axes of the equal cylinders intersect at an angle
16a?
a, it will be found that the volume common to both is
8sina’
16a’
and the surface
8. The volume of the figure generated by a straight line
which moves parallel to the plane of yx and constantly passes
through the axis of # and through the perimeter of a circle
whose plane is perpendicular to the axis of y and center in
that axis, is 2 1 za’b, a being the radius of the circle and b
the distance of its center from the origin.
9. To find the volume contained between the co-ordinate
planes and the surface
we NEE Nae
The section of this surface by a plane parallel to wy at F
2
distance x’ from it, is the parabola vee J! =l]-— Wee
c
and the area contained between this curve and its two tangents
245
ab
6
—
—
3 » e e .
(1 - /*) ; consequently integrating this from z’=0
Cc
, “ 1
to x = c, we get the required volume = iG abe.
; x y° 3"
10. The volume included by the surface act ra +—=1
a . a
2n
is arabe.
11. The volume intercepted in the positive compartment
c!
of vy by the surfaces vy = az, 2 +y’ =", is }—; and the
a
area of the intercepted part of the surface vy = az is
Ta ce?) 3
erat: e fereahe| eT
6 ( +5)
12. To find the differential coefficients of the volume
and surface of a solid, bounded by two of the co-ordinate
planes and a plane parallel to the third, and by a plane
through one of the co-ordinate axes, and by a given curve
surface.
Let DMQ (fig. 40) be a curve surface, QD being its
trace in the plane of za. Draw planes MP, Np parallel
to yx, and MA, NA planes passing through Az. Let
a, y, x be co-ordinates of the point W and ¢ =tan z HAP,
therefore y= wt; also let r+h, (w~+h) (t+ e) be co-ordi-
nates of the point K. Then AAHP = 2x’ t,
h
and area KH = (wx + hy - 2° =he («+ | :
Now the volume DMP may be considered as a function
of # and ¢; and by subtracting the increments arising from
their separate variations from that arising from their simul-
taneous variations, it may be shewn that
246
hence, taking the limit of both sides, by making hf and e
vanish, in which case vol. MKmk, becoming ultimately a
prism whose base is HK and altitude MH, =xhe (a+ 5h),
we have
a Vi
| dn dat
Also, if S’ = area of the surface DMQ, by a similar process
we may shew that
surf. MN ds
“ he © dedt
and taking the limit of both sides, in which case
UR
+ &c.;
h
surf. MN = HK secry = he (w+ 3) sec 4,
dS Ne —tds\? Gy
i Vv <= = WW —— — -
we have HE @ SEC ty Vv + ( ) +
dx dy
In applying these formule to examples, the substitution
of xt for y gives
av ds
ei YL DAS (yy Fee
Tite Me EES
integrating with respect to «#, considering ¢ as constant,
between the limits 2 =GN=¢(¢), «= GN’=\, (t) (these
values being obtained from the equations to the bounding
cylinders AN, A’N’) (fig. 42) we obtain quantities, functions —
of ¢ only, which when multiplied by d¢ are respectively the
ultimate values of the volume and surface of the wedge
PM’; and these integrated between the limits =m, t=m’,
will give the volume and surface contained between two
planes inclined to zw at angles whose tangents are respec-
tively equal to m and m’.
EOE (2, 07)
13. <A sphere is pierced by a cylinder the diameter of
whose base is equal to the radius of the sphere, and their
surfaces are in contact; to find the volume and area of
the surface of that part of the sphere which is intercepted
by the cylinder.
2447
Taking the center of the sphere for origin, and planes
drawn through the axis of the cylinder and perpendicular to it
for those of xv and wy, and using the formula of Prob. 12, we
have
RV es ae Bie.
=0s=a\/a—-—-v -—y=v7V/a-—-(1+2%)2';
Pee J y V/ lier ts)
3 3
peSlimog, 90 Bie 2 \etea tau foo
dt 714+¢ Sll+ 2 aa epi?
a
from v= 0 to x = AN = acos* MAN = siete (fig. 44) ;
2
V at yrds : egies @
ie = —< tans a ob
3 V4 Pat? 7 a+e) ¢
and integrating between the limits ¢=0, t= 0,
3 3
vol. ABPM =—{" 2} =e aaa
Now vol. ABCD ="— ; therefore the part of ABCD not
9g 3
comprized in the cylinder =—— . Consequently, if the sphere
be pierced by two equal cylinders, the part not comprized in
2
them = Aaa
Again, for the area of the surface
as Ry: dz\?
dx
dx dt
(j=) <« Cage nk IE Sa
dy a 5 Ve a(t 2).
lel arte Vea FOFF) = 0 : -ami
dt Tee 1+ (1+#)8
taking the integral between the limits # = 0, # = ra as
cs
.S= a? (tan4 + 5 os
+ C,
/1 + =) +
248
and taking the integral between the limits ¢=0,f= 0,
area of surf. BDP =a (z _ 1)
2
Now area of surf. BCD = a therefore part of BCD not
intercepted = a?; and if the whole sphere be penetrated by
two equal cylinders, the area of the surface not comprized
in them = 8a? = 2 (diameter)’.
14, To find the volume of any wedge of an ellipsoid
intercepted between two planes passing through an axis.
(fig. 41)
av / oY? wi MT lineat®
——_ = &@2 = C# se aoa ae 1-—-@ Gaace
dadt a
dV betes ad he ie okG
BP Whi Pree “(5+5) ioe ee
a BP oe
1
integrating between the limits 7=0, v= MQ =
lea’:
a Be
foram the ploncice ees a eee
or in the plane of wy, — saag8 em at ait paimaas
be t
= —— tan? a = vol. of wedge ABQC,
b
and making ¢ infinite, vol. ABDC = =. ze
Ea yol fof whnlenel i neord ease
15. To find the volume of a spherical sector which is
inclosed by the spherical surface, and a conical surface having
the center of the sphere for its vertex and one of the circles of
the sphere for its base.
249
Let a be the radius of the sphere, and 2a the vertical
angle of the cone; then by the formula of (Art. 118),
3.3 dV 3 3
SP Woe NBL cow
dddp 8 ° d¢
between the limits @=0, 0=a; therefore integrating again
between the limits ¢ = 0, @ = 27,
Q7a°
volume of sector = ee (1 — cosa).
16. The volume contained between the co-ordinate planes
and the surface which is the locus of the projection of the
origin on the tangent planes of wyx = a*, may be shewn by
3
9a
polar co-ordinates to equal tk
17. To find the area of the surface of an ellipsoid.
Let DR (fig. 41) be a section parallel to the plane of
vy at a distance =ccos@ from it, and BR a section through
the axis of x, inclined to the plane of s# at an ZW, and let
tan = = tan @; and let S = area of surface BDR; then S is
a function of @ and @, and if DR’, BR’ be sections cor-
responding to the angles 0 + 60 and @ + d¢, we shall have
ad’ § ees aa? of surf. RR’
dpd0 dopo
Now if OR =p, Or = p+ Ops and ry = inclination of tan-
gent plane at R to wy we have ultimately
area RR’ =(its projection on wy) sec y = pop ow sec x.
Q
. ee 3 '
But the equation to RD is 2 + a = sin’@,
? cos? ? sin?! ? cos? :
al a + —— = core (1 + tan? ~) = sin’,
Layer a ae and Gye eon Gee 88 :
cos Vi cos Vy
ay od; ultimately ;
, area RR’ = ab sin O cos OS Ped sec y;
eer
+ aon" ab sin@cos@ — Nf, - ie
=absin@\/1— Su. cos’ @ + y sin’ p} sin? @,
a— Cc 5? — ¢?
making a gli a sean
and observing that w= pcos\=asin@cos¢, andy =bsin@sing;
this can be integrated only by elliptic transcendants and gives
the whole area
Q2ab
ae ge ee ee I cp ete a
Ferg leF@+@ - EC}
CAML
where a cos a = ¢, and the square of the modulus = est) .
b* (a — c*)
If the ellipsoid differ little from a sphere, making
p. (cos ~)” + v (sin P)’ = uv
(which is a small quantity in the case proposed) and expanding,
ai ab sin 6 $1 — Lu (sin 6)’ HE? (ai 6)*-&
= — — —— n = K,
dpdé 2 2.4 re
.”. integrating between the limits 9 =0, 9=4z, (Int. Cal. p. 126)
ds 1 1
FES en So Tae oR ete
and integrating again between the limits 6=0, P=47,
we find the area of the surface of dth of the ellipsoid
aw ab 1 1
ae me 2a es ies 4.
Gee 3 ihesjl I Rees:
where P}= 3 (uty), Po =p? Ft ae ——-; &c.
251
and in general P,, is the coefficient of "in the developement
of (1 — wx)72. (1 — ve)73.
18. ‘Lo find the area of the surface of an oblique cone on
an elliptic base.
Let the perpendicular VW (fig. 47) upon the base from
the vertex of the cone fall upon CA the major axis of the
ellipse, CM =f, VM=h; w«=asingd, y = bcos @, co-ordi-
nates of P, BP=s, VPT the tangent plane along VP,
cutting VMP’ in VQ, VH a perpendicular upon PT,
S'= area of surf. BVP; then ultimately
=e op = surf. VPP’ = triangle VPQ = 4PQ.\VH=1VH.0s;
s CITE: YH L/h (a? — cc’ sin’d) + 0° (a —fsin d)’,
as will be found upon calculating VA, since © = / a —c'sin’ —c’sin <0
where c’ = a’— b*; this can only be eral by elliptic func-
tions, unless 2 and f be so related that b°f?=c’(h? +6?) when
the quantity under the vinculum is a perfect square, and
awabf
ins
whole surface of cone =
19. To find the whole area of the surface which fs the
locus of the projection of the center of an ellipsoid on its
tangent planes.
Employing polar co-ordinates we have (Art. 120)
a’ 3 aq J ; ae :
ee = a ry sin@, since == Pe ELOumiam). 255),
where 7” is the radius vector of the ellipsoid at a point where
the normal makes an angle @with the axis of zs and its
projection on the plane of wy makes an angle @ with the axis
of w, and r is the perpendicular on the tangent plane to the
252
ellipsoid at that point; consequently by Prob. 3, p. 234, we have
a’ s
TVR
= sin 9 /(a’ sin 0 cos p)* + (B’sin 6 sin )” + (c’ cos 6)”.
But for the surface of an ellipsoid with semiaxes a’, b’, c’
we have (Prob. 17),
as
dodo
which coincides with the preceding expression if
=sin 0\/ (a’b’ cos 0)” + (b'c' cos @ sin 0)’ + (a’c’sin @ sin@)?
, ‘
ab=c’, be =a’, ac =0';
on Cla sO Clee eeL
or a4 = “eas ee Cc = - °
a b Cc
Consequently, as the limits of the integrals are the same, the
area of the surface which is the locus of the projection of the
center of an ellipsoid (with semiaxes a, 6, c) upon its tangent
planes, is equal to the area of the surface of another ellipsoid
ac ab
: ; be
with semiaxes —, —, —.
= UaeG
ProspiEMs oN SEcTION V.
The following are examples of finding the equations to
curves in space, and drawing tangents, normal planes, &c.
1. A sphere is pierced by a cylinder the diameter of
whose base is equal to the radius of the sphere, and their
surfaces are in contact; to find the nature of the curve formed
by their intersection.
Take the center of the sphere for the origin, and planes
drawn through the axis of the cylinder and perpendicular to
it, for those of zx and wy respectively; and let 4N =a,
NM =y, MP = (fig. 44) be the co-ordinates of a point P in
the curve; and dB =2a. Then because P is in the spherical
253
surface, vw + y® + 2” = 4a°, and because M is a point in the
semicircle a + y° =2aax; therefore the equations to the pro-
jections on wy and sa, or the equations to the curve, are
y=2ax—- 0°, 8° = 4a" -2aa...(1);
the latter representing a parabola BD, vertex B, and axis
AB; if we eliminate wz, we find 4a’y’ = 4a’z” — x‘ for the
equation to the third projection on yx.
Differentiating equations (1), we find
dy a-@ d’y a ods a d’ x a®
dx Tee. dat ens eda" 3 |
Hence the equations to the tangent P7' at a point wysx, are
a—
Yen —
wv a
(a'-@), x -g=-—-(a'-2);
%
and the co-ordinates of its trace 7’ on the plane of wy, making
% = 0, are
3
? ,
e©=404—-% Y= 7
The equation to the normal plane is
v—x@2+ rapa! Yy) -=(s/-2) = 0;
and the equation to the osculating plane
(y —y)y — (e — x) x= (a — a) fax’ + (a a) yf.
Also if s = length of arc DP,
toe wie ie a) (=) -! Ee) ian
da Q2v 2a—2
let w = 2acos'@, .. —_ Dp) are: — $sin’d
therefore, integrating between the limits 0 =i7, 0=0,
arc DPB = elliptic quadrant semiaxes 2a WA 2, and 2a
2. To find the equations to the Helix.
254
Whilst the rectangle ABCM (fig. 45) revolves uniformly
about its side AB, the point P moves uniformly along the
parallel side MC, and generates a curve called a helix.
When the rectangle is in the plane of zw, let P be at
M, and let the velocity of P= times the velocity of M,
“*. PM=n.arc DM; also lett dAN=a, NM=y, MP=s
be co-ordinates of P, and 4M =a,
xv es
- g=mnacos-!—, and y=\/a@_-a’;
a
which are the required equations; the former may also be
written ¢ = ma sin! J
a
Cor. Since the corresponding increments of DM and
PM are always in a given ratio, the curve DP always cuts
the generating line CW of the cylinder on which it is traced
at the same angle; and the line touching DP is always inclined
to the plane of wy at a constant angle a whose tangent = 7 ;
therefore also the length of the helix
x
DP =seca.arc DM =asecacos™!-.
a
3. Let A (fig. 36) be the pole of the great circle «By
of a sphere whose center is C; and as the quadrant AB
revolves uniformly about AC, let the point P move uniformly
along it; to find the equations to the locus of P.
When the plane ACB coincides with za, let P be at A,
and let B’s velocity = times that of the point P,
- arc Ba=n.arc AP;
also let CN=a, NM=y, MP=2s, be co-ordinates of the
point P, and AC =a,
| g
- tan? 2 = ncos7 -, and 2? +y?+2° =a’;
v a
which are the required equations. If m=1, these equations
may be transformed into
av=x /a— 2°, ay = a’ — 2’,
255
Cor. To find the area of the spherical surface included
between the co-ordinate planes and the path of P.
