UNIVERSITY OF CALIFORNIA
AT LOS ANGELES
Price Three Shillings and Sixpence.
PART FIRST OF
AN
ON THE
APPLICATION OF
The Jllgebraic Analysis
TO
BY W. S. B. WOOLHOUSE.
lv Part 2, denominated "GEOMETRY OF THREE DIMEN-
SIONS,'* will he given the derivations of the formulae relating:
' fj O
to lines and planes in space, curved surfaces, curves of double
curvature, &c.
And Part 3, will comprise numerous miscellaneous pro-'
blems, elucidating the varied and extensive application of the
formula? and principles laid down in the preceding Parts,
LONDON:
SOLD BY GEO. GREENHILL, STATIONERS' HALL, LUD6ATP
STREET; AND ALSO BY w. GLENDINNING,
NO. 25, HATTOX GARDEN.
1831.
Entered at Stationers 1 Hall,
AX
ELEMENTARY TREATISE
ON THE
APPLICATION
OF
TO
G-SOMETrLT".
BY W. S. B. WOOLHOUSE.
SOLD BY GEO. GREENHILL, STATIONERS* HALL, LUDGATE
STREET J AND ALSO BY W. GLENDINNING,
NO. 25, HATTON GARDEN.
1831.
[ENTERED AT STATIONERS' HALL.]
Pi nlted by VV. Foidyct, Newcastle.
*St A Library
v
PREFACE.
J. HE decidedly great* advantage of the Modern Mathemati-
cians over the Ancients, has almost entirely arisen from the
7 introduction and refinement of the Algebraic Analysis, united
with the Differential and Integral Calculus; and particularly
from the truly elegant and systematic mode which has been
\ adopted in their application to problems connected with
Geometry.
It is but recently that the plan of determining the relative
geometrical positions of points, by means of their ordinates
y related to fixed axes, has engaged the attention, generally, of
our English writers. By means of this simple and ingenious
^ contrivance, almost an unlimited power has been acquired in
4 the resolution of physical problems, which require the aid of
' Geometry, and the differental and integral calculus has been
employed with surprising efficacy. In many cases, wherein
the greatest effort of the imagination would be inadequate to
afford any clear or correct reasonings, the relations of the fi-
gure and the necessary conditions can be fully and distinctly
put down by means of analytical equations; and the several
operations, which are generally attended with much faci-
lity, depend, in a great measure, on the general principles of
elimination.
The following concise introductory treatise was commenced
with a view of setting forth, in the most simple and perspicu-
ous manner, the principal formulae, which may be found use-
ful in the solutions of the various descriptions of geometrical
and physical problems; and furthermore to condense them
into one small volume, with a progressive arrangement, so as
afford the utmost facility in referring to such principles as
may be applicable to any particular subject.
CONTENTS.
PART 1.
SECTION I.
Definitions and First Principles .
ART. PAGE
1. A Function defined. .......1
2. The values of Functions depend on the values of the quan-
tities involved. ........
3. 4. Notation, &c. of Functions. ..... 2
5, 6. Description of the Axes of Co-ordinates.
7. The reason of the positive and negative Signs of the
Ordinates. -----.--3
8. Invariable Ordinates are sometimes distinguished by traits
9. The equation of a Curve explained. -
10. Rectangular Axes most generally useful. ... 4
SECTION II.
Equations of the First Degree.
11. Two forms used for Equations of the first Degree, viz. :
ax-}-by-\-c=:o, and y = m x + h. - -
12. Equations of the first Degree determine straight lines. - 5
II. CONTENTS.
ART. PAGE
13. The Formulae may be readily transferred so as to appertain
to either form of the Equations. - 5
14. The Equation of a straight line making a given angle with
the axis ofx. __._--- -6
15. Determination of the Intersections of a straight line with
the Axes of Co-ordinates. - - - -
10. The Equation of a straight line in terms of the portions cut
off from the Axes. - - - - - - ' 7
17. The Case in which a line passes through the origin.
18. The Cases in which a line is parallel to one of the axes. -
19. The Equations of the Axes themselves are y o, x o . 8
20. The condition requisite for parallel lines. -
SECTION III.
Formula, fyc. relating to Straight Lines.
21. The Form of the Equation when the line passes through a
point which is given. - - - -
22. The Equation of a line passing through a given point, and
inclined at a given angle with the axis of x. - -9
23. Determination of the Equation of a line drawn through
two given points. - - - -
24. Determination of the intersection of two given lines.
25. Expression for the length of aline joining two given points. 10
26. 27. Expressions for the inclination of a line with the axis
ofx. - - - - _ _
28. Expression for the perpendicular from the origin on any
proposed line. - - - _ - 11
29. Expressions for the inclination of two straight lines. -
30. The condition of parallelism determined by putting the in-
clination of the lines rr o . - _ - 12
31. The condition necessary for two lines to be perpendicular
to each other. - - - . _ _
32. The same condition when the equations are of the form
y =r mx + h . - - - - _
33. The equation of a straight line traking a given angle
with another straight line. - - - -
CONTENTS. ^* 111.
ART. PAGE
34. The same when it also passes through a given point. - 13
35. Determination of the equation of a straight line, perpendi-
cular to a given one. - -
36. The equation of a line perpendicular to another given line,
and also passing through a given point. - -
37. Expression for the perpendicular on a given line, from a
given point. -
38. Expression for the perpendicular, from the origin on a
given line. - - - - - - 14
39. Expression for the perpendicular distance between two
parallel straight lines. - - - -
40. 41. Expression for the distance between a given point, on
a given line, and its intersection with another given
line. - - - - - - 15
SECTION IV.
Transformation of Co-ordinates.