Let AB’ be a position of the quadrant very near to AB,
meeting the curve in P’, and a section of the surface through
P perpendicular to AC in p; then if ACP=0, ACP’=0 +60,
we have the increment of the area dw BP = PP BB’ = PpB'B
ultimately ; hence (Int. Cal. p. 182.)
oS = BB’.PM=n.P'p.PM=n.a00.acos0, ultimately,
or on = na’ cos 0.
Therefore area of surface 4a BP = na’ sing.
This result may of course be obtained by means of the
formula of Art. 113.
Also if V be the volume contained between the co-ordinate
planes and a conical surface whose directrix is the path of P,
we have, making 4 BC# = qd,
d2V
=a sin@;
dpdé
dV 3 3
°, Ee 6 Rec) = 4 cosh: from gouCP Shh to@=90°:
da 3 3 n n
; 3
ee a from @ = 0 to @ = BCa,
5 n
or volume Bed Ce sin 0,
3
4. To determine the curve to be traced on the surface
of a sphere so that its length shall always be equal to m times
the ordinate x of the describing point.
Let x, y, x be the rectangular co-ordinates of the de-
scribing point, and r, @, @ the polar co-ordinates, then we
are required to assign a relation between @ and @ which will
2
1 a. a Ss 2. 2 e 20 ® ° t t 5
give s = ”2Yr COS GU, OF do =nr sin’G, since 7” 1S constant 5
therefore (Art. 132)
256
} dq * dp eee
] 2 a B. a 2 —_—- «= 2 —_
+ sin @ ( ) m sin’@ or amass nen a8 sin*@? — 1;
.O@+C= eo + sin7* Ga (Integ. Cal. p. 120),
n—
J +C=ncos™! +sin-!
NB x
VA eee r/n?— J 2 1/7 ~ 2
the equation to the projection on the plane of yx; the
constant C' may be determined if any point be given through
which the curve is to pass.
or sin~!
5. To find the line of greatest inclination at any
point of a curve surface; that is, among all the straight
lines which touch the surface, to determine that whose in-
clination to the plane of wy is the greatest.
As all these lines must lie in the tangent plane at the
proposed point, the line required is obviously that which is
perpendicular to the trace of the tangent plane on wy. Now
the equation to that trace is
, Pro ad ’ ! Pe
—-y=-—--(w-a)--3; “yY -y=-(e-@
y F i y at )
is the equation to a line perpendicular to it, passing
through the point (a, y), and is therefore the equation to
the projection on the plane of wy of the line required ;
the other equation
v-e= A gst (2’ — x)
is obtained by combining the former with the equation to
the tangent plane (in which the line in question lies) so as
to eliminate y/ — y.
6. Hence we can determine the curve of greatest in-
clination on any surface, that is, one whose tangent is always
inclined at the greatest angle to the plane of vy.
Suppose y = (wv) the equation to the projection of the
O54
curve, then the equation to the line touching the projection at
a point (a, y) (which must be identical with the equation to
the projection of the line of greatest inclination) is
’ dy, d
goers Hed
“da p
If, therefore, we determine p and g from the equation to
the surface, independent of x, and substitute them in the
above equation and integrate, the result will represent a
curve on which if a cylinder be erected, it will intersect
the surface in a curve having the property that its tangent
is always inclined at the greatest angle to the plane of vy.
Ex. Let the surface be of the second order represented
by the equation
A & By
A B ; 2 D oe ect hg perp a ar ie $
ities: ae & ? Exige ©GR
B e ,
os = = a or, integrating, y4 = Cx.
wv v
Hence the curves of greatest inclination are of double cur-
vature, and are projected horizontally into parabolic curves
if 4 and B have the same sign; if, however, the point of
departure, which determines the constant C’, be in a prin-
cipal plane, z# for instance, we have y=0 when @@=@,
and therefore C’=0; hence the curve of greatest inclina-
tion is the principal section in zw since its equation is
reduced to y = 0.
7. On a given surface to determine the curve of con-
stant inclination, that is, one whose tangent line is always
inclined at a constant angle to the plane of vy.
Let x = f(a, y) be the equation to the surface, p and q
the partial differential coefficients of x; also suppose y = ¢ (a)
to be the equation to the projection of the curve of constant
inclination, and yy the angle at which the tangent line at a
point wyx is inclined to the axis of x; then (Art. 131)
ds dz i (=
2 Ser te =)
OT dae da’ aa Tpuata sp Ge
17
258
Be oy Gaal la.
or + {——]) = tan —};3
dx Y \de
but, vw, y, x being co-ordinates of a point in the curve, we
have z = f(#, y) with the condition y = f(a) ;
dz d
dx da
dy Wh dy\?
ane t * aad = 1 a 3
hey (p Ht =] fj (5*)
and it remains to substitute for p and q their values in terms
of wand y, and to integrate this equation. We shall then
have the required relation y = @ (a).
d dy dy
Sf (@%y) ti ae) sia? hd aye
8. When the given surface is one of revolution, the
equation to the projection of the curve of constant inclination
on a plane perpendicular to the axis, may be conveniently
expressed by polar co-ordinates ¢, p, so that |
w= pcosd, y=psin d.
2 2
As above we have 1 + (5”) = tan’ cy :
\aAP xv
dd\? ds\?
“. 1+ 9° a = tan’. eA :
2 2 aot?
which may also be put under the form (") ag la 2
dp} p’—p
p being the perpendicular from the origin upon the tangent
to the required projection. Now from the equation to the
: dz.
generating curve x = f(o), we can find 4. in terms of p, and
hence obtain the differential equation to the required projection.
Ex. Let the surface be a paraboloid, and the origin
in the center of its base, ce its altitude, @ its semi-latus
rectum 5
“. (€— 2) 2a= 9’, AE
259
or, p’— p’ = a’ cot’ y,
the equation to the involute of a circle whose radius = a cot ry.
If the surface be a right cone, the projection is an
equiangular spiral.
9. To determine a curve traced on a surface of revolu-
tion, which always cuts the generating curve at the same
angle.
Let DP (fig. 48) be the curve, EN its projection on
a plane perpendicular to the axis of the surface; CPG
a section of the surface through its axis meeting the curve
in P, PN=s, DP=s, ON=p £AON=d®, and a the
constant angle CPD at which the curve cuts the meridian ;
also let CQP’, a meridian plane very near to CPG and
inclined to zw at an angle @ + od; cut the curve in P’,
and a section of the surface through P perpendicular to its
axis in Q; then ultimately
Ie ) d
sin CPD = 55, =P? or sina. — =p,
d t dye
or ee geene. J + (| >
ee dp
by integrating which (since =f (p) by the equation to the
generating curve), we obtain the equation to the projection.
Ex. Let the surface be an oblate spheroid ;
2 pA
, a —e
to integrate this, make —, P
8 Rol 1h
2
, and the result is
=U
260
If the surface be a right cone, the projection of the
curve is an equiangular spiral; if a sphere radius a, the
equation between the radius vector and perpendicular on
the tangent to the projection, is
1 cosec’a cot’a
— —__ a
10. To determine the shortest line that can be drawn
between two given points on a surface of revolution.
Let OC (fig. 48) be the axis of the surface which take
for the axis of z, and a plane AOB perpendicular to it for
that of wy; DP a portion of the shortest line=s, wv, y, 3
the co-ordinates of P, ON=p, 2AON=9,
d.s\2 (apy* :
pos el EP, a) +f ()}*,
if s =f(p) be the equation to the generating curve, and
dz ‘
dp =f (p);
= Fay VA + ap? + {f' (wv)? = LV,
‘ d
putting p=a, p=y, <P =p, in order to adapt the expres-
P
sion to the usual formule of the Calculus of Variations.
dV dV x’
Hence io =10; Pp ee
dy
dp 1/1 + a*p* + Sf" (a) }?’
but tS bodys + &c. = 0, for a maximum or minimum; there-
ev
fore in this case P = a constant,
wp ap ds
RN erene ae Tomar! Tid Tem oy aft
+ ap’ + if(a)} p dp
ds Cc F 1
p+ ae = 5 or sin CPDc< ON” as shewn in Prob. 9.
Hence the shortest line always cuts the meridian at an
angle whose sine varies inversely as the distance of the
261
point of intersection from the axis; and the equation to
its projection on a plane perpendicular to the axis of the
surface is
2 am Mel bial 9 aol A
gE,
BOP ENG dp
e being the distance from the axis at which the shortest
line cuts the meridian perpendicularly. Also for the length
ds p dd
of the shortest line we have — =
dp c dp
Ex.1. Let the surface be a right cone, and mz =pV/1 —m’,
the equation to the generating line, then
d Poe ae
a £ is p =esecm(p+a),
the es st to the projection, which is therefore an elliptic
spiral.
Ex. 2. Let the surface be a sphere, and let DP be
the arc of a great circle, then
sin CD
ie
sin n CP
sin D;
if now the meridian revolve round OC,
] 1
o ——
sin CP ON
sin Poc
.
>
hence an arc of a great circle is the shortest line that can
connect two points on the surface of a sphere; this result
may be obtained from the formula.
Ex. 3. To determine the shortest line on the surface of
an oblate spheroid.
Let CEB (fig. 49) be the plane of the equator, PE, PN
262
two meridians, 4B a portion of the shortest line cutting PE
perpendicularly at 4. Let the point M be determined by the
angle @, so that CT'=p =asin0, ’'M=bcos@; and let A
be similarly determined by the angle a, so that its distance
from the axis = a sina; then by substitution we get
ay OND cos’ 9 + 6? sin’ @
= sin@ ,
sin’ @ — sin’ a
Now, since a is the least value that 0 can assume, let
cos 9 = cosa cos Wy,
i d in
Pees ; ee. 4. beet Vals b? + (a? — b°) cos’ a cos” Ws
‘ cos a sin Vr ON: cos a sin Wy
() Dre _Ja cos’ a + b* sin’ a — (a” — b*) cos’ a sin’ Wy
= aft —c’sin* Wy,
therefore s = arc of an ellipse, amplitude yy, whose semiaxes
are a’ =/a? cos?a + b? sin? a, b, and eccentricity =c. At the
point B, 6 = 90°, therefore y = 90°, and AB =a quadrant of
the ellipse; also
asina
P
ZL ABE = 90 - a.
sin PMA = = sina at the point B,
For the projection on the plane of the equator,
dd : ds dp _ sina ds
since p° — =asina
do dp’ ms dw ~ asin? @ dy’
aD a a V/1—c' sin?
adi, asina 1+msin*
, where 2 = cot*a,
2 2
1 a r 1
i ere | AE + m sin’ hy)
~ asina ENE TE — ¢c’ sin’ PW
‘ p= {( +o) IT, (7, W- SF},
employing the notation of elliptic functions (Integ. Cale.
p. 208.)
Cor. If the eccentricity of the generating ellipse be
very small, let
a oe
b?
=e, and cos’acos* Wy =m;
dp sina \/b? + (a? — b’) cos*acos’*W sina Ry/ Leh me
diy a 1 — cos’ a cos’ vy l—m a rey
sin a 1-m l1—m)(3+m) .
7 1) - Ee 4) EMO ACA) wee
1—m™m Q 8
sIn a sina o6 ¢& COSs’a
= 2 ae a (« - a ee cos’ yy — &e.] :
1 — cos“ acos Vy 2 4A 8
tan sing 38e é . sing ;
. p=tan( : 4 —y (6-5 | +o(ys "| sinacosta+ &e.
sina 2 4, 16 2
The indefinite continuation of the curve will manifestly
produce an infinite number of spires similar and equal, and
contained between two parallels each at a distance = b cosa
from the equator.
11. If a uniform and perfectly flexible thread of given
length have its ends fastened in a surface of revolution whose
axis 1s vertical, and rest upon that surface supposed perfectly
smooth, to find the curve into which it will be formed.
Taking the axis of the surface for that of z, and a plane
perpendicular to it for that of wy, and proceeding as in Prob. 10,
we have, # and & denoting constant quantities,
to (noe) = J, (h +2) Ae + p* a4 + ta =a minimum ;
264
do dp\** (dz\*
o* % h —_ = k 1 2 —— (=)
Qa ee +e (7) +z)
which, since we have x = f(p) by the equation to the gene-
rating curve, is the differential equation to the projection of
the required curve on a plane perpendicular to the axis of
the surface.
12. The lines of greatest inclination on the surface
\/ : . : :
Seif 7) are given by the intersection of the surface with
@
the cylinder x + y* = a’, where a is a parameter.
13. The tangent line at the point wyz to the curve
2 2 2 2 v4
@ Yy 20 x y &
ne ae ee Ore +—-1
a? 2 a a ¢
x-v yy Y-y 2 2-8
1S > et Fe Ser °
a 6° a-—@ a c
PROBLEMS ON Section VI.
1. Ir a surface of the second order having a center be
cut by a plane which always passes through a fixed point, to
find the locus of the centers of all the plane sections.
Let Aa’? + By’ + Cz’ = D be the equation to the surface,
z—c=A(w@7—a)+B (y-b) (1)
the equation to the cutting plane, a, b, c being the co-ordinates
of the point through which it always passes; therefore, elimi-
nating y, the equation to the projection of the section on xw@ is
Jez -c—dA'(w-a) + BO? =D. (2)
z
. B
24 Cx?
A x + et
265
Let h, hk, J be the co-ordinates of the center of the section,
then h and 7 are the co-ordinates of the center of the projec-
tion; if therefore in equation (2) we substitute # +h for a,
and x +/ for x, the projection will be referred to its center
as origin, and therefore the coefficients of wv and x’ must
each vanish;
B
B”?
pete i(k 0) AP =1(B bee cy AR = 0,
B , ,
Cl + Fe id — 0) + Bb- A (h—a)} =0.
Multiply the latter of these equations by 4’ and add it
to the former,
: : Ah
ee Ah+CAl=0, CM pint reat
imilar] B re
sImMarly, = ie
Hence substituting these values of 4’ and B’ in (1), and
expressing that it is satisfied by h, k, 1, we get
Ah(h-—a)+ Bk (k- 6b) +Cl(l—-c) =0,
the required equation, which represents a surface of the second
order similar to the proposed one, the co-ordinates of its center
being 4a, 3b, de.
It is manifest that the two surfaces intersect in a plane of
which the equation is daw + Bhy+Cez =D.
2. To find the locus of the middle points of chords of
a surface of the second order that has a center, all passing
through a given fixed point.
Take the given point for the origin, and two conjugate
diametral planes which pass through it for the planes of z@
and wy, and a plane parallel to the third conjugate plane
for that of yx; then the equation to the surface will be
of the form
ax’ + by? +c2x°+2a v+d=0,
266
Let v= mz, y = nx, be the equations to any chord; then
combining these equations with the former, we have
(am? + bn? + c) 22 + 2a"ms+d=0,
in which equation the values of = are the co-ordinates of the
extremities of the chord; therefore the co-ordinates of its
middle point are
”
am , , ,
3 vV= MB, Y=H=ns.
,
CS Se Sy ee
am? + bn? + ¢
Hence, eliminating m and n, the required equation to
the locus is
an oy by” a ce” + ie 0;
which represents a surface of the second order, similar to the
proposed one, and similarly situated and passing through its
center and through the origin.
8. The curve of contact of a surface of the second
order, and of a cone the vertex of which is given, is a plane
curve.
Let aa’ + by +2? 4+2a'a +2by + 2e2+d=0,
be the equation to the surface which may represent all surfaces
of the second order. Then the equation to the tangent plane
at a point vys is
(ax+a)X+ (by+0') V+ (ex4+e) Z+ar+by+cx24d=0.
Now because the cone and surface touch one another, at
the points of contact their tangent planes coincide; and since
the tangent plane of a conical surface always passes through
its vertex, if h, k, 1 be the co-ordinates of the vertex they
must satisfy the above equation ;
“ (avt+ayjh+(byt+0)k+(exte)l+av+by+e2+d=0,
which gives the relation among the co-ordinates wv, y, x of the
points of contact, and shews that they all lie in the same plane.
The curve of contact is therefore a plane curve resulting from
267
the intersection of the plane whose equation we have just found
with the given surface.
Cor. If the surface have a center, the line joining that
point and the vertex of the enveloping cone, passes through
the center of the curve of contact.
Let C (fig. 52) be the center of the surface, 4 the vertex
of the enveloping cone, Q2#S the plane of the curve of contact ;
and let any plane drawn through 4 and C cut the surface in
the line of the second order PQG, and the cone in the straight
lines 4Q, AS’; then C is the center of the section PQG, and
AQ, AS are tangents to it. Also, by a property of lines of
the second order, QS, a line joining the points of contact,
is an ordinate to the diameter PG which passes through the
intersection of the tangents; therefore QS’ is parallel to the
tangent at P, and is bisected in V, and CV.CA=CP’.
Since therefore every chord of the curve of contact drawn
through the point where a line joining the vertex of the cone
and the center of the surface meets the plane of contact, is
bisected in it, that point is the center of the curve of contact ;
also the plane of contact is parallel to the plane touching the
surface at the point where the same line meets the surface,
and its distance from C is determined by the equation
CV.CA = CP’.
Similarly it may be shewn, that if a cone envelope an
elliptic paraboloid, the line drawn through the vertex parallel
to the axis of the surface passes through the center of the curve
of contact; and the portion of it between that point and the
vertex is bisected by the surface.
4. If the plane of contact of the enveloping cone of
a surface of the second order always pass through a fixed point
or a fixed line, the locus of the vertex of the cone will be
respectively a plane or a straight line.
Let 2’, y’, x be the co-ordinates of the fixed point through
which the plane of contact always passes; then they must
satisfy its equation,
“. (av +a)hs (by +0)k 4+ (c2’4+0)l4+ a a'+b'y'+0'x’+d=0,
268
which gives the relation among the co-ordinates h, k, 1 of
the vertex of the cone, and shews that the locus of the
vertex is a plane; and that that plane is the plane of
contact with the given surface of a cone whose vertex is
in the given fixed point, for the equation may be put
under the form
(ah+a)a'+ (bk +0 )y'+(cl+c)x+adh+bk+cl+d=0.
Again, let v=mz+ fy y=nz+g, be the equations to
the line through which the plane of contact always passes.
Then, in order that the plane may contain the line, we
must have (Art. 23),
(ah +a')m+ (bk +b')n + (cl + c’) =0,
(ah+a)f+(bk+0)g+ah+0k+cl+d=0,
which are the equations to the straight line in which the
vertex moves, its co-ordinates being h, k, /.
5. If the plane of contact of the enveloping cone of a
surface of the second order always touch a surface of the
second order, the locus of the vertex will be another surface
of the second order.
Let the plane of contact always touch a surface of which
the equation is
me+ny +72? 4+2m'e+2n'y+2rx24+5=0......(1)3
then the equation to a plane touching this surface at a point
VY 1s
(ma+m')a'+(ny+n')y + (retr)s 4+m’'et+n'yt+r's+s=0,
which must be identical with the equation to the plane of
contact, namely,
(ah +a')a'+ (bk +b)y +(cl tc) +adh+Vk+cl+d=0;
met+tm ah+a ny+n bk+d
t
id eth oleae. eee en | Reet oy ley a. cree.
metny+re+s ah+bk+cl+d
me ees —_________
,
TZ+7 el +c!
269
and it remains to eliminate v, y, x between these equations
and equation (1). The result will be found to be
m , n’ , ” r , , ‘
{i (ahta)+ (b+ U) 42 (el+e)—Whatkrcl+d|
m2 mn? “2 1 1 1
rar. nea aide on h 4\2 yeni 2 a /\2
(= eee 5) > (a +a) +7 (bk +b) +i (else)t,
which represents a surface of the second order.
Cor. If the surfaces are concentric and
av+by+ex,=1, ma’+ny? +72’ =1,
their equations, the above result is reduced to
2 2 2
Cie TO ae
—h?+—kh?+-—P=1.
m n r
6. If two surfaces of the second order have each a prin-
cipal section in the same plane, the projection of their curve of
intersection on that plane is a curve of the second order.
Suppose the plane of wy to contain a principal section of
each surface, then no odd powers of z can enter into their
equations, which will therefore be of the form
Av’ + By? + Cx? +2Cay + 2A’a+ 2B"y + D=0,
aa’ + by? + cz + 2Qcaey +20" a4 +2b' y+d=0,
and if we multiply these equations respectively by ¢ and C
and subtract, we shall find the equation to the projection of
the curve of intersection of the surfaces on the plane of vy to
be of the second degree in w and y.
Hence if the axes of two surfaces of revolution intersect,
since the plane containing the axes will be a principal plane of
each surface, the projection of the curve of intersection on that
plane will be a curve of the second order.
7. To find the diametral surface of
Avy + es + yz) = vy,
270
which passes through the middle points of a system of chords
equally inclined to the three co-ordinate axes.
Here the equation to any chord, if aB-y be its middle
point and m4/3 = 1, will be
w-a=y-B=2s-y=mr,
- a}(at+mr)(B+mr)+(atmr)(y+mr)+(B+mr)\(y+mr)}
=(a+mr) (B+ mr) (y +mr),
or m7 +(at+B+y—3a)m' r+ faBtay+By—-2a(atB+y) mr.
+{aBy -a(aB+ay+By)$=0,
or Ar? + Br? +Cr+D=0; and if p, —p, q be the roots,
—Ag=B, —-Ap=C, Apq=D; «. AD=BC;
. aBy —a(aBt+ay + By)
=(a+B+y —-3a)$aB+ayt+ By -2a(at+ B+y)}
the required equation.
8. The sum of the squares of the projections of an
ellipsoid on any three planes at right angles to one another,
is equal to the sum of the squares of its projections on its
principal planes.
Let the equation to the surface referred to any three
rectangular axes through its center be
av’ + by +c22 + 2ay%4+20ax+2cuy=d, (1).
Since at those points in the surface for which
cz +ay+ba=0, (2)
dz dz
dav dy
are parallel to the axis of x; hence, eliminating = between
(1) and (2), the equation to the projection of the ellipsoid
on the plane of ay is
become zero, the tangent planes at those points
a (ac — b°) + y? (be — a”) + 2ay (ec —a'b’) = de;
271
.. square of area of this projection
a’ dc” nw’ d’c
~ (ac — b”) (be — a”) - (ec’- ab’)? = re
(S,5 S25 83, being the roots of the cubic in Art. 146) therefore
sum of squares of projections on the co-ordinate planes
w (a’ B? af ary? + By’)
where a, 3, y are the three semi-axes of the ellipsoid.
wd’ (a+b+ec)
a 8 82 83 *
9. If O be a given point in a surface of the second
order and OA, OB, OC any three chords passing through O,
mutually at right angles, the plane ABC will always pass
through a fixed point in the normal at O.
Let the equation to the surface be
au’ + by? + cz? + cay +c'2s=0,
the co-ordinates being rectangular, and the axis of z a normal
at the origin. Let the surface be referred to three new rect-
angular axes ;
2. a(ma’ +ny +r2')+ b(m'a' +n'y' +17'')?+ c(m' a’ +n"y' +7"s")?
+0 (ma! +ny' +73) (ma +n'y' +9'x') +0" (ma +n’y' +72’) =0.
Let h, k, 1 be the parts of the axes of a’, y’, x’ intercepted
by the surface
M9 oe c'm m’) a em” = 0,
“. h (am? + bm? + em
k (an? + bn? + cn”? + cnn’) + cn" = 0,
Lar + br? + er? err’) +67 = 03
therefore the equation to a plane passing through the ex-
tremities of h, k, 7 is
Ri wiloistetnk
and referred to the original axes, it is
metmy+m’s netnytn’s retrytr’s
rey | 1g Ie i
Il
pd
i)
272
and putting «=0, y =0, to find when it meets the axis x,
we have
” A Ad
\=1 (a+b+c) +c’ =0
— + —+—]j =1, or x(a c= 0,
(s+ E45 ’ C) +
consequently the plane cuts the normal in a fixed point.
10. An infinite number of surfaces of the second order
may be made to pass through the sides of a given quadri-
lateral in space.
Let ¢=0, w=0, v=0, w=0, be the equations to four
planes OAB, BAC, CAO, OBC (fig. 11) forming a tetrahe-
dron; then the equations to the four edges AB, BC, CO, OA
forming a quadrilateral in space, will be
t=0 u=0 w=0 v=0 (1)
=01 > ep = Ole tell Pele
If therefore the equation to a surface of the second order be
tw =p.Wv, it is evident that every one of the four straight
lines (1) will be situated in the surface throughout its entire
length ; and as uw is arbitrary, an infinite number of surfaces
may so pass through the same four lines.
11. <Any tangent plane to a two-sheeted hyperboloid will
cut off from the asymptotic cone, a cone of constant volume ;
and the point of contact with the hyperboloid will be the
center of the plane section of the asymptotic cone.
12. If through any point of a hyperboloid of one sheet
the two generating lines that intersect in that point, and a
diameter of the surface, be drawn to meet any tangent plane
of the asymptotic cone, a pyramid will be formed of constant
volume.
13. The locus of the centers of all sections of the surface
Ax’? + By? + Cz’ =1 that are at the same distance d from
the center has for equation
(Aa? + By? + CC) = (Aa? + By? + C? x”) &.
14, The area of a section of an ellipsoid made by a
273
plane through its center such that a perpendicular p to it
makes angles a, (3, yy with the axes of the ellipsoid,
mabe awabe
Pp \/ a2costa + b’cos® (3 + c°cos*ry
15. The locus of the intersection of tangent planes to
an ellipsoid drawn at the extremities of a system of conjugate
diameters, is an ellipsoid concentric with the original one, and
having its axes greater in the ratio of / 3 to 1.
ProBLEMs ON Section VII.
1. ‘To find the equation to the cylindrical surface which
envelopes a given oblate spheroid.
Let the center of the spheroid be the origin, and the
plane of sw that which contains the axis of the spheroid,
and a line through the origin to which the generating line
of the cylinder is always parallel; therefore the generating
line in any position will be w=mz+a, y=.
Let c? (a + y’) + a’2” =a’c’ be the equation to the
: dz dz
spheroid, therefore the equation m——+n-——=1 becomes,
dx dy
since n = 0,
mca + a?x =0, which together with c? (a + y’) + a’s” =a’c*
are the equations to the directrix; and if between these and
the equations 7 = mx +a, y = 3, we eliminate x, y, x we find
3? (a2 + mc’) + aa? = a? (a? + mc’).
Therefore, restoring the values of a and 3, the equation
to the surface is
y? (a + m?c’) + a? (@ — mz) = a’ (a? + mc’).
If we make x=-—c, we shall obtain the equation to
18
O74
the curve in which the cylinder meets the horizontal plane
on which the spheroid stands; ~ this gives
yy («+mc)
@ atmo”
which is the equation to an ellipse, the origin, that is, the
point on which the spheroid stands, being the focus. Hence,
an oblate spheroid placed in the Sun on a horizontal plane
stands in the focus of its shadow.
2. If a cylindrical surface be circumscribed about a
surface of the second order of which the equation is
Aw’ + By’ + Cz’? =1, and if R be the length of the semi-
diameter to which the cylinder is parallel, and A, uw, v the
angles which it makes with the axes, the equation to the
cylindrical surface is
Av’ + By +Cz? -1= RF (Axcosdy\ + Bycosp + Cz cos pv)’.
3. To find the equation to the conical surface which
envelopes a given surface of the second order.
Let the given surface be Av? + By’? + Cz’? = D, then the
equations to any generating line through the vertex (a, b, c)
will be
and to find the intersection of this with the surface we have
A(a+ilry?+B(b+mr)’?+C(e+nr)?-D=0,
and since the values of r must be equal, we have between
l, m and nm the relation
: (Aal+ Bbhm + Cen)?
= (AP + Bm’ + Cn’) (Aa? + BB? + Ce* — D),
therefore substituting for 1, m, m their values from (1), 7 dis-
appears and we get the equation to the enveloping cone
{Aa (wa) + Bb(y - b) + Ce(z -c)}?
= {A (w-a)? + B(y—b)* + C (w—c)*t (Aa? + BL?+ Cc? D)
O15
which reduces at once (agreeably to Cor. Art. 165) to
(Aa? + By’ + Cs -— D) (Ad? + Bb? + Ce? - D)
= (dav + Bhy + Cex — D)’.
It is evident that in the same way the enveloping cone to
any surface represented by an algebraic equation F(a, y, x) =0
may be determined, by finding the equation of condition between
l, m, and m that F(a +/lr, b+mr, c+ nr) =0 may have a
pair of equal roots, and then eliminating /, m, and n by means
of the equations (1).
The cone touches the surface along the curve in which it is
intersected by the plane represented by Aaw + Bhy+Cezx=D
obtained by substituting for p and qins—c=p(v-—a)+q(y—8).
Also if S'= AC (fig. 52) the length of the line joining the
vertex of the cone and the origin, and R = CP the portion
of that line which forms a semidiameter of the surface, it
may be shewn that
Aa tb be + -Cu = DP.
which may be substituted in the above equation to bs
conical surface.