42. The trans formation of the axes is sometimes useful to sim-
plify equations. - . - - 16
43. To transfer the origin to a given point, the axes retaining
a parallel position. - - - -
44. To transfer an equation to two other rectangular axes,
making a given angle with the former, and proceeding
from the same origin. - - - -17
45. To transfer to two other rectangular axes, making a given
angle with the former, and having their origin at a
given point. - - - - 18
SECTION V.
Equations of the Second Degree,
46. The general equation is of the form
Ax" + By + Cry + ax -f by + c o.
IV. CONTENTS.
ART. PAGE
47. The first three coefficients A , B , C are independent of
the position of the origin, which is affected only by the
values of the last three constants a, b, c. - - 19
48. The origin transferred to the centre of the curve. -
49. The axes transferred to the principal diameters of the
curve. - - - - - 20
50. The equations of the ellipse and hyperbola. - - 21
51. The cases in which the general equation determines an
ellipse. - - - - -
52. The case in which the general equation determines an
hyperbola. - - - - - 22
53. Expressions for the squares of the principal semi-diameters.
54. Equations from which the values of G, A", B" and u are
to be found. . - - -
55. The values of the co-ordinates of the Centre of the curve
determined by the general equation. - - _
56. When the equation is of the form
Ax* + By* + Cxy + c = o,
the origin of the ordinates is at the centre of the curve.
57. For any curve the Sum of the Coefficients of x* and y* will
be the same in all positions of the axes of co-ordinates. 23
58. A case in which the general equation determines a straight
line. - .
59. Cases in which the locus is impossible. - .
60. A case in which the general equation determines only a
single point. - - _ _
61. Recapitulation of the preceding transformations. _ __
62. The sum of the reciprocals of the squares of every pair of
semi-diameters, of a curve of the second order, which
are perpendicular to each other, is equal to the same
constant value. - - _ - 24
63. When 4 AB C 4 o, the centre of the curve is infi-
nitely remote from (he origin. - _ - 25
The origin transferred to a point x'y' in the curie.
64. The axis of x made parallel to the principal diameter of
the curve. - - - _ _
65. The assumed point x'y' in the curve taken at its vertex.
C6. The determined curve is a Parabola. - -26
67. The equation of the principal diameter. . - 27
CONTENTS. V.
ART.
68. In the particular case wherein a V B b *J A the locus
is a straight line^ - - - -27
69. When C is negative, u < ^- ; and, when C is positive
2
TT
>_.--
70. The equations of the principal diameters of a curve de-
termined by the general equation. - - 28
71. The equations of the principal diameters when the origin
is at the centre of the curve. - - - 29
72. The formula tan 2 u =. _ - applies to both diameters.
A - B
73. Formulae for the values of the principal semi-diameters. - 30
74. Expressions for tho values of the principal semi-diameters
when the origin is at the centre of ihe curve. - 31
75. Discussion of the particular descriptions of a curve of the
second order from the values of the constants which
belong to its equation. - - -
Arrangement of the several cases. - - - 33
SECTION VI.
Formula for Curves, fyc., involving the Differential and
Integral Calculus.
76. Determination of the point of intersection of two indefinitely
near positions of a variable straight line. - - 34
77. The line being supposed in motion, this point is the centre
of instantaneous rotation at each position, and its locus
is the curve to which the line is always a tangent.
This curve may be determined by eliminating the
variable from the resulting values of x 7 and y>. -
78. How the point of intersection of two indefinitely near
positions of a variable curve may be similarly de-
termined. - . - - - - 35
vi. CONTENTS.
ART. PAGE
79. Expression for computing the area of a curve, viz :
ydx. ' - - - - -35
80. Expression for the computation of the length of a curve, viz :
81. The tangent, normal, subfangent and subnormal of a curve
described. - _ 36
82. The equation of the tangent to a curve at any proposed point. :
83. The equation of the normal at any point. - - 37
84. Expression for the perpendicular from the origin upon
the tangent. - - -
85. Expression for the perpendicular from the origin on the
normal at any point. - - - -
86. The values of the Subtangent, Subnormal Tangent and
Normal. - - - - -
87. On Asymptotes. - - 38
88. How to ascertain whether a curve be convex or concave to
the axis of x at any proposed point. - 39
89. To ascertain whether a curve be convex or concave to axis
of y at any point. - - - -
90. The condition which takes place at a point of contrary flexure.
91. 92, 93, 94. Expressions for the Radius of Curvature of a
curve at any proposed point,. - 40-42
95. How to determine the locus of the centre of the circle of
curvature, or the Evolute of any given curve. -
96. The normal at any point of a curve is a tangent to the
evolute and, estimated from the point of contact, is the
radius of curvature at that point. - - 43
97. The length of the arc of the evolute between any two points
is equal to the difference between the radii of the cor-
responding circles of curvature. - - _
98. Any curve may be conceived as described by the unwind-
ing of an inextensible thread from off its evolute. The
qurve as contradistinguished from the evolute is called
its involute ; and hence the involute of a curve is
thus described. - - - - 44
99. A curve can have but one evolute, but may have an inde-
finite number of involutes. - - .
CONTENTS, VII.
ART. PAGE
100. How the inrolute of a curve may be determined. - 44
SECTION VII.
Formula; appertaining to Polar Equations.