Cor. If instead of referring the surface to its axes,
we refer it to a system of conjugate diameters, one of which
passes through the given vertex and is taken for the axis
of z, the equation to the plane of contact will be Ces -D=0;
and the equation to the enveloping cone will be found to be
ree, t
(x — c)? = (5 ~ Fa (Aw? + By’).
4. If a cone envelope an elliptic paraboloid, the line
joining its vertex and the center of the curve of contact is
parallel to the axis of the paraboloid, and bisected by the
surface; and the volume of the cone is “rds of the volume
3
of the enveloped segment of the paraboloid.
5. If lines be drawn from the center of an ellipsoid
parallel to the generating lines of an enveloping cone, a
18—2
onG
conical surface will be formed intersecting the ellipsoid in two
planes parallel to the plane of contact.
6. One of the most useful applications of the general
method of determining conical surfaces, is to find the stereo-
graphic projection of a curve upon any plane; that is, the
trace upon the plane, of a conical surface, called the optical
cone, whose vertex is the place of the eye, and directrix the
given curve.
If we take a line drawn through the place of the eye,
perpendicular to the plane of projection for the axis of 2,
the equation to the optical cone will be
feeb ash
F+C +C
which results from the general equation to conical surfaces,
by making a= b= 0, and changing the sign of c, that is,
supposing the vertex to be situated below the plane of vy,
If in this equation we make sz constant, we obtain the
equation to the stereographic projection of the given curve
upon any plane perpendicular to the axis of =; and if in
the result we make c infinite (so that the projecting lines
may all become parallel to one another and _ perpendicular
to the plane of projection), we shall find the equation to
the orthographic projection, which is the sort of projection
so frequently employed in every part of this subject.
When the curves to be projected are situated on the sur-
face of a sphere, the term stereographic is usually con-
fined to the case where the eye is situated in the pole of
the great circle whose plane is the plane of projection;
and when the eye is in the center, and the plane of pro-
jection is a tangent plane to the surface, the projection is
called gnomonic. We shall now illustrate the method by
the following examples.
7. To find the stereographic, gnomonic, and orthographic
projections of any circle of a sphere.
Let the center of the sphere be the origin, and a plane
277
perpendicular to the plane of the circle to be projected,
the plane of zw; then
P+y+e =r, s=axrt+d,
are the equations to a circle of the sphere, forming the
directrix of the optical cone ;
also w=a(z+c), y=B(e+0),
are the equations to the generating line; hence, eliminating
Xv, Y, x, between these four equations, we find
acat+b bh+e b+e
oe! P2e yerrmees $A ah ef! ) a :
5) aT 1—da l-—adaa
“. (b +c) (a’ + PB’) + (aca + 6)? =7* (1 - aa)’;
or, restoring the values of a and #3,
(b+ c) (a? +y)t+laca+b(z+o)P=r(z+e-az)’,
the equation to the optical cone, from which those to the
projections may be deduced, as follows.
In stereographic projection, the plane of projection passes
through the center, and the eye is situated in the surface of
the sphere, therefore s = 0, c =7;
“ (b+r) (a +y) +7 (aa +b) =r (* - aay’,
2ar 2 Pp — OD
vC=r
y +b Pee Oe
or w+ +
° ° ° Ue Te er ae ee
the equation to a circle, whose radius = oa Vr (1 +a’) — 0,
Pak
ar
r+b
and distance of its center from that of the sphere =
In gnomonic projection, the plane of projection touches
the surface, and the eye is situated in the center of the sphere,
therefore s = 7, c=0;
Rw +y)+Prar (r-azy’,
or a (0 — ar’) + By’? = 7 (7 — 0’) - 2rav;
278
the equation to an ellipse, hyperbola, or parabola, according
as b>, <, or = a’7".
In orthographic projection c= ©, # =0;
. vw +y + (ax + b) =2°, the equation to an ellipse.
It is to be observed that the optical cone, since it meets
the surface of the sphere in a plane curve, will meet that
surface again in a plane curve (Art. 78); so that the stereo-
graphic projection relative to any point, of any circle of a sphere
upon the surface of the sphere, is a circle.
In the same manner it may be shewn that the stereo-
graphic projection of any plane section of a paraboloid of
revolution upon a plane perpendicular to its axis, isa circle,
the eye being placed in the vertex; or that the stereographic
projection of any section of an oblate spheroid upon the plane
of the equator, is a circle, the eye being placed in the pole.
We may employ the same method also to compare any angle
on the surface of a sphere with its different projections ; this,
however, may be done more simply by geometrical con-
siderations, as will be seen in the following problem.
8. The gnomonic, orthographic, and stereographic pro-
jections of the rhumb-line, which is a curve of double curvature
traced on the surface of a sphere so as to cut all meridians
under the same angle, are three of Cotes’s spirals.
First to find the gnomonic projection.
Let y Ba, (fig. 55) be the plane of projection perpendicular
to BC, T¢ its intersection with a plane touching the sphere at
a point P in the plane of za, then T¢ is parallel to By; Pt
a tangent to the rhumb-line at P, Q the projection of P de-
termined by producing CP to meet the plane wBy. Then
ZTQ¢t is the projection of T'Pt;
A ES Me
CB
and tan 7'Q TQ TQ TP TQ tan 7'Pt CQ tana,
if a =the constant angle at which the rhumb-line cuts the
279
meridian. Let BQ=p, p=perpendicular from B on Qt
which touches the projection of the rhumb-line at Q, BC =a,
ei p atana Li cosec? q/\* cot’a
Jip V/p+a ep p” rs ate?
the equation to one of Cotes's spirals.
Secondly to find the orthographic projection.
Let # Cy be the plane of projection (fig. 56), T?¢ its inter-
section with a plane touching the sphere at a point P, situated
in the plane of za, then 7'¢ is parallel to Cy; Pt a tangent
to the rhumb-line at P, N the projection of P, thenz TN¢
is the projection of 7'P¢;
Ls Gene 0) Ege, fate iy GP
t = SS ee ee SC >| . f = 3
and tan 7'N¢ TN Ty’ TP pn’ T'Pt py ane
or if CN = p, p= perpendicular from C on N# which touches
the projection at JV,
p atana Dee cosec’ ai cot? a
ye RU ar rR ca
the equation to the hyperbolic spiral, as already found p. 260.
Thirdly to find the stereographic projection,
Let CO = CP be perpendicular to the plane of projection
«Cy (fig. 56) ; join PO meeting the plane of projection in Q,
which is therefore the projection of P, and 2 7'Qt of T'Pt.
Then Z TPQ =complement of CPO =complement of COP
=PQT, .«. PT=QT;; and since 7J'¢ is perpendicular both to
PT and QT, .. 2TQt=T7Pt=a; therefore since the angle
between the tangent and radius vector is constant, the curve is
an equiangular spiral, its equation being p = p sin a.
Gor. It appears, that if an angle a on the surface of a
sphere have one of the arcs which contain it, a meridian, and if
ry denote its gnomonic, w its orthographic projection, and @ the
are of the containing meridian intercepted between its vertex
and a plane through the center parallel to the plane of projec-
280
tion, then tan y=sin@tana, tanw= tana; also tana
sin
= /tanw. tan ry, that is, the tangent of the stereographic
projection is a mean proportional between the tangents of the
orthographic and gnomonic projections of the same angle.
9. To find the equation to the conoidal surface gene-
rated by a horizontal straight line which constantly passes
through a Helix, and the axis of the vertical cylinder on
which it is traced.
The equations to the Helix are (Prob. 2, p. 254)
w
e+y=a’, z=nacos'—;
a
let the equations to the generating line be s=, y= aa,
then eliminating a, y, 2, a +a°’2 =a’,
& ] 1
Te ee ig V1 a
therefore, restoring the values of a and £, the equation to the
surface is
1
ie cose =natan—'aq;
Z= na tan) ;
c
This is the surface presented by the inferior superficies of a
staircase, attached to a vertical column round which it winds.
10. To find the equation to the surface generated by
the revolution of any straight line about a fixed axis.
Let the fixed axis be taken for the axis of zx, and let the
equations to the directing straight line be
vw=Azth, y= Brr+k,
and those to the generating circle x = 2, a + y° =a sethen
eliminating a, y, x, we get (A468 +h)’?+(BB+k) =a, or,
restoring the values of a and #, |
(Az +h)? +(Bs+kye=a'’s+y’,
or vw +4? — (47 + B’) 2 -2(AR+ Bh) 2 =h' + k’,
281.
which is the equation to the surface, and manifestly represents
a hyperboloid of one sheet with its center in the axis of gz.
If in this equation we make w = 0, or y = 0, in order to find
the equation to a section through the axis, the resulting
equation represents a hyperbola; therefore the surface is a
hyperboloid of revolution. Also if, retaining one of the above
equations to the directrix, we change the signs of the second
member in the other, the equation to the surface is not altered;
therefore the same surface may be generated by two different
straight lines, and through any one of its points two straight
lines may be drawn so as to entirely coincide with it. This
agrees with Art. 73.
If we take for axis of x the shortest distance of the
directrix from the fixed axis, the equations to the directrix,
since it is parallel to the plane of yz, will be r=h, y= Bx;
therefore putting 4=0, k=0, the equation to the surface
(reckoned from its center as origin) becomes
e+ yy? — Bs? = h’.
11. An oblate spheroid revolves about any diameter, to
find the equation to the surface which envelopes it in every
position.
Let w=mz, y= nz be the equations to the diameter,
c (a + y’) + ax? =a’ the equation to the spheroid ;
then the equation
(y—nz)p—(# —mz)q+my —nx=0
becomes (my — nx) (a? — c?) =0; so that the equations to
the directrix are
C(#+7)+asx =a, nx=my,
and the equations to the generating circle,
s+me+ny=2, wt+y +2 =a;
eliminating w, y, x between these four equations, we find
a? (a ~ c°) (m? + n°) = (Bae Va =a)"
282
therefore, restoring the values of a@ and #, the required
equation is
a? (a + y® + 3? — 0?) (m? + 2’)
=S[(st+me+ny)/ae-@ —cV/a —a —y — #°)
12. The locus of the normal to the surface y = # tannz
along a generating line is a hyperbolic paraboloid.
PROBLEMS ON SeEctTion VIII.
Tue following are some examples of finding the equations
to twisted and developable surfaces, and envelopes.
1. To find the equation to the twisted surface of which
the directrices are two vertical circles having the opposite sides
of a horizontal parallelogram for diameters, and a straight line
passing through the center of the parallelogram perpendicular
to the planes of the circles.
Let the center of the parallelogram be taken for origin and
its plane for that of wy, and the rectilinear directrix for the
axis of y; then the equations to the three directrices will be
c=0 2=0,
y=-b (#-a)+xX?=7,
y=4+b (#+a)?+2? =2".
Also let the equations to the moveable line be
w=a(y-B), x=y(y- 8),
so that it already fulfils the condition of meeting the axis of y,
and one of the parameters consequently is eliminated; then,
expressing that it passes through each of the circles, we have
CC eae ane (1);
12 (6 B) 42h" 4 oO = Be ey Cees
283
therefore, by subtraction,
B (ba? + aa + by’) =0;
and rejecting the value 8 = 0 which would make the moveable
line always pass through the origin and so generate an oblique
cone, we have
a” + 9° +o = Qe» (2)s
by virtue of which, either of the equations (1) gives
| aa (b®? — B’) = b(7* — a’)...(3);
and it remains to eliminate a, 9, yy between (2) and (3) and
the equations to the generating line. Substituting the values
of a and vy given by the latter in (2) we find
b a? + 2° a w
=yt+- and then a = —— ———.;
PAL, Gia alas b wv + 3?
therefore, substituting these values of a and @ in (3) and re-
ducing, we find for the equation to the surface
faany + b (a + 2°) }? = br? ax + b? 2? (7? — a’).
2. To find the equation to the developable surface gene-
rated by a straight line which constantly touches a Helix.
The equations to the Helix being
x ns
v=acos—, y=asin—,
na na
at a point for which x = a the equations to its tangent are
a Late ne
v — a cos—— = — — sin — (z — a),
na n na
hag Oi 1 a
y — asin — = — cos — (#—- a);
Nad nm na
a i oeet a ar\/ a + 2_ ge — aw
, @ CoS — + Y sin — =a, and tan — Eyed da Yn & Gs
na na
na y - a
284
also, adding the squares of the above equations, we find
1
2 2 2 \2
¢ + = ys eS, :
vty —a 5 (% -a)
therefore the equation to the surface is
——_——,, a/xv+y—-a—2x
n/ x +y°—-a@’=2%—-—natan"' (oy eae :
y—a
3. To find the nature of the curve traced upon a cylin-
drical surface, when the surface is made plane.
Let aw, y, = be the co-ordinates of any point in a curve
traced upon a cylindrical surface; then the equations to the
generating line which passes through that point are
zg —-zs=m(a'—-2), y—-y=n(a'—-2@);
and the equations to the tangent line of the curve are
ped pike its Maenacly Be
Pitts pe CED yy =— (a -2);
let 7 denote the angle between these two lines, y the in-
clination of the generating line to the axis of w, and s the
length of the curve; then (Art. 30)
cost = cosy ( La +1)
i= cosy (m— +n — tT.
TO da
Now when the cylinder is developed, the generating lines
preserve their parallelism, and may therefore be taken for
the ordinates ; let a line perpendicular to them be the axis
of the abscissee, and a, (3, the co-ordinates of the point in
question ;
d d d
then “P = cos , bes oP — cosy (m S24 nts 1); also
1+ 68+ (= 65) +8) ~ Ge +2)
285
: ds\* :
each being the value of (=) ; and if between these and
wv
the two equations to the curve we eliminate a, there will
remain a differential equation between a and 3 which will
be that to the required plane curve. But if the latter
equation be given, that is, 8 = f(a), we may eliminate «
between the two above equations, and there will remain a
differential equation between a, y, and zs, which, with the
given equation to the surface, will be the equations to the
required curve of double curvature.
Cor. If the curve on a plane be a straight line, the
angle 7 is constant ;
Pets (m dz dy : "ds
eee CO == —— Se
Y an 2) en } COS 2
ve
is the equation for determining its nature, when the plane
is applied to a cylinder ; ae curve is called the Helix,
and we have for its length, by integration,
cos y (ms + ny + #) = cosi(s + C).
It is manifestly the shortest line which can join two points
on the surface of the cylinder.