101. Definitions. - - - - - 45
102. The polar equation of a curve explained. - -
103. Radii vectores, &c. which are invariable are sometimes
distinguished by traits. - - -
104. It is often useful to transfer expressions involving rec-
tangular co-ordinates into such as shall involve the
radius vector and the polar angle. - -
105. How to transfer expressions involving the rectangular
co-ordinates xy and their differentials into such as shall
involve the radius vector and the polar angle. - 46
106. The same when the origin or pole is required to be at a
given point ; and also when the polar axis is required
to make any proposed angle with the axis of x. -
107. How to reduce expressions involving the radius vector and
the polar angle into one involving rectangular co-
ordinates. - - - - -
108. Formula for the length of the arc of a curve, when the
polar equation is known. - - - 47
109. Expression for the perpendicular from the origin on the
tangent at any proposed point. - - -
110. How to find the sectoreal area contained between the
curve and any proposed radii vectores. - - 48
111. Expressions for the inclination of the tangent with the
radius vector drawn to the point of contact. -
112. Expression for the polar subtangent. - - 49
113. Expression for the polar subtangent in terms of the radius
vector and the perpendicular on the tangent. -
114. Expression for the radius of curvature in terms of the
radius vector and the polar angle. - - 50
viii. CONTENTS.
ART. PAGE
115. Expression for the angle contained between any two radii
vectores, when the equation of the curve involves the
radius vector and the perpendicular on the tangent. - 50
116. Expressions for finding the length of any arc of a curve,
and also the sectoreal area between any two radii
vectores, when the equation expresses the relation be-
tween the radius vector aud the perpendicular on the
tangent. - - - - -
117. Expression for the radius of curvature in terras of the
radius vector and the perpendicular on the tangent. - 51
118. To find the equation of the evolute. . -
119. How the involute may be determined. - - 52
THE
MODERN APPLICATION
OF THE
ALGEBRAIC ANALYSIS
TO
GEOMETRY.
SECTION I.
DEFINITIONS AND FIRST PRINCIPLES.
ARTICLE 1. ^ Function is any analytical expression,
containing one or more variable quantities, combined or
not with constant quantities ; it is called a function of
the variable quantity, or quantities, which it contains.
Thus a; 2 -\- a x, \f (a 2 a; 2 ) are algebraic functions
of a?; tan x, which denotes the length of the circular
arc whose radius is unity and tangent a 1 , is a trigonometric
function of x; and a- 2 -{- y 2 - -j- x y, a x -\- by x ^ are
algebraic functions of # and y.
2. It hence appears that functions of any quantities have
their values entirely dependent on the values of those quan-
tities. For when the values of any quantities are given, the
values of any functions in which they may be involved become
immediately determinable.
B
o. A funrtion of any variable or variables is generally de-
noted by prefixing one of the characters JF, f, <p, \J/, &c.
Sometimes functions of single variables are distinguished by
their capitals; thus X, X', X", Sec. being taken to represent
functions of a-j and Y, Y, Y' t &c^ functions of y.
4, functions are said to be the same when the variable
quantity, or quantities, which they contain, enter into their
respective expressions in exactly the same manner. Thus
.r 2 -j- ax is the same function of a; that y* -\- ay is of y, and
a,' 2 -J- 7/ 2 -\- x y is the same function of x and y that n 2 -j- v 2 -[-
wt? is of M and r. Supposing <p to denote the characteristic
of # 2 -}- fl:r, that is, supposing x 1 -^- ax to be indicated by <p ar,
the expression ?/ 2 -j- ay will thence be indicated by <$y;
Similarly if a? -J- y 2 -j- a?/ be represented byy* (.r, y), the
value of M 2 -j- v 2 -|- wt; will hence be denoted byy (, v).
, *'2
Also assuming \J/ (.r, ?/) for ar 4- by - 3, by the same
c
transfer of notation %J/ (M , ?;) will denominate the expression
,2
+ 7 iy
b v .
c
5, In order to express geometrical positions algebraically
points are referred to what are called the axes of co-ordinates.
These axes of co-ordinates are two fixed
indefinite right lines O #, OB, taken on
a given plane and parallel to two given
lines so as to form a given angle A O B.
The point from whence they pro-
ceed is called the origin*
If any point P be assumed in the same
plane and the parallelogram PD' OD completed, the portiops
OD, OD', taken from the origin along the axes, are called the
co-ordinates of the point Preferred to the axes O A and O B.
These ordinates are usually denoted by x and y, viz : O D
by x and O D' or D P by y and OA is hence called the
axis ofx and O B the axis ofy.
6. When the axes O A, O B are per-
pendicular to each other, they are called
rectangular axes ; and O D, D P are
then called the rectangular co-ordinates
of the point P.
7. If the position of a point be on the contrary side of the
axis O A its ordinate y will have a negative value ; and if
it be on the contrary side of O J?, or if the point D be oa the
contrary side of the origin O, we shall similarly have its ordi-
nate x negative. For the ordinates are then estimated from
the origin in the opposite directions,
8. Particular ordinates which appertain to fi.xe<J points are.
distinguished thus, x' y', x" y', &c. ; and the points thus de-
termined are called the points x' y', x" y\ &c. A point on
O A has y' = o and is denoted by x' o, and a point on O li
is denoted by y' o.
9. If the ordinates x and y &r e so related as to fulfil a given
equation in which they are involved with constants, we shall
have particular values of y for each particular value gf a\
Thus let
be the equation which connects the ordinates x and y,f (,c, t/)
denoting some given function.
Then if a series of ordinates Q />,
O D', &c. be assumed as values of-r,
the ordinates D P, Df P', &c, which
are the respectively corresponding
values of y, will be determined by
the solutions of the proposed equa- o D
t'on; and the point P is hence limited to a certain curve
line P P'. This curve line is said to be represented or
indicated by the given equation, which is usually called the
equation of the curve.
4
10. Rectangular axes are more extensively used, and, in
most cases, are more simple in their application than those
which are oblique. Sometimes, however, in particular geome-
trical investigations, the axes may, with great advantage, be
taken parallel to certain given lines in the figure; yet, as
rectangular axes are generally of more utility, we shall con r
fine ourselves to the consideration of them in the subsequent
investigations.
SECTION II.
EQUATIONS OF THE FIRST DEGREE.
11. The equation of the first degree is of the form
17 I ^-| \
I J I '""" ' * * * * \ /*
wherein each of the constants a, b, c, may be either positive
or negative.
By solving for y, it gives
a c
y =. x ;
and hence, assuming
a , c
m __, h =. ,
b b
it becomes
This form of equation, in which the constants m and h may be
either positive or negative, evidently then comprehends in it
all cases whatever of the fast degree, the same as the above
equation marked (1).
12. Let C' be any assumed inva-
riable point in the locus
determined by the equation
y =. mx -\- h,
whose ordinates OG = x', GC' = y'
will hence satisfy the equation
y' == mx' -|- h.
Let also P be any other point in the locus whose ordinates
O D, D P fulfil the equation.
y = m x -j- h ;
and, by deducting the above, we derive
y y' =. m (x #').
But, by drawing C'D'H' parallel to O A the axis of <r,
it is clearly seen that 01 x' =. GD = C'D' and y y' =z
D' P ; therefore
D P= rn.CZ>'
and .'. m =. _- . =. tan _ PC' 11.
\-s ./-/
Since this property applies equally to all points P whose
ordinates fulfil the proposed equation, the determined locus
is evidently a straight line through the point C' making an
angle with the axis of A) whose tangent =. m.
o o
Similarly the equation
a x -j- b y -}- c o
determines a right line intersecting the axis of x at an angle
whose tangent ==. , (see article 1 1).
13. It hence appears that all equations of i\\e first degree
determine straight lines ; also, (1 1 ,) that these equations may
be reduced to either of the forms
ax -\-by-\-c-o .. (1,)
y = m j? -f h . . . (2.)
The latter of these is the more simple in its application, in
consequence of its involving only two constants, in, h. How-
ever, as equations will not always reduce to this form without
fractions, we shall generally deduce the several formula- for
both cases. And it may be observed that formulae applying
to the former may be easily rendered suitable to the latter
by substituting
m for a ; 1 for b, and h for c,
as the equation would then become
m x y -J- h zz: o
or y == m x -\- h t
agreeing with the latter equation.
It may also be observed that, vice versa, formulae belonging
to the latter mode of equation may be appropriated to the
former by substituting, (11,)
~ for m> and ~ for A ....
b b
14. Since, (12,) m is the tangent of the angle which the line
makes with the axis of #, by calling this angle , the equation
of the line may be expressed thus :
y^=.ap tan u -j- #,
15. When y z= o, the point P must obviously be situated
on the axes of x, and will therefore determine the intersection
of the line with O A. Putt/ = o in the equation
ax | b y J c zz: o
and let x" be the corresponding value of x and we shall
thence have
c
X ,
a
that is,
Qi = _,
a
I being the intersection of the line with the axis of or.
Similarly, by taking x = o, >ve find the intersection on the
axis of y to be determined by
10. By dividing the equation of the line by c and changing
the signs, it becomes
a b
--*--y-i = o,
which is hence equivalent to
L+ *L_i=. f
x' y
x y
Or ^ + 7 = 1 -
This equation may therefore be regarded as determining that
straight line which cuts portions from off the axes of x andy,
estimated from the origin, respectively equal to at" and y".
17. When c = o, or h = o, the equation becomes of the form
ax -{- fey = o
which is satisfied with a? = o, y = o.
This shews the line to pass through the origin. In this case
we have x" = o, /" = o.
18. When a = o, the equation is
i y -|- c =. o
/*
or y z= , a constant value.
6
In this case the value of x is arbitrary; and therefore the
line is parallel to the axis of x at the distance on the
b
side where y is positive.
If b =. o, or the equation be of the form
a x -{- c =z e,
we shall have
x = and i/ arbitrary ;
and consequently the line is parallel to the axis of y at the
distance on the side where ,r is positive.
a
8
, 19. When the equation becomes simply b y =. o, or y r= o,
it evidently determines the axis of x ; and similarly when it is
x =. o it indicates the axis ofy.
20. We have seen, (1 2,) that the tangents of the angles which
the straight lines
a x j- b y -f- c n= o,
y = mac -\- h
make with the axis of x are respectively and m.
b
It iherefore appears that equations, which have the values
of , or of m, the same, determine parallel lines. Thus
b
aa?-\-by-J(-c = o,
a x -j- b y -j- c' o,
n a x -j- n b y - {- c" r= o
represent parallel lines.
And similarly, the equations
?/ := ?n cV -j- h,
y =.mx -\- h f
determine parallel lines.
SECTION III.
FORMULA, &c. RELATING TO STRAIGHT LINES.
21 . To express the equation of a straight line which shall
pa$s through a given point.
As one condition is here imposed upon the line, one of the
constants a, &, c will become a function of the other two and
the invariable ordinates x' y' of the given point, since they
have to satisfy the condition
a x' -|- b y' -j- c o
Hence eliminating c, by taking its value (.??' -|- by'),
the equation of the line, a .r -j- b y -|- c =. o, becomes
9
aat-{- by (ax' -\- by')=o,
or a (x on') -\-b(y y')=o,
which determines a straight line passing through the pro-
posed point ao' y'
Or it may evidently be expressed thus :
?/ ?/' = w (# - <),
m being the tangent of the inclination of the line with the
axis of x.
22. Cor. It hence appears that the equation
y y' = (a? a?') tan 01
determines a straight line passing through the point OB' y' and
making the angle u with the axis of x.
23. To determine the equation of a straight line passing
through two given points.