4. To find the nature of the curve traced upon a
conical surface, when the surface is made plane.
Let a, b, c be the co-ordinates of the vertex of a conical
surface, v, y, x co-ordinates of a point in a curve traced upon
it, and 7 the distance of that point from the vertex. When
the cone is developed, the generating lines become radii
vectores, and if @ be the angle which x forms with some fixed
: ds\’
radius vector, equating the two values of (==) » we have
a
ESE ACI ale ee MOEN Nee LAE
ae (a) + (3) fo uo a) +13 (5) :
gun niente) (yi 2b)" 6 (iC)
from which, together with the two equations to the curve
we may eliminate «, y, x, and there will remain a relation
286
between x and 0, the equation to the plane curve when the
surface is developed; or, if » = (0), and the equation to the
conical surface be given, we may eliminate r and 0, and so
arrive at the remaining equation to the curve of double
curvature resulting from the application of a plane on which
a given curve is traced to a given conical surface.
Cor. All curves which make a constant angle with the
generating line of the cone are characterized by the equation
dr
Ber ak i, and become equiangular spirals when the surface
s
is developed. Also all curves which become straight lines
when the surface is developed, and which therefore are the
shortest lines which can connect two points on its surface, are
d 7 A gps Sak
characterized by the equation aes /r — a’, a being the
de dex
length of the perpendicular let fall upon the line from the
vertex as origin, or
1 ee Bag SE a é 2 2 (Jilin yoda
zs (5=) ey (54) aaV? See et
or finally, by developing this equation,
Ot) eed FOS ee 3 Otter 0 \. es dz\* (dy\?
Ca ea) + (ye 9s) = & {1+(3) + (5) .
When the surface is a right cone on a circular base, the
problem admits of a very simple solution, as will be seen
in the following instances.
5. A given curve being traced on the surface of a right
cone whose base is a circle, to determine its equation when the
surface is developed.
Let P (fig. 58) be a point of the curve situated on the
generating line CR, Q its projection on a plane perpendicular
to the axis; then our object is to find a relation between
CP =r, and the angle which CP makes with CA, when the
cone is developed ; call this angle 0, and let
CQ =7', QCx = 6’, m=cosec ACD;
287
now, when the cone is developed, @ is subtended by a circular
are, length AR;
~.0.4C =0'.AD, or, 0 =m@;
also CP.sin PCD = CQ, or, est
if, therefore, f(7’, 0°) = 0 be the given equation to the locus
of Q, F(=, m8) =0 is the equation to the curve when
the surface of the cone is developed.
Conversely, if f(r, 9) =0 be the equation to a curve traced
in a circular sector, and the sector be formed into a right cone,
the equation to the projection of the curve on a plane perpen-
,
dicular to the axis of the cone will be f( mr", =) = 0.
m
Ex. 1. <A right cone being intersected by a sphere whose
center is in its surface, to determine the equation to the curve
of intersection when the cone is developed.
Let B in the plane of xa be the center of the sphere (fig. 58)
BC =c, a = its radius, a = 4 vertical 4 of the cone, then
(c sna — w)*?+ (ccosa—32)?+ y= a
is the equation to the sphere, and 2° tan?a = 2 +y’ is the
equation to the cone; therefore, the equation to the pro-
jection of their intersection on the plane of wy is
:
c+ (a + y’)§1 + cot? a} -2ca sina — 2¢ ts Sey a+ y? =a’,
sina
and its polar equation is, making w = 7" cos 6’, y = 7’ sin 6’,
c? + 7” cosec’a — 2c7'sin a cos 0’ — 2cr’ cosec a cos’?a = a.
But m = coseca, r=? coseca, 0 =m@;
therefore, when the cone is developed, the equation to the
curve of intersection is
yr? — 2cr fsin? a cos m@ + cos? a} +c-—a@’=0.
Ex. 2. If a semicircle be described upon the bounding
288
radius of a quadrant, to find the equation, to its projection
upon a plane perpendicular to the axis, when the quadrant
is formed into a cone.
Tv ° ay"
Here m= 27 => ae 4; also the equation to the semicircle
is r = a cos 0, therefore the equation to its projection is
, Sl 1 1 ,
r=7acos;6.
6. To determine the shortest line on the surface of an
ellipsoid.
av 7
Let — + a + — =1 be the equation to the surface; then
a
1H q e e
p=- bia a q= = es and the equations to the shortest line
as &
Ly eae ving ad’ x ;
F papel waka y By, 7 =<" where a” = a &c., the arc s being
Bs s
an independent variable; also we have 3’ sol +qy’ which gives
wx yy | 88 dle ne
—— hk a = 01), ands a
v wo y y"" 2 3"! y pe ar y 2 )
ag EY ate pee SE ae Tee | ey
a B Y x z 3° f @)
rom yw ru ”
REAPER M EA od ig Fc
a p + Po Be *y”
Hence multiplying this by equation (2) so as to eliminate
ys"
x
and integrating, we get
Be A Bee ois ae :
Ce eeeaiad
oP) NAG: iG Cc
between which and equation (1) together with the equation
to the surface if we eliminate one of the variables, we shall
obtain the required differential equation of the first order
to the shortest line, or geodesic line as it is called.
?
289
7. The direction of the shortest line at any point of an
ellipsoid is determined by the condition that the product
- of the semidiameter of the surface which is parallel to that
direction, multiplied by the perpendicular from the center on
the tangent plane at the same point, is invariable.
For if a semidiameter D be drawn parallel to the tangent of
the shortest line at vyz, wv’, y’, x’ will be the cosines of the angles
which it forms with the axes of the ellipsoid, and we shall have
Sy ad | eg a sean
73 +—_= alii chats Coolie aii i
from Prob, 3, p. 234, P being the perpendicular from the
center on the tangent plane at wyz; therefore the result of
the preceding Problem becomes
PxDe=C.
8. The following are examples of Envelopes, when the
equation to the series of surfaces contains only one parameter.
Ex. 1. To find the equation to the envelope of all right
cones of a constant volume, whose axes are in the same
straight line, and bases in the same plane.
eee
b
The equation to any cone is =@ , where
a = altitude, 6 = radius of base, and the origin is in the center
of the base. Let its volume always equal that of a hemi-
sphere, diameter 3c;
wah’ 7 (8e)? PM aon 27 ¢% J a+ y?
%3 : Saeco or ety eae | Le (i-Y=*"),
differentiate with respect to the parameter 6,
3 3
et — — + ee Ja +y, or b == Val ays
a a (1 =) = a , the equation required.
= 2b 276 2) 9c
ADV ae carr, Cr abr
19
290
are the equations to the characteristic, that is, determine the
radius and position of the center of the circle, in which
the cone, the radius of whose base = b, is intersected by the
consecutive cone of the same volume.
Ex. 2. Ifa series of planes, passing through a fixed point
in the axis of x, have their traces on the plane of wy all of the
same length ; to find the equation to the developable surface
formed by their intersections.
If c= distance of the fixed point from the origin, a = length
of the trace on the plane of wy, and a= the angle which it
makes with the axis of x, the equation to the plane is
v
bask
acosa asina ¢
which being differentiated with respect to a gives wtan’a = y;
therefore substituting in the equation to the plane, we get
Ps 1
— + — (#3 + y3)§ = 1,
c hua
the required equation to the surface.
The equations to the characteristic in this case are
x y
-sePa+-=1, == tan’®a.
a c av
Ex. 3, If the center of a sphere whose radius =a,
describe a curve in the plane of «xy, to find the equation
to the annular surface which envelopes the sphere in every
position.
Let a and £3 be the co-ordinates of the center of the
sphere in any position, and y=f(a#) the equation to the
curve which it describes, therefore (3 = f(a) ;
. (w-a)’+ fy -fla)P+=a
is the equation to the surface, and by differentiating with
respect to a,
w-at fy—f(a)}f’ (a) =03
these are the two equations to the characteristic, whose
291
position and magnitude depend upon a; and if we elimi-
nate a between them, we find the equation to the envelope.
The characteristic in this case is manifestly a circle of con-
stant radius whose center is a point in the curve which
forms the axis of the surface, and whose plane is perpen-
dicular to the tangent line at that point, as is expressed
by the above equations; hence its nature is entirely inde-
pendent of the curve whose equation is y= f(x); it is also
the curve of greatest inclination, for its tangent line at every
point is perpendicular to the intersection, with the plane
of wy, of the plane touching the surface at the same point,
therefore its equation is (p. 257) are The remaining
equation is that to the envelope since the characteristic is
situated upon it, which we obtain by observing that the
normal always meets the plane of zy in the curve which forms
the axis of the surface, and the length is equal to the radius
of the generating sphere ;
.=eVYltps+g=a, (Art.110) or #1 4+p'+q) = a3
this might also have been obtained by eliminating the
arbitrary function from the two equations to the characteristic ;
hence the integral of the equation z°(1 + p® + q*) =a’ is re-
presented by the system of equations to the characteristic.
Ex. 4. If the vertex of a right cone describe a given
curve in the plane of wy, and its axis be always perpen-
dicular to that plane, to find the equation to the surface
which touches and envelopes it in every position,
Let a and £ be the co-ordinates of the vertex, y = f(«) the
equation to the curve which it describes, therefore 3 = f(a) ;
a = tangent of the angle which the side of the cone makes
with the plane of wy; then the equation to the surface is
Pv
(@- a)? + fy -f@}= 55
differentiate twice successively with respect to a,
. Sy-fla)t f'(a)+a—-a=0, fy-f(a) $f" (a)-1- ff (a) }?=05
19—2
292
from which equations we may deduce as above the equation
to the envelope, and the equations to its characteristic, and
edge of regression.
The tangent plane to the envelope at any point is also
the tangent plane to the generating cone, and therefore makes a
constant angle yy with the plane of wy; but sec y= V/l+p'+qs
therefore, since a=tan-y, the differential equation to the
surface is
p+g=a’;
the integral of which is of course represented by the system
of equations to the characteristic.
Hence, if S’ denote the area of any portion of the surface,
as!
dudy
that is, any portion of the surface bears a constant ratio
to its projection on the plane of wy. The characteristic in
this case is evidently a side of the generating cone, and
therefore perpendicular to the intersection of the tangent plane
with the plane of wy; therefore its two equations are
any +p +q= secry, or S = A, secy;
dy q 2. 2 2
da p’ Pq = 2s
hence the curve of greatest inclination of all surfaces generated
in this manner is a straight line, inclined at a constant angle
to the plane of vy.
9. The following are examples of envelopes when the
equation to the series of surfaces contains two parameters.
Ex. 1. To find the equation to the surface touched by
a series of planes so drawn, that the product of the per-
pendiculars let fall upon any one from two fixed points is
invariable.
Let the line joining the two points be the axis of «, 2a its
length, and the origin in its middle point; z=Ax+ By +e
293
the equation to one of the planes; then the product of the
perpendiculars upon it will be found to be
ce — A’ a” :
eT ae b° suppose, or c’= 6°(1 + 4°+ B’) + A’a?.
Hence, differentiating the equation to the plane successively
with respect to A and B considering ¢ as a function of
A and B by virtue of the above equation, we have
A
i Ce) or ates GHA Se
Cc
B B :
OD alin 12 a or y = — b’—; and consequently x =— ;
r) Cc Cc
9
xv” . y i" 2 (P4+D)4+VP B+ ;
* a? +6 BB - Cc ey
the equation to a prolate spheroid, whose axis coincides with
the line joining the two given points.
Ex. 2. Let the equation to a surface be
™m
v y” on
ay a Ys bn +
a” Pie
to find the equation to the surface which it touches in all the
positions it can assume, subject to the condition a”+ b"+c"=k"
a constant quantity.
Differentiating the equation to the surface successively with
respect to a and b, regarding c as a function of a and b by
virtue of the equation of condition, we have
a™ gm q"-1 a” gm a”
ake yas iC apemirian as Re er e e
y” 2m br-} y” g™ b”
m+1 m+l atts a 0, OF Fh Dn a
b c c Cor'C
Pare me Se Sa Ot ot et pee gen
ete Aa ae Toe TGR as xe Om A nt J ee
a b Cc c c AY»; k k
294
mn mn mn
Z\ m+n Cc -nilarl U\ m+n a” y\ m+n b”
or y. = ie 5 similar af ie = ie 9 4 — ke 9
mn mn man mn
=, BM TB ymin 4 ymin — fe™+", the equation required.
If m=n=1, the equation becomes \/ 8 + / a + V/y = / kes
and agrees with Prob. 10, p. 238; ifm =n=2, itiszs+a@+y=hk,
which shews that if a series of surfaces of the second order
have the sum of the squares of their axes constant, they are
all touched by a plane.
Similarly, if the equation of condition be abe = k’*, it may
3
aye
be shewn that the equation to the envelope is vyx = (=) ;
and if m=1, we find wyz = Ss for the equation to the surface
touched by a series of planes which form with the co-ordinate
planes a pyramid of constant volume. ‘The surface repre-
sented by the equation wyx =a constant, has also the property
that the tangent plane at any point 2, y, x, forms with the
co-ordinate planes a pyramid of smaller volume than any
other plane drawn through the point of contact, the equation
to the tangent plane being
} ,
moth Ld poe leg,
Ue Pm os
Kix. 3. To find the equation to the surface touched by a
series of planes which cut off from a given right cone, an
oblique cone, such that its volume is constant, or such that
the transverse axis of its elliptic base is constant.
If the volume of the oblique cone be constant, then by a
property of conic sections the transverse axis of the elliptic base
is also constant, suppose it’... = 2c, and let 6 = semi-vertical
angle of the cone; then it will be found that the equation to
the touched surface is z* = cot” (a + y’ +c’), the origin being
at the vertex ; which belongs to a hyperboloid of revolution
about the axis of the cone, and to which the surface of the
295
cone is an asymptote, the 3 axes of the generating hyperbola
being c and ccot 6; also the point of contact is the center of
the elliptic section.
In like manner if a series of planes cut off from a parabo-
loid of revolution a segment of constant volume, the surface
touched by them is a similar and equal paraboloid about the
same axis,
Ex. 4. When from three straight lines meeting in a point
a plane cuts off three segments such that the product of two
of them divided by the third is constant, the plane constantly
touches a hyperbolic paraboloid.
Ex, 5. To find the equation to the surface which is
always touched by a plane whose equation is
me+ny+iz=v,,.,..(1),
m and 2 being independent parameters, and J and wv functions
of m and m determined by the equations
Differentiating (1) with respect to m and n, and taking
account of (2), we have
7) dv n dv
vex (-7 ree y+e(-7) ==.