Let x' y', OB" y" be the co-ordinates of the two points ; as
these points are situated in the line, the constants m t h t must
evidently fulfil the equations
y' = m x' -[- A,
y" =maj" -f- A;
from which we find
x" at' at" at'
As the line passes through both of the points so' y', ao" y" its
equation, (21,) is hence
or
wherein x'y',x" y" are the ordinates of the two given points
and a? y any point whatever in the line.
24. Given the equations of two straight lines to find the
co-ordinates of the point where they intersect.
Let a b c, a' b' c' be the constants which belong to their re-
spective equations ; and let x y be the ordinates of the point
c
10
of intersection j this point being posited in both lines, its
ordinates x y must satisfy both of the equations
ax-}-by-{-c=iQ,
c' =. o.
fee 7 c\)' r ,, ac' ca'
oc t - ana w = - . ,
ab' ba' ab' ba'*
which determine the required point.
If the equations of the lines be of the forms
y c= m x -j- ft*
y = mf x -f- h'
we shall hare
h h f __ m' h m h'
m w'' ' m m'
25. The co-ordinates of two points being given to find an
expression for their distance, or the length of the line which
joins thein.
Let xy,x'y' be the given ordinates; then drawing lines
from the two points parallel to the axes, viz : one parallel
to the axis of oc and the other parallel to the axis of y, we
shall obviously have a right angled triangle whose legs are
x _ #' t y y' 5 and thus the square of tbe required line
is found equal to
(^-^
or i= v $(*
26. The equation of a straight line being given to express
its inclination with the axis of x.
The equation of the line being
its inclination ca with the axis of x is, (12,) determined by
a
tan &==. ,
b
which gives
cos M z=
tan * ) V ( 2 - fr 2
11
and sin = tan u
_ _
(I + *a 2 w) ^ (a 2 -|- fc 5 )*
27. When the equation of the line is of the form
y = m at -j- h,
we have, (14,)
= TO and .'. cosw=
28. Cor. Letx" determine the intersection of the line with
the axis of #,as in article 15, and we shall evidently have t for
the perpendicular from the origin on the proposed line,
p = x" sin u. Hence, substituting 1 the value of x" t (article
15,) we have
*29. Given the equations of two right lines to find their
angle of inclination.
Suppose 6c,a / b' c' to be the constants contained in their
equations and let i denote the required inclination ; also let
<y, a' be the inclinations of the two lines with the axis of x.
Then, (12,)
tan u-=. , tan u' = ;
b b'
and we evidently have
tan u tan &/
i =. u u' and .". tan i =:
1 -j- tan ca tan a/'
Therefore by substitution
a' b b' a
tan t=
aa'
Hence also
1 a a' -j- b b'
COS I =
V 4- tatf i> V { ( 2 + & 2 ) (a A H- 6/2 ) \
sin i=cos iXtani= "
30. Cor. 1. If the lines be parallel, i = o and sin i = o ;
.*. a' b b' a = o
a' a
or F = T'
agreeing with (20.)
31. Cor. 2. When the lines are perpendicular, cos i = o
and hence
a a' -f- & o' = o.
32. Cor. 3. If the equations of the lines be
y = m x -\~ h,
y = w' # -f- ^'
we shall have
tan u = m, tan ' = m' ;
m m x
^aw i =
1 -4- m mf
cos i =
*;$*;;
"it * 7/t
* == ./'5/i 4-^2) /f 1^/2)?
When the lines are parallel, sin i = o,
And when they are perpendicular, cos i = p, and
33. To determine the equation of a straight line inclined
at a given angle with a given straight line*
In the last proposition, article 29, let a o c be the constants
to the equation of the given line and a' b' & those of the one
required. Then
tan */ = tan (, - Q = - tan * tan .
1 -J- tan u tan i
or, substituting , , for tan u, tan u',
13
.. a -\- b tan i q cos i -j- b sin i
b' b a tan i b cos i a sin t "
Hence the required equation is
(a cos i -j- & sin i) x -j- (b cos i a sin ?) y -\- e" = o,
wherein c" is an indeterminate constant whose value may be
found from another condition.
34. Cor. 1. If the line be required to pass through a given
point x' y' its equation, (21,) will therefore be
( cos i -\- b sin i) (x a;') -\- (b cos i a sin i) (y y'} = o.
35. Cor. 2. When the line is perpendicular to the proposed
one its equation will be
b x a y -\- c" =. o.
36. Cor. 3. When the line is perpendicular to the given
one and also passes through a given point OB' y' its equation
will hence be
b (x x') a (y -* y') = o.
The general equation of a straight line which is perpendicular
to one determined by the equation
y =. mr -\-h
is y =. x -J- h' ;
m
and, when passing through a given point x' y' is
y y* = (* a?').
37. To express the length of the perpendicular on a given
line from a given point.
Let ax -{-by -f-c = obe the equation of the line and x' y f
the ordinates of the given point. Then, (36,) the equation
of the perpendicular is
b (x arO -^a(y--y') = o,
from which, together with the above equation of the given
line, we find the point where it is intersected by this perpen-
dicular to be determined by ordinates x y whose values are
6 2 x' a b y' ac a 2 y' a b x' be
o 2 -f 6* a 2 -f 6 a "'
14
Now, (25,) the length of the perpendicular 1* =
V {(*-*') 8 +G/-2> / ) 2 ?-
Hence by substitution
- -,_ a(gM-ty / + c) ,_ a(a
, D --
and P = - !
V ( 2
If the equation of the line be of the form y = m x -\- h t
substituting m for a, 1 for b and A for c, (see 13,) and we
shall have
P = -
38. Cor. 1. When the given point is the origin, we have
x' = o, y' = o ; and hence, for the perpendicular from the
origin on the given line, we have
P=
'
/ / 9
v let
T \"^
or P =
V (I + * 2 ) "
39. Cor. 2. By (20,) the equations
ax- \- b y -\- & =. o
represent parallel lines.
Now, (38,) the respective perpendiculars from the origin are
*i' _
the difference of these gives the perpendicular distance of the
said lines =
V (a 2 -f 6 2 "j
Similarly the perpendicular distance between the paral-
lel lines
y = m x -j- h,
y z= m x -\- 7i A
15
is equal to
A<^> V
V (1+V
40. To express the distance between a given point x' y'
on a given line and its intersection with another given line.
Let the equation of the given line, which passes through
the point x'y', be
a' ( x _ */) _|_ V (y _ y'] o ;
and that of the other given line
ax-\-by-\-c-=zo
Let #T/ be their point of intersection and its ordinatcs will
be the same in both equations. The latter one being put
in the form
a(x x'}-\-b (y 2/0 -f ( x' + by' -f c) = ,
we shall, by means of it and the former, find
a' 1) b' a a' b b' a
Therefore, for the required distance,
= v (^-
Otherwise,
Let p be the perpendicular from the point x' -g' on the line
ax -{-by -f- c = o, Then, i being the angle of inclination
of the lines, we obviously have
p-=. D sin i
!>=-?-.
sin i
Hence, substituting the values of p and sin i already laid
down, (37, 29,) we get
a' b b a
41. Cor. When the equations are
y =. m x -\- h for the intersected line
and y y f = m' (x x'} for the line passing through tire
16
point x'y'i substitute m, m' respectively for a, of and 1 for
b t b' t and h for c, (13,) and
D = nx' + k-y' v (l + m ^
m m'
SECTION IV.
TRANSFORMATION OF THE AXES.
42. With the view of expressing any particular lines or
curves, being- the loci of points, by algebraic equations, we
are manifestly at liberty to assign to the origin and the axes
any positions whatever, relative to the said loci ; and hence,
when the equation of a locus is complex, it becomes some,
times useful to assume another position of the axes which
will reduce it to a more simple form. This transformation,
which is called the transformation of co-ordinates, is effected
by expressing the original in terms of the new co-ordinates,
of any point, which will, of course, be ready for substitution
in any equation or formula, appertaining to the former axes,
so as to produce the equivalent involving the new co-ordi-
nates. For these operations the three following propositions
are necessary.
43. An expression involving the two rectangular eo-
ordinates being given to Jind the corresponding expression
in terms of the co-ordinates when the origin is transferred to
a given point, the axes retaining a parallel position.
17
Let O' be the given point to which
the origin is to be transferred ; and
let its position referred to the axes
OA, OB be OG= a, GO' = b; also
let the position of the point P related
to the new axes O'A', O'B' parallel
to OA, OB, be x'y' viz O'D' = x'~ and D'P=y'. Then is
O D = OB = O' D' -f O G = x' -f a,
which substituted for x and y will give the expression re-
quired, wherein ab, the ordinates of the new origin O', will
be given constants and x'y' the ordinates of Preferred to
the new axes.
By this means we transfer the origin O to a point whose
ordinates are x=.a,y^z:b,
44. An expression involving the ordinates of a point
referred to two rectangular axes being given to Jind the
corresponding expression when the point is referred to
two other rectangular axes making a given angle with the
former and proceeding from the same origin,
We shall omit the axis of y in the figure for the sake of
simplicity, since it is sufficient to bear in mind that the positive
ordinates y extend from the axis of
x upwards. Let OA be the original
and OA' the new axis of x ; then are
OD, DP, and OD', D'P the co-ordi-
nates of P. Draw D'H perpend icu- D ir A -
larand D'K parallel to OA ; and let the ordinates OD', D'P
which refer the point P to the new axes be x'y'. Then,
assuming the given /_ A'OA = _ D'PK =. u, we shall have
OH=.x' cos u and DH= KD' = y' sin u, the difference of
which gives
OD -=.x~ x' cos u y' sin ....(!);
also D'H=, KD =. x' sin u and PK = y' cos u, which added
give
PD = y^= as' sin u -\- y' cos u . . . . (2).
D
18
These values of OD and PD introduced instead of a? and y
will produce an expression involving x'y' and the given
angle u.
It must be here observed that the new axis OA / of x is taken
on that side of OA on which the ordinates y are positive ;
when taken on the contrary or under side of OA, the angle &>
will have a negative value.
45. An expression involving the co-ordinates of a point
related to two rectangular axes being given to deduce the
corresponding equivalent in terms of the ordinates of the
same point referred to two other rectangular axes making a
given angle with the former and having a different origin.
Let O'A' be the new axis of a? ;
I'D' perpendicular to it from the
point P, and O'er parallel to OA.
Denote the position of the new
origin O' by a/>, viz:
GO'=.b; let the position of P with respect to the new axes
be x'y', that is O'D' x', D / P=y / - and, as before, denote
the given angle of inclination A'Oa. by o>.
Then, (44,) the co-ordinates of Preferred to O'a as an axis
of x are
O'd =. us' cos cj y' sin u Pd^=.x f sin u -\-ycosu;
and hence, (43,) the values of the original ordinates are
OD = x=.x f cos u y' sinu-\-a. .. (1),
PD =. ii =, x' sin u -f- ?/' cos u>-\-b. . , . (2).
/ i / v~/*
By substituting these instead of x and y in the proposed
expression we shall get an expression, involving x'y' with
the new additional constants a b and the given angle a, which
will be the one required.
19
SECTION V,
EQUATIONS OF THE SECOND DEGREE.
46. The general form for equations of the second degree,
being those in which the ord mates XT/ are involved to the
second power, is
A x 2 -f B y* -f- Cx y -f- ax -\- by -}- c = o
wherein each of the constants #, J9, C, a, b, c, may be either
positive or negative.