But differentiating (3) with respect to m, and putting for
dv é
—— the value found above, we have, making
dm :
m n ia
eg bg toe A
(v° he a’)? (v" = b*)? (v" = c’)? >
1 1 & &
- Z of (e4- *) 3
al (hee a Co & ;
similarly, 2s) tae A ie = *) v;
296
therefore, multiplying by m’ and n’, and adding,
m” n* P—1 ?—1
+5 = A (mr t+ ny +
Pe 0
o” —-q? a
4 ee z J y
a s—a- 4(F-0)o, apo 4(Z-0)e.
Therefore multiplying the squares of these three equations by
, m*, n”, and adding,
A= A (a +y' + 2° 4+v°—2v°)v* by (1) and (2),
or 1= A (7 — wv’) v*, making r? =a? + y' 4+ 2”.
pes lv Sm ay wt ea ra
Hence from (4) =—, VOCS
y— or gz—lu v~ zu — lv
and adding unity to each side of these equations we find
Yr —c Ps ce sv—lr
rP—v g—lv’ vw xv—lv?’
aie aie C* % v (xv — Ir’)
therefore eliminating z — Jv, we have ———. = —____ ;
2 2 2 2
Yr —C r—v
. nilarl a? @ v (wv — mr’) bey v(yv —nr"’)
similiar => — = —_—___—_— ;
y 72 — QQ? yp? — a? y? — Bb? y2 —
therefore multiplying by z, x, y, respectively, and adding, we
have the required equation to the surface,
Cc gy? a” uv’ b? y”
v
2 2
+ ——. + =" -,, = = ("0 - rv) = 0.~
(ee | ga ar = 2 ae | )
10. To find the singular points of the surface
a? x’ b? y Cc 3°
Pic? Cea Tae
or u=7"(a?a?+b°y’ +0") — a2a?(b’ +07) —b’y?(a? +0”) —c°s" (a? +b’) 40°’? =0,
a, b, and ¢ being in descending order of magnitude.
297
We must have (Cor. Art. 108) ze = 0, ie = 0, = meri
or w {a2(r?— b’ - c?) + a? a’ + by’ + c?2°} = 0,
y {P(r — a? — oc) + a? a’ + Dy’ + c?s"t = 0,
sf{e(r-a@-B)+ aa +b y+ &x*t =0.
We can neither suppose the three variables to vanish together,
nor two of them to vanish together, for by neither of these
suppositions is the equation «=0 satisfied. Also if we
make w or x vanish separately, the values of the other co-
ordinates become Sat deh but if we make y = 0, we find
a? — b? Bae
vw=+c - ee Ea
- a’ —
which are real values and satisfy w=0; and at the point
corresponding to these values, we get
d? a—-B du
Bese Wits GG RAY Sima saountrse 200)
a’ sate? BP—c du
——— | C. SSS SS
d 3° a’—c? dady ’
au a? + ¢ = a eae EE ea
SI hens ah ie te VG . = Ponca)» S
consequently the equation to the tangent cone at the singular
point, taking that point as origin, becomes
2
uv? ae, a a +c os
Boe ace 7 "PP * /(@- b?)(b?-c) ac
Also for the distance of the singular point from the center
we have, substituting its co-ordinates in (1),
2-h R-@&
+ :
Poee gt FS
=0; which gives 7? = b’,
If fig. 60 represent the intersecting ellipse and circle that
form the principal section of the surface in the plane of az,
then the intersections of those curves, one of which is R, are
298
the singular points, and the two sheets of the surface meet
in them.
11. To shew that tangent planes can be found which shall
touch the surface in the preceding Problem in circles whose
GEA)
Ae
radll ob
The section by «z is an ellipse and a circle intersecting one
another represented by the equations
(a +2 — b’) (a?x 4 Ce? ac?) = 0 (fig. 60).
Let PQT be a common tangent to this ellipse and circle,
then CQ the perpendicular upon it = 6, let zQ7'C = 0, then
2
CT =bcosec@, T7'Q= bcotd, TP = — cot 6, and cot@
6 oe Cc e ee
= yf ——,,- We shall begin by determining the intersection
a-—
with the surface of a plane perpendicular to s@ passing
through the common tangent PQT'; for which purpose we
must, in the equation to the surface, change 2 into — x cos 9
+ bcosec @, and x into wsin @ (Art. 101) which gives, dividing
by 6° and observing that a” cos’ 0 + c’ sin’ 0 = BD’,
2a?
(a + y? + cosec’?@ — 2bacot) (a +4" +a’ cosec’™ — yp weot 0)
2 2
a’c
—2° (2 +
) yeye
2a”
“t oe ( +c’) a cot 8 — ab’ cosec® 9 — a*c’cot?@ = 0.
From this we get by successive reductions, taking notice of the
value of cot 0,
g
(a +)? + (a +’) {G + b°) cosec” 9 — 2x cot 0 (2 oe =)
+ a°b’ cosec’O — 4.2.a7b cot @cosec?@ + 4a°x* cot’ @ — (x* + y*)(a" +c’)
2
+ x°(a°—b") (1 ae,
207 2 .
7a) one (U' +c") cotO—a’b’cosec®8 ~a*c’cot*O=0,
299
or (wv +9’)? + 2(a° + y’)acot 6 {0 cot 0 — & (< ee -)|
a
ek egy. ey ay:
+ (a cot 0)* — 2(acot 6)° (5 ae -) v® + aa’ cot’@ ¢ ae =
a a
or | a’+y? , a AL
v+y +tacot@iacotd—wx ae = 0,
a
which shews that for the assumed position of the cutting plane,
the two curves of intersection become united in a circle (and
consequently the plane touches the surface at every point in
the circle) whose equation is
a Ob
y? = acot@ (; + -) 2 — a’ cot?@ — 2”
2
“= ( cot 8 — v) (@ — bcot 0) =(PT —- x)(« - QT),
so that PQ is the diameter of the plane circle of contact; also
22
by the ellipse CP’ = a’ +c? — —
2A
1
. P@ =a - B+ 8 - ae =a (a - BY - e).
ProspLeMs on: Section IX.
1. Asa first example of this theory, take the curve of
double curvature, resulting from the intersection of a sphere
and cylinder, considered in Prob. 1, p. 253.
Its equations are y°=2av—a"*, x°=4a°—2aa, and the
equation to the normal plane, as we have seen, is
To obtain the equations to the line in which this is inter-
300
sected by the consecutive normal plane, we must join to it
the derived equation with respect to #, namely
1 a-«a« dy a dz
u(-f- GB) EE
or, by substitution, 2 +4 — =0,......(2)5
fois
and to find the equation to the surface generated by the per-
petual intersection of the normal planes, we must eliminate
v, y, and x, between these two equations, and the equations
to the curve.
Yuen Pee (ve
Now from (2) (-%) == or — =2(%) :
x x
3
, 2a a
. v a—- @
and (1) may be written — ey hiss eee
x %, a x
é
nq Nal uagip eae a
& av Xv Xv 3
or — =(2-2) +%(1-=) + (%) = 0.
x a a % a} x
é 4 4
e e v e e
Hence substituting for — its value, and reducing, we find for
a
the surface generated by the intersection of the normal planes
2 eT ET Te, 2 ee
2(e, - yp/ 23 -y)= 8/85 — y 3.
a
Al 1 - 2 2g
sO +y° +a ay | a +2)
Ad dA ") eel A ih de a a
y+ @')+ y's ~2"y)'= (F5) (loa +52) 5
(2a 4+ 2x)?
.. radius of curvature = —7—="—..
\/ 100+ 3H
2. On a given surface, to trace a curve of such a nature,
that its involute shall be a plane curve.
The curve of constant inclination, that is, one whose tan-
gent line is inclined at a constant angle to the plane of ay, will
have this property. For if s denote the length of the are, inter-
301
cepted between the plane of wy and a point whose ordinate
is x, and + = the constant angle which the tangent line makes
f ; dz
with the axis of x, then rF = COS y ;
Ss
*. S=scosy, ands =~2 secy
= the length of the tangent line between the point of con-
tact and plane of wy; hence if a thread be applied to the
curve of equal inclination, and then be unwound from it
beginning from the plane of wy and be kept stretched in
the direction of a tangent, the extremity of the thread will be
always in the plane of wy, and will describe a curve which is
manifestly the involute of the projection of the curve of equal
inclination on the same plane; for the projection of the tangent
line will touch the projection of the curve, and will be of the
same length. The equations to the curve of equal inclination
were found at page 257.
Ex. Let the surface be a right cone and the origin in~
its vertex, so that p = x tana,
2 Fe eA oes Ue,
cotta = cot! y 5» or Pp =pV/1- (tana cot y)’,
the equation to an equiangular spiral. Hence it follows that
this curve cuts the generating line of the cone under a con-
stant angle, that is, it is the conical helix. Hence if a thread
be applied to the surface of a cone, according to a helix traced
on it, and then be unwound beginning from the vertex, its
extremity will always be found in a plane perpendicular to the
axis through the vertex, and will trace out an equiangular
spiral on that plane, which is also the involute of the pro-
jection of the conical helix on the same plane, and therefore
similar to it.
3. If a uniform and flexible string be suspended from
two points in the surface of a vertical cylinder whose base is a
circle, it will form itself into a curve of double curvature, such
that the involute lies on the surface of a sphere.
302
Let A (fig. 61) be the lowest point of the catenary, 4B
the radius of curvature at that point; take the horizontal plane
drawn through B for the plane of wy, and the axis of the
cylinder for that of z, and let CM, MN, NP be the co-
ordinates of any point P in the curve. Draw NT" per-
pendicular to the tangent PT’ and join C7’; then PT is
perpendicular to the plane C7'N, for a line drawn through NV
parallel to PT’, and therefore coinciding with the tangent
plane of the cylinder would be at right angles both to 7'N,
CN, and therefore to the plane passing through them; and
CNT is a right angle, therefore
CT? = CN’ + NT® = CB’ + BA’ = CA’,
because the perpendicular upon the tangent from the foot of
the ordinate, is constant by the nature of the catenary. Hence
the locus of 7’ is on a sphere radius CA, and since CT'P is a
right angle, therefore PT' touches the sphere. But PT'=arc AP;
if therefore a thread be unwound from AP beginning from A,
and be always kept stretched in the direction of a tangent, its
extremity will trace out a curve situated on a sphere and the
thread itself will be a tangent to the sphere.
4. In a curve of double curvature, if S'Y be a perpen-
dicular on the tangent at any point P from a fixed point S,
and SY" be a perpendicular on YY’ the tangent to the locus
of Y, then, 2 being the perpendicular from S$ on the plane
through YY’ and the middle point of SP,
SY! = SP’ (SY? -— h’).
In fig. 55 bis take Q, R, points of the curve on opposite
sides of P and very near to it, and let qg, 7, be the points of
intersection of the perpendiculars from §' on the chords PQ,
PR produced; these points lie on a sphere diameter SP.
Through O the middle point of SP, and through the line qr
draw a plane cutting this sphere in the circle gYr. Now let
Q; R, move up to and coincide with P; then Pq, Pr both
coincide with PY the tangent to the curve at P, and qr
coincides with YY’ the tangent to the locus of Y; and SY’
303
being the perpendicular from the vertex of a cone upon a tan-
gent to the circular base, it is easily shewn that
SY‘ = SP*. (SY? —i’).
Hence the normal plane to the locus of the foot of the per-
pendicular on the tangent to any curve from the pole, always
bisects the corresponding radius vector.
ProBLEMS ON SECTION X.
WE shall next apply the results obtained in Section X. to
several particular cases.
1. To find the equation to a plane which shall have a
contact of the first order with a surface at a proposed point.
Since the equation to a plane (which here takes the place
of V=0, Art. 212) contains only three arbitrary constants,
‘a plane can in general have only a contact of the first order
with a surface completely given. We have therefore
s=Av+By+c, P=A, Q=B,
from which there results between the constants A, B, c, and
the co-ordinates x, y, s of the proposed point in the surface,
the relations s =Axv+ By+c, A=p, B=q; subtracting
therefore the value of x from that of x’, and putting for A
and B their values, the equation to the plane having a contact
of the first order with a given surface at a point vyz, the same
as that to the tangent plane, is
s—#= p(w —ax)+q(y'—y).
2. The equation to a sphere is
(2 - a)’ + (y' - BY’ + (@ - 7) =,
and since it contains only four arbitrary constants, a sphere
cannot generally have a complete contact of the second order
with a surface. Suppose it required only to have a contact
of the first order ;
304
then ia SE, inne?
dx s—y dy em—+
Hence there results between the constants a, 3, y, 6, and the
co-ordinates wv, y, %, of the proposed point of the surface the
following relations,
(7 —a)*+(y — B)?+ (2 -y)P= oe, —
G—-a y-p
— P> —_— —___ =
e-¥ s—¥
or w-a+tp(e-—y)=0, y-B+q(e- +) =9,
which equations shew that the centers of all the spheres lie in
the normal to the point of contact.
q>
r)
Also (z- iF l+p+q = 0, ors— ee
mt z Vit pre
po qo
and az=@&#-4+ SS 9 Bp=y+— ———..
ASE rig \/ Le rae
Hence the number of spheres, which may have a simple
contact with a surface at the same point, is unlimited; the
radius of the sphere being assigned, the co-ordinates of its
center are given by the above equations.
3. To find the radius of curvature of any normal
section, at a given point of an oblate spheroid. (Art. 229.)
Let /= 2 PGA, PG being a normal (fig. 67); and let the
section PQ make an angle a with the meridian, and its radius
of curvature = R;
; a a(l-eé
‘+, radius of curvature of meridian PO = p = eed :
(1 — e’ sin?/)3
c a
radius of curv. perp. to meridian PO’ = p’ LIP
1 — e’ sin®
pp. a 1-é
iz p sin’a + p cos’a oY Ay esin?] 1 ~ e? + e’cos’acos*]
4. To find the umbilici of an ellipsoid.
These are points (Art. 226) at which, if the ellipsoid were
305 a
generated by a circle of variable radius moving parallel to
itself, the plane of the generating circle would become the
tangent plane. Let M (fig. 67) be one of these points which
is necessarily in the principal section containing the greatest
and least axes, and let AZ7' be the tangent at MW; then
c a—b ec av
tanta / Oo
F oe GPa = (Art. 104)
of 7 = A c 9 6
if CN =a, since the equation to BPM is x = —\/a?— a’;
a
hence the co-ordinates of M, and of the three other umbilici
similarly situated in the other quadrants of the principal
section, are
a — b amy
e=+ta ‘ 2 0° z<=tec
— C™
a a” —
Also the radius of curvature at an umbilicus
since it is the radius of curvature of the principal section cor-
responding to these co-ordinates.
5. To determine the lines of curvature of an ellipsoid.
9
~~
A Wik
Let the equation be —+5 4 —-—=1;
+3
dx y v ne yy
poly dp az OY chi PB 'sF
dp ne 2 dp a
ate Ora y)3"8 dy ape
OG, sep ty MY 2»
ERIE Na ee
therefore, by substitution in the equation of Art. 231,
a(B-y) avy eS)
20
306
+ {B(a-~7)w*—a(B-y)¥"-aB(a-B)} —" - Bla-y) ay =o,
dy\* dy
A sonrs 2 — Ay*® — B) — -: =O:
eae ey) gre : De + ?
tate) Sig Gk ea
B(a-) Cary
which will be both positive if we suppose the plane of pro-
jection, that is, the plane of wy to contain the greatest and
the mean axis, so that a, 8, y are in descending order of
magnitude.
calling >
; ; { d LU >
To integrate this equation, let d =» UE being a
v Yy
function of v;
x? 2
* vy (4 3 - 1) +5" (@*- dy'- B) =o,
or Aa’ue—y’?+u(a’? —-Ay’® — B)=0,
Bu
o P(l = au (1 _ 1 OF i= oe
y(1+ Au) =a?u(1+ dAu)— Bu, or y=2°u my
du B du
: tent d
Hence differentiating 2y = 2UUu + x ain Guna 7?
r ek Fe i 0, since 2 dy g
OC e — = ;
(1+ Auyf ’ Scar oa
. oa : : of 2°
. u =C, which gives y c (a a)
the complete integral, C being the arbitrary constant.
Also taking the other factor # = j , we have
+ Au
y? =u (0° - #\/B) = 7 (@- VB) (VB -2),
or (yA) + (@- VB) =0
307
the singular solution, which resolves itself into the two
mir ee BP
y =0, Pa aly Sora asn ale
a — Cc
and indicates the umbilici of the surface (Prob. IV.), the normals
at which, as we know, are intersected by the normals drawn
from consecutive points in every direction on the surface.
6. To construct the projections of the lines of curvature
of an ellipsoid upon the plane containing the greatest and the
mean axis.
First, to determine the constant in the complete integral,
let a, y’ be the given co-ordinates of the point through which
the lines of curvature pass,
A ey B Ayiiti Bi whol gh
te Meaeee 10) oe Cais tases Sa
Ay?+B— ax 1 Aa LI ee Li a
oe a pag Ave + BE ah aday
Ay” + B+x? ;
and Lig hea sre 3G? (Ay? +B +a) — 4Ba”,
Hence, we see that C has two real values one +, and the
other —, and that 1 + AC is always +; hence the complete
integral represents an ellipse when C is —, and a hyperbola
when C is +; and therefore the projections of the lines of
curvature passing through any point on the surface of an
ellipsoid upon the principal plane containing the greatest
and mean axes, are an ellipse and hyperbola.
In the first case, if m and m be the semi-axes of the ellipse,
is the equation by which m and m are connected; in the
second case, if m and m be the semi-axes of the hyperbola,
n* : B Win Ae
Wee AC? Bt BS
20—2
308
Hence the semi-axes for each ellipse in the projection are
the co-ordinates of a point in the same given hyperbola, and
for each hyperbola the semi-axes are the co-ordinates of a
point in the same given ellipse; these are called the auxiliary
ellipse and hyperbola; they are concentric with the ellipsoid,
and their semi-axes major and minor (which are the same for
both curves, and coincide in direction with those of the
ellipsoid) have the following values
SJB =o oa ana M/E nent
a Cc” a
Hence we have the following construction for the pro-
jection, upon the plane containing the greatest and mean
axes, of the lines of curvature of an ellipsoid. Let ACB
(fig. 68) be this plane (that of wy), and with the axes
found above construct the auxiliary ellipse and hyperbola
GO, HO. From any point J in the hyperbola, and 7 in
the ellipse, let fall perpendiculars on the axes CA, CB; and
with semi-axes CN, CM construct the ellipse JZN, this
is the projection of a line of the first curvature; and in the
same manner the projections of any number of lines of the
first curvature may be constructed. Again with semi-axes
mi, ni construct the hyperbola »X, this is the projection
of a line of the second curvature; and similarly any number
of projections of lines of the second curvature may be con-
structed.
As the points J and i approach O and G, the ellipse and
hyperbola continually approach the lines CO, CB, and are at
last confounded with them; therefore CO, CB are the pro-
jections of lines of curvature, and consequently the curves in
which the surface is cut by the principal planes of wz, yx are
lines of curvature; also if in the equation to the auxiliary
hyperbola we make m = a, we find n = b, therefore the ellipse
AB is included in the construction, and is itself a line of
curvature. Hence the intersections of the surface with its
three principal planes are all lines of curvature. Since CO
~ Bb
a= wih 72 , and is therefore <a, the point O falls within
a
309
the ellipse 4B; it is the projection of an umbilicus of the
surface (Art. 235) round which the lines of both curvatures are
symmetrically disposed, and towards which they all turn their
concavities.
If the plane of projection contain the mean and least axes,
i.e. if -y, B, a be in descending order of magnitude, then
APRA EC. of Ne y-B
are still both positive, so that this
case is resolved into the preceding.
7. To construct the projections of the lines of curva-
ture of an ellipsoid upon the plane containing the greatest
and least axes.
If we suppose the plane of projection, that is, the plane
of xy to contain the greatest and least axes of the ellipsoid,
we shall find the construction easier, In that case a, y, 3
are in descending order of magnitude, and
suppose; B still remains positive. Hence
B
Wiens (Jab pawensit os fee k :
y (« Fa)
{Ayy? - Bra? &/ (Ay?- B+ay—4Ayay?}.
Hence both values of C have the same sign, and are
both negative
Ayy” a’?
B Lent 4. e. if
re
— 2 Aw?
{Z 12 Ae 2
fies 219 ait Cen ene
a- 23 a-Ba
which must always be the case, because the given point being
(2 tela
sah Hae 1.
eee a
if
in the surface of the ellipsoid, we must have
310
Hence both projections are ellipses; and if m and n be
the semi-axes of any ellipse, we have
n? C. om? B m An?
a. es n= —_ Aga
a Peri er Rawr
which is the equation to an ellipse whose semi-axes are
ome a? — Bb 7 fone
VAPELVA ond V2 at olor
a—-¢
In this case, therefore, the semi-axes of each ellipse of
the projection are the co-ordinates of a point taken on another
given ellipse, which is the same for all.
Hence let BA (fig. 69) be the principal section of the
ellipsoid containing the greatest and least axes; and concen-
tric with it describe the auxiliary ellipse Y.Y with the semi-
axes just found, and which we observe are greater than the
corresponding semi-axes of the ellipsoid @ and 6. From any
points J, i let fall perpendiculars on OX, OY and with these
as semi-axes construct the ellipses MN, mn; then MN, mn
are the projections of lines of curvature. If in the equation
to the auxiliary ellipse we make m =a we find n = J, therefore
the principal section is itself included in the construction; and
if from the points 4 and B we draw tangents meeting the
auxiliary ellipse in D, the perpendiculars dropped from every
point in DY will serve to construct the projections of lines of
the first curvature, and perpendiculars dropped from the points
in DX will serve for the construction of the projections of
lines of the second curvature.
It is not difficult to shew that if AY be joined, all the
ellipses in the projection are touched by it, and consequently
they are all inscribed in the same equilateral parallelogram.
This however must appear from the singular solution of the
differential equation to the lines of curvature, for that re-
presents a curve which always touches and envelopes the curves
rized in the complete int 1. Now the factor v?— ———_—
comp omplete integra Oo (14 dat
311
B
(p. 306) which in this case becomes a — G-Auye? put equal
to zero, gives a value of w which substituted in the equation
u 1 a
iv at eee (ee B);
gives ¥ r, 6 VB)
2 = wu — ——_—
y 1 — A,
and this equation represents the four chords which pass
through the vertices of the auxiliary ellipse, and of which YX
is one touching AB at an umbilicus of the surface.
The conclusion from the whole investigation is, that if the
two lines of curvature which pass through any point on the
surface of an ellipsoid be projected upon the principal planes,
the projections will be an ellipse and hyperbola, or two ellipses,
according as the plane of projection does or does not contain
the mean axis of the ellipsoid.
Cor. 1. Since the equations of condition which charac-
terize umbilici always make the differential equation of the
lines of curvature identical, and since that equation in the
present case is
dy\? dy
A om *_ Ay’ — B)— -—wy=0......(1),
vy (52) + (@ ig Bisa ey (1)
we must have wy=0, a? — Ay? — B=0; but we cannot sup-
pose w=0, for that gives y® equal to a negative quantity ;
therefore we must have
2 BP
y=0, «= 2B, or «= aL!
6 » 9
a? — Cc’
which equations indicate four points placed symmetrically in
the four angles of the principal section containing the greatest
and least axes, and of two of which O (fig. 68) is the pro-
jection. ‘These expressions for the co-ordinates of the um-
bilici agree with those previously found by considering that
they are situated at the extremities of the diameters which
pass through the centers of all the circular sections of an
ellipsoid.
312
Cor. 2. In constructing the projections of the lines of
curvature of an ellipsoid upon the plane containing the
greatest and the mean axis, we saw that for each of the
umbilici the lines of curvature passing through it were re-
duced to a single one, viz. the vertical ellipse projected
into COA. ‘This also appears from the differential equation
cs e d e
to the lines of curvature; for since oe for the particular
wv
values of w and y belonging to an umbilicus, assumes the
O ae? :
form Bere must, to obtain its true value in that case, recur
to the next derived equation according to the principles laid
down in Art. 235; this gives, not writing down the terms
2
which involve —~
dx?’
dy\* dy\* dy
Aw - _— p>“ -—y=0;
= (=!) Ay (=) pee dx J
but at an umbilicus when y= 0, a= £1/B this equation is
decomposed into
if wv dx
is the only admissible value, which when integrated and cor-
rected so as to pass through an umbilicus, gives y=0, and
indicates the vertical ellipse which is projected into COA.
d dy\? d
an, 4 (=) +1=0; therefore = 0
8. The lines of curvature of an ellipsoid are the curves
in which it is intersected by homofocal surfaces of the second
order.
Let the equations to a given ellipsoid and to another
homofocal surface be
2 2 2 z 2
eh ap tis PMA Dee
mg Ta ta Oe Gaolegad pam de)
then the plane of wy being supposed to contain the greatest
and least axes of the ellipsoid, the constants a”, b%, c”® are
subject to the conditions
2 2yeee 2 Ae 2 PE Pak. ae er 2 Qian 12 a
a baa." — b*, a —~c= a4? — 07, 1c bv c* —bee)
313
which amount only to two distinct equations, so that one of
those constants is arbitrary. ‘The co-ordinates of all the points
of intersection of (1) and (2) by Prob. VI., p. 237, satisfy the
equation
ae y” 3?
eat ppt poe?
which shews that some of the constants a”, 6”, c’® are negative;
and from equations (3) it must be either 6” alone, or b” and c¢”
together that are negative. Hence the curve of intersection
of (1) and (4) which is the same thing as that of (1) and (2)
has for its projection the equation
@
; art peaiom 2 e-—-B)=1, (5
on J (8-8) =1, ©)
which represents an ellipse since 6” is always negative. Let
X and Y denote the semi-axes of this ellipse, then
2 2
; Bae eck =a? — b,
a” b?
so that the semi-axes of (5) are subject to the same condition as
are the semi-axes of the projections of the lines of curvature of
the ellipsoid by Prob. VII.; consequently the curves in which
2 2 2
o ~ ~
e ° e e v e ° 6 .
the ellipsoid is intersected by — — 7 + —; = 1, in which a” is
oe Yas
arbitrary but 6” and c” subject to the conditions a” + 6” = a? — b’,
a® —¢c® =a’ —c’, are its first lines of curvature. Similarly the
2 2 2
. ° . dh J] S$
second lines of curvature lie in the surface — — co oe 1;
where a” is arbitrary, but 6” and c” subject to the conditions
a’ 4b? =a?—0, a? +c* =a’ —c’. And equation (4) shews
that those lines of curvature are likewise situated on a conical
surface of the second order concentric with the ellipsoid.
314
9. To determine the radii of curvature of an ellipsoid.
. of ies :
Let the equation be — + B 4+ —=1,then taking the values of
a w
the differential coefficients from Ex. 5, we have
’ 2 2 2
ee) t= — x (~) (=) _#
G4 pyt=- laa (2) (E) @- a},
ry) (x)' aaa
oe appa acd A 2? a BY lc 9
ae ra «)° apt
spate #+(2) (f) @- 99},
But in taking the sum of these three equations, the co-
2
efficient of (~) within the brackets is
g
a are
S+5- (Sef) (C48) (.-£-8)-o off)
(+ pi 2pqs+ + eee =
Pa fetBry- Way} =o
also rt — s? = (45) {(B-y)(a- 4) - wy} = : oe
a“ y 1
md taped S (54 B +3) yee
D being the distance of wy from the center, P the perpen-
dicular from the center on the tangent plane, and 4?=a+6+ y3
therefore by substitution in the equation of Art. 233 and re-
duction we have
(4 @- x)? + tf - D*) (z’ — #) + + Be = 0.
,
But Rea~(/-) /ispaeg = -2=9%,
z
315
A’ — D’ ary
ag 3) ageieaie 2 =
kt P R+ pe = 0 (1)
Let R’, R” denote the values of # in this equation, then
A2 — D?
P ?
abe\?
- tore 2 , Reg
R.R (=) h+
which is the simplest form to which the result can be reduced ;
the first equation shews that at points where the tangent planes
are equidistant from the center, the product of the radii is
constant; and the second that at points which are themselves
equidistant from the center, the sum of the radii varies inversely
as the perpendicular on the tangent plane. Hence the volume
of the ellipsoid may be expressed by the two radii of curvature
at any point, and the perpendicular on the tangent plane, for it
4cabe
_4 7pm ./PR
SSS =— PS RR’.
Also the radius of curvature at an umbilicus (since the two
radii become equal to one another)
b 2 2 Gb?
pein (= +5) = —, as before.
Pew iy ac
The equation (1) for determining the two values of R may
be presented under a different shape, leading to several im-
portant results; and may be investigated by the following
direct method that is equally applicable to all surfaces whose
equations are of a particular form.