Let us in the first place transfer the equation to two other
rectangular axes parallel to the original ones and having
their origin at a point whose ordinates area/>; and, (43,)
by substituting x -{- x* and y -j- y' for x and y t we shall find
the corresponding equation to be
A (& + 2 x'x + x) + B(y*+2y'y + y'*)
-J- C(xy -|- y'x -f x'y + x' y')
+ a (x + a-) 4- b (y + y ') -f c = o ;
which arranged for # and y becomes
^^ + By* -f C^y
+ (2 4*' + Cy' + a)x+ (2 ^/ + Cx'
47. The first three coefficients A,B,O stand unaffected
with the new constants x',y', by which we observe that they
are independent of the position of the origin ; and hence the
position of the origin of any equation of the second degree
depends entirely on the values of the three last co-efficients
a b,c.
48. We may now assume the values of the two ordinates
x' y' at pleasure since the position of the new origin is entirely
arbitrary; and consequently, by the principles of algebra,
we may fulfil any two possible conditions which involve
them; let us therefore put the coefficients of x and y each
equal to nothing, viz :
20
and thence
'
lience also, by substitution, the last term
C*
or by assuming
it becomes =
IAB C*
The equation is thus transformed into
-- 2 = ... (a),
in which the fourth and fifth terms are wanting.
49. Let us now transfer this equation to two other rec-
tangular axes inclined at an angle u with the former and
retaining the same origin ; and , (44,) substituting x cos u
y sin u and x sin u -\- y cos w for x and y, we get for the cor-
responding equation
A (x* cos 2 u -|- y 2 sin 2 u 2 <r y cos u sin u)
-\- B (a? sin* u -f- y 2 cos 2 u -\- 2 x y cos u sin u)
-\- C $ at? cos u sin u if- cos m sin u-\- xy (cos 2 u sin 2 u) ?
, G
h 4^jS C 2 "
which arranged for x and y, observing that *
cos 2 u siw 2 ca =. cos 2 u and 2 cos u sin u =, sin 2 <y,
becomes
(A cos 2 u-\-B sin 2 u -\- Ccos u sin u) x 2 -j- (A sin 2 ca -\- B cos 2 <a
C cos u sin u) y 2
-\- $Ccos2u (A-B)sin2w$ xy
G
-4- - rr o.
r IAB C*
21
By taking iLe value of u so as to exterminate xy,
Ccos 2 u (A B) sin 2m = o
and tan 2 <u = - ,
AB
which reduces the equation to
(A cos 1 u -j- B sin 2 u -\- Ccos u sin u) x 2 -f- (A sin 2 u-\-B cos^w
Ccos u sin u) y 2
! &
4- -- =0;
r 4AB C 2
and it hence appears that every line of the second order may
be referred to two determinate rectangular axes so that its
equation shall be transformed into the above form. By
assuming
A cos 2 u -{- B sin 2 u -\- C cos u sin u =, A" t
A sin 2 u -\- B cos 2 en Ccos w sin u = B\
it becomes
A"
__ = o . ..(b).
7 2
50. Now if the principal semi-diameters of an ellipse and
hyperbola be denoted by a', b', and the former be taken for
the axis of a? and the origin at the centre, their equations will
be as follow :
For the ellipse
* + ll = 1, or b' 2 x 2 + a /2 y 2 a' 2 b* = o ;
a' 2 fe' 2
and for the hyperbola
Jl = 4- 1, or b'W a' 2 ij 2 IE a' 2 b' 2 =
/2 /2
o
the under sign representing the conjugate hyperbola,
The signs may be all changed if necessary.
By means of these two equations and the foregoing trans-
formed equation, (6,) we deduce the following particulars
relative to the general equation.
51 . 1st. When A ,B are both negative and G, 4 A B C 2
have the same sign, the equation determines an ellipse ; and
22
when A",B are both of them positive andG ? and 4 AB C 2
have different signs, the locus is also an ellipse.*
52. 2nd. When A",B"are of different eigns and G not
= o, the locus is an hyperbola.
53. 3rd. In each of these cases the squares of the principal
semi-diameters are equal to
G __ +G
A (4 AB C 2 ) ' B (4 AB C 2 ) '
the under sign being for the ellipse and either sign for the
hyperbola.
54. 4th. The values of G,A",B" are determined from
the equations
G = C a b A & B a 2 + c ( 4 A B C 2 ) . . . ( 1 ) ,
A" =. A cos 2 w -\- B sin* v-\-C cos u sin a, } ,v
B" -= A sin 2 u -\- B cos 2 v-~C cos u sin u }
wherein w is the angle included between the original axis of x
and the principal diameter of the curve.
55. 5th. The position of the centre of the curve is de-
termined by
x' = cb 2Ba > _
" ' *
4 AB C*
56. Gth. When the equation is of the form
wherein the fourth and fifth terms of the general equation are
wanting, we have a=o,b=zo and thence ae' = o, y'=.o
which therefore shews the origin to be at the centre of the
* For the immediate values of A", B" see article 73.
23
curve. This agrees with equation (ft,) article 49, where the
origin is transferred to the centre.
57. 7th. By adding the equations (3), article 54, \ve find
A 1 +B" A-\- B.
Hence we see that, whatever be the position of the axes of
co-ordinates, the sum of the co-efficieuts of a? and ?/ 2 will be
the same.
58. 8th. When G=zo and also A" and B" of different
signs, the general equation defines a straight line.
For in this case the transformed equation (ft), article 49,
becomes
which gives
!L=v-~;
and this value is real when A", B" have different signs.