10. To find the two radii of curvature at any point of
a surface whose equation is of the form
f (#7) +) +x) = 0.
In this case the quantities w’, v’, w’ vanish in the formula
of Art. 217, and we have
‘
TA APE Ss 1 oie
Pu+tmv +n'w = ous + V* + W? = P suppose,
316
which must be a maximum or minimum by the variations of
l, m, m under the conditions (Art. 24)
Pamin?=1, 1U+mV+nWe=0, (2)
by virtue of which m and m may be considered as functions
of 7; hence differentiating the above three equations with
respect to 7, multiplying the two latter by indeterminate
coefficients } and «, adding them to the former and then
d dn
making the coefficients of a and Pr vanish, we get for
finding J, m, m the equations
lu+«U +aAl =0
mv +«V+Am=0> which give P+rA=0;
nmw+kW+rAn =0
=P o— PP 0 — P
‘ob pa, es
U Ve W
consequently from (2) we get the equation furnishing p, p’ the
maximum and minimum value of &
: 1 Seo
(since P = aU + V* + W*),
U° V? Ww? i
ht gaa Oar eres a Te
x’ y° 3°
In the case of the ellipsoid — + = + —=1, this becomes
ham ao eey
x y° Prd
BGeiRp) a PCB) ay CReaR gy a
p being the perpendicular on the tangent plane at wyz.
We shall next deduce some results both with regard to
the radii of curvature and the lines of curvature of an ellipsoid,
from applying the general equation to lines of curvature given
in Cor. Art. 231, to that surface.
. @&
317
11. To find the equations to the straight lines which
touch the lines of curvature at any point of an ellipsoid.
The equation of Cor. Art. 231 when applied to the ellipsoid
2 2 2
—+>+—=1 gives (considering each co-ordinate as a func-
¥4
a
‘ fer Gee
tion of a new variable ¢ and making sates: &c.)
y¥x(y—- Beta (a-y)y+ay(B-a)x=0,
yy ee’
casera
Now the equations to the tangent line at vy are (Art. 124)
OY — ye Lee
ap eee Se pas
a’ y’ 2!
,
to which must be joined SETS as
a
therefore eliminating w’, y’, ’, the tangent lines to the lines
of curvature at a point wy are determined by the intersections
of the cone and plane
(Y-y)(Z-32)(y—- B)vt+(X-2)(Z-2z)(a-y)y
+(X-«)(Y-y) (6 -a)%=0,
(¥-«)"4(¥-y) 54 (Z-s)==0,
a B ye
the cone having its vertex in the proposed point, and the plane
being the tangent plane to the surface at that point.
12. The tangents to the lines of curvature at any point
of an ellipsoid are parallel to the axes of the diametral section
made by a plane parallel to the tangent plane at that point.
OR diy? > 3?
Detect B + — =1 be the equation to the surface, ryz
a 5.
tide sie. oC ele ZZ j
the point in it, and —- +— ee the equation to the
a
cutting plane; then r= X* + Y?+Z? is to be maximum,
318
X, Y, Z being taken so as to satisfy each of these equations ;
therefore proceeding as in Ex. 10,
(a+A)X+Kv=0, (B+A)V+ny=0, (y+A)Z+nx=0, (1)
and eliminating \ and « we get
YZ(y—-P)v7+XZ(a-y)y+ XY (B-a)x=0;
; woaekw | oY: ZZ t ; :
which with — + nes +— =0 determine two straight lines
a
6
which are the axes of the diametral section; and these lines
are evidently parallel to the tangents to the lines of curvature
determined in the preceding Problem. Also for the magnitude
of the axes we get from equations (1)
7+rA=0, < (a=) = B=) =ly-0)3
ERE coe it, Uolss y y a ee
“a(a-r) B(B-1r) yy-n ”
which determines ”.
13. The radius of curvature of any normal section of an
ellipsoid = D’ +p, D being the semidiameter of the surface
which is parallel to the tangent line of the normal section,
and p the perpendicular from the center on the tangent plane.
For comparing the above equation for finding the axes of the
diametral section, with that for finding the radii of curvature
in Problem X. we have #p =r, which shews that the radii of
2 2
curvature at any point of an ellipsoid are equal to aa and —,
p
p being the perpendicular on the tangent plane at that point,
and 4A and B the semiaxes of the diametral section parallel to
that plane. Also for the radius of curvature R of any normal
section at the same point inclined at an angle @ to a principal
section, we shall have (Art. 222)
1 (cos’@ sin’ @ p
rae e* =) a
319
14. For different points taken along the same line of
curvature of an ellipsoid, the product of the diameter of the
surface that is parallel to the direction of the line of curvature
at any point, and the perpendicular from the center on the
tangent plane at the same point, is invariable.
As in Problem XI., the equations to the lines of curva-
ture are
a (sy'— ys’) a + B (wa! — xa’) y+ y (ya’— wy’) x'=0, (1)
ae yi ee 0.
Ce Tas ame cy
Hence eliminating by cross multiplication, and taking » to
denote an arbitrary multiplier, we get
a’ = B (#2 - Bees ie y (yx - vy) 5
= Boye (“Ss het =)- Bryw (= +e +),
2 2 2
or putting = + ie a = =u, and \=pn.aBy we get for a’,
o res
and similarly for y’ and 2’,
x a old Baye dhs ore/’ par x
v=—uU——U y¥=—-t -—-—4& s=—uw—-—UuU, 2
lu Pte a ad a eg aca (2)
Hence multiplying these equations respectively by wv”, y”, x
and adding, and supposing the arc to be independent variable
EST
so that a’ a” + y’y” + ’3" = 0, and observing that
“v
320
or PD = C as in Prob. VII. p. 289. Hence all properties de-
duced from this equation for the geodesic line of the ellipsoid,
will hold also for the lines of curvature.
Also this process leads to the integral of (1) and conse-
quently to the determination of the lines of curvature. For
equations (2) give
, , : ,
VU ; u SU
a 9 2Y mB en! 3 Of a
uU+ ma uw+uld Ut ey
hence multiplying by i ; Bi and "e , and adding and dividing
a Y
by w’ we get
2 2
‘ x
v
——_—_— + s Ee
a’(w+ua) SP (wtup) xy (u+py)
which by substituting for w its value may be reduced to the
form
x - 3
SEL Ses Ve EP a- ., -- es SS ee Paes ol Es + ee ee
ataByn B+raByp ytaByu
which shews that the intersections of confocal surfaces of the
second order are lines of curvature, agreeably to Prob. VIII.
=1,
15. The curvature of the shortest line through any point
of an ellipsoid varies as the cube of the distance of the tangent
plane at that point from the center of the surface.
Since the osculating plane of the shortest line contains the
normal to the surface, the radius of curvature of the shortest
line will be the same as that of the normal section of the
surface which passes through its tangent, and therefore will
Woe Ge :
equal Soe ae rs PD=C by Prob. VII. p. 289. In the case
of a surface of revolution, the perpendicular on the tangent
plane coincides with the perpendicular on the tangent line to
the meridian, and consequently the radius of curvature of the
a?b?
meridian = —_, and the curvature of the shortest line is
proportional to the curvature of the meridian through the
same point.
321
16. If a’, a’, be the parameters of the lines of curvature
that cross one another at any point of an ellipsoid (that is, the
semi-major-axes of the confocal hyperboloids on which they
are respectively situated) and @ and 4 7 — @, the angles which
the geodesic line through the same point makes with the lines
of curvature, then the value of a” sin®@ +a” cos’ @ is the
same for every point of the geodesic line.
It is shewn Prob. VIII. that the lines of curvature which
cross one another at any point of an ellipsoid are the curves in
which the ellipsoid is intersected by homofocal hyperboloids
of one, and of two sheets, respectively. If therefore we take
for the equation to the ellipsoid
i, 2 oe
at ow de ao:
we may consider the two lines of curvature at any point to be
determined by the hyperboloids whose equations are
we y 2 “ y?
“Sy eee g fone A
a” Pea? f—a”
= -a toa Le
gre BFL 2 cig 7—*
and a’, a’, are called the parameters of those lines of cur-
vature. If we subtract each of these latter equations from the
equation to the ellipsoid, we get
ue y” Poa
2, /2 (2 par (pe a, + 73 2) (,,/2 ae (2),
a”a (a -—b)(h -—a*) (a&-Cc)(a*-©&)
a y =
Es ata 6%) (62. 052) (abot) (Gia) © Hats)
But between the co-ordinates wv, y, x of a point in the ellipsoid
(1),.and 7° the square of a semi-axis of the diametral section
drawn parallel to the tangent plane at that point, we have by
Prob. XII. the relation
P x’ y rf
Gof) @-A@F-F) | @-A@oésr)
which compared with (2) and (3) gives for 7° the values
a’ —a’®, and a’ —a’™”. Now let P be the perpendicular from
_ the center on the tangent plane at ays, and 4, B, D the semi-
_ diameters of the diametral section, respectively parallel to the
lines of curvature and the geodesic line passing through zyz ;
then
21
a
322
AB? ‘ par vir 2. | rapes
se = 4’ sin’ 0 + B’ cos’@ = (a — a”) sin’? 0 + (a — a”) cos’O
(ABPY?
or a’? — a” sin’ @ — a’” cos’*0 = (PD) = a constant,
since ABP is proportional to the volume of the ellipsoid, and
PD is constant by Prob. VII. p. 289.
The result a” sin? @ + a” cos’@ = constant, which is the
integral of the differential equation of the second order to the
geodesic line on an ellipsoid, was first given by Jacobi; and
the result PD = constant, as well as several other Problems
here given, are due to Joachimsthal (see Crelle’s Journal,
Vol. 26).
17. The tangents to the lines of curvature at any point
of an ellipsoid bisect the angles between the tangent lines to
the two circular sections through the same point.
For if the diametral section be drawn relative to that
point, the tangent lines to the circular sections are parallel to
the equal diameters of the section, and the tangents to the
lines of curvature to the axes of the section; and in an ellipse
the equal diameters are equally inclined to either axis.
18. To find the radii of curvature at any point of a
paraboloid.
Pr 2
Let the equation to the surface be — + = + 2%= 0; then
a
it may be shewn, as in Example 9, that the maximum and
minimum radii of curvature at a point wysx, are
va Sula genbisnee
" ape 2
a+b—2% y |
a GB Jab (S48 41).
Hence if the radii be bestia on the axis of x, the sum
of the projections = a + b — 22.
19. In surfaces of the second order, the two curvatures
are in the same direction at every point of the same surface,
or they are in opposite directions at every point.
323
The curvatures are in the same or opposite directions
at any point of a surface, according as r¢ — s’ for that point
is positive or negative. But, substituting for 7, s, ¢ their
values from
2 1
ee ey we find rt—s? = X, ; ;
Camny 3.8 iry x apy
a quantity which can never change its sign by the variation of
2, y, x, and will always have the same sign as ary; therefore
for the ellipsoid and hyperboloid of two sheets, it will be
positive, or the two curvatures at every point of the surface
will be in the same direction ; and for the hyperboloid of one
sheet it will be negative, or the two curvatures at every point
will be in opposite directions, Similarly, for the elliptic and
hyperbolic paraboloids the curvatures are respectively in the
same, and opposite directions, at every point.
20. If # be the radius of absolute curvature at any
point of a curve defined by the intersection of two surfaces
u,=0, U,=0; and r,7,, be the radii of curvature of the
sections of w, = 0, w,.=0 made by the tangent planes to
U,=0, U,=0 respectively, at that point; it is required to
express R in terms of r,7,, and @ the angle between the
tangent planes.
Let OP, O,P (fig. 54 bis) be the normals at P to the two
surfaces, PV, PV, the radii of curvature of the sections of each
surface made by the tangent plane to the other; then if VO.
V,O, be perpendicular respectively to PV, PV,, by Meunier’s
Theorem O, O, will be the centers of curvature of the normal
sections; and if P# be perpendicular to OO,, R is the center
of curvature of their intersection; .«. PR=R PV=*r
PV,=r, OPO, =0, OPR = a suppose, O,PR = 6-a,
PVcosa rcosa cosa. sin@
SLE RO 8 008, rear a ott EW R — )
cos(@—a) sin@ cos@cosa_ sin@sina cos@sin@_ sin@sina
> = = qi —4+ ——_ = —_—___—_—
ite v; R R r POA
1 1 2cos@ 1
SS = oo ke 7
Pr rv; r,
21. The locus of the focus of an ellipse rolling along
a straight line is a curve such that if it revolve about that line,
the sum of the curvatures of any two normal sections at right
angles to each other will be the same at every point of the
surface generated.
Suppose the ellipse to touch the line ON along which it
rolls in P; join SP and draw SN perpendicular to ON, then
SP is a normal to the locus of S, and if ON=2, NS =y,
ds
and the arc of the curve described by S =s, we have SP=y—;
faa tel Nie Ca aR or aiteknaan
WP SN” ds yy
eae cL aM Pai de ae Oe
ds°’ ds y’ r y
if r be the radius of curvature at S' to the locus of S;
ts 2a (25+) =2 08g, ee ae
Ses SP. %2-a
which shews, since SP and r are the two radii of curvature at
SS’ of the surface of revolution, that the sum of the curvatures
of any two normal sections at right angles to one another
is the same at every one of its points.
22. If # be the radius of curvature at a point whose
distance from ‘the axis =p, of the shortest line traced on
a surface of revolution; and if the radius of curvature of the
meridian at that point = r, and the part of the normal inter-
‘ : 1 1 1 ] c
cepted between it and the axis =m, then — = — + e _ | =
) Kier Tee
e being the value of p where the shortest line cuts the meridian
at right angles.
23. If a line of curvature be a plane curve, the tangent
planes applied to the surface at its various points will all make
the same angle with its plane.
Jas! Necle teulp. 362 Strand .
Hymeas Analytical Geometry
Jos! Neele teudp. 352 Strand .
a a
Plate 3
Ja*Nede se. Burleigh Str. Strand
Hymers Analy tical Geomeby
Plate 3
Ja? Neele se Burleigh Str, Strand
>
rar
bs
S
ne
Sd
E
3
5
%
=
S!
8
S
Flate 4.
ale sc. Burleiah Sta: Sand
JatNve
Plate &.
Jos Neele teulp. 862 Strand.
Liymers Analytical Geometry
Plate 4.
56 bis
Jost Neele teulp. 862 Strand.
&
3
&
J]
f
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Hymers Analytical Geomary r
y See Jost Neele fiulp, 852 Strand.
= Sp wee
Cambridge.
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