59. 9th. In the two following cases it will be found that
no real values of x and ?/ can possibly fulfil the equation (ft) ;
and consequently that the equation can have no locus.
First. When G and 4 AB C 2 are of the same sign and
A", B" both of them positive.
Second. When G and 4 AB C 2 are of different signs and
A",B" are both negative.
60. 10th. When G-=.O and A", B have the same sign, no
real values of x and ?/ can satisfy the equation (ft,) except
the particular case ot'x=o,y=:o. In this case therefore
the locus is the single point corresponding with the new
origin x' ' y'.
61. 11 th. It appears that by changing the position of the
origin to the centre x' y'
the equation
A ofi+By* + Cccy -\- ax -j- ft t/ -f c = o
24
is transformed into the form
*4
wherein h = .
Also, that by taking two other axes of co-ordinates making
an angle with these so that tan 2ou= _ , the equation
1 A Jo
A 'x* -4 B t/ 2 -j- C x y | h o
becomes of the form
wherein A' -\- B" = A -}- B and the constant h is unchanged.
62. 12th. Let oo",y" be the two semi-diameters of the
curve
which coincide with the axes of co-ordinates to which it is
referred, and they will be determined by taking first y~o
and then x = o in the equation, the results being
h ,, h
~A' y ' ~B'
Let also a / ,ft / be the principal semi-diameters which coincide
with the axes to which the equation
appertains ; and we similarly have
a /2 = A, J'2 __A
A" B
Hence as A" -j- B " = A -\- B, we have
That is the sum of the reciprocals of the squares of any two
semi-diameters, of a curve of the second order, which are
perpendicular to each other, is the same ; and, in reference to
the general equation, is =
25
63. When 4AB C 2 = o, we have, (55), x'y' both of
them infinite which shews the centre of the curve to be infi-
nitely remote from the origin. It becomes hence necessary
to consider this case separately.
Let
bo the general equation in which 4 AB C' 2 = o.
Then, transferring the origin to a point x'y', the correspond-
ing equation, (46,) is
Ax'~ -\- B y'- 2 + Cos' y' -|- a x' -\- b y' -f- c) = o.
Let x' y' determine some point in the curve, so that
Ax' 2 -\- By'*-{-Cx'y'-\-ax'-\- by'^-c = o,
and the equation becomes
Atf + B f- A- C 'of y
+ (2 A x' -\- C y' + ) as + (23 ;;' -f CV + b) y = o.
But, since 4 AB C' 2 o and .; i'J= 2 \f JIB, we have
Atf 4- B if + C a? y = (.r // ^ -[-?/>/ ) 2 .
Hence the reduced equation is equivalent to
(xV A+yV ^) 2
+ (2 J^ + Cy' + ) a- -|- (2 5 / + C/* x + 5) y = o.
C4. We shall now, as in article 49, transfer this equation
to two other rectangular axes proceeding from the same
origin and making an angle u with the former; and, (44,)
putting x cos co y sin u and x sin u -}- y cos u for x and y t
the resulting equation is
^ (cos u \f A -\- sin u \/ B) <r (sira w \/ ^4 cos u *J B)y\ 2
-\-{(2Ax'-\-Cy'-\- a) cos a -{- (2 B y' -\- Ccc' -f fe) *in $ #
J (2 ^a?' + Qy x +) sin w (2 By'-\-Cz'-\-b) cos *>}y = o.
Let a satisfy the condition
cos u \f A -\- sin ca A/ B = o,
which will give
= \/ ,coswi=:
26
and thence
sin u \f A cos u V" B =. */ {A -f- B) ;
V
and (ZAx'-\- Cy' -f a) sin w (2 # / -f C x' -f 1) cosa> =
a \S A -4- b \/ B
The equation thus becomes
65. We have, (63,) assumed x'y' to determine a point in
the curve, but not restricted ourselves to any particular point ;
we may therefore take this point where the curve is intersected
by a straight line whose equution is
/^i / n i V A-4-b A/ B
x \f A-\-y \f B - } r - - IU - = o,
2
by means of which we shall have
which reduces the equation to
B b f
. x = o,
/ /? i ?>\ 9 a \ B b \f A
(A -4- JB) y- _i
But the equation of a parabola, whose parameter is p, taking
the origin at the vertex and the principal axis for the axis
of a 1 , is
2/ 2 =/Kv or y* px 0.
Hence the following particulars:
66. 1st. When a V B b V A not=;o, the locus is. a
Parabola whose parameter is equal to
27
n V B 1> V A
67. 2nd. According- to article 12, the equation
/ /i i / o i a V A-4-b V E
x y A-\-y A/ .B-r- - IL l - -=.
defines a straight line inclined to the original axis of x at an
A
angle whose tangent V aud which is therefore equal
H
to u, the inclination of the axis of the curve, with the axis
of#; this line, (65.) also passing through the vertex x'y\
it must coincide with the axis of the curve. Therefore the
above equation properly represents the principal diameter of
the curve; by uniting it with the original equation we may
hence find the co-ordinates x'y' of its intersection with the
curve, or the vertex.
68. 3rd. lfaVBbVA = o, or aVB = l\fA,
the equation (c) gives simply
y = ,
which shews the locus in this case to be a straight line cor-
responding with the new axis of a\ the equation of which
is given, (G7).
69. 4th. The equation AB C 2 =o giving C 2 r=
+ 2 \f AE, the values of the constants A,B must have the
same sign to make C real, that is, they must be either both of
them positive or both negative ; and hence we may consider
them both positive for, when negative, they can be made so
by preliminarly changing all the signs of the original equa-
tion. If, under this consideration, C be negative we shall
have C = 2 V AB instead of -f-2 V AB ; in this case,
the foregoing operations hold good by either substituting
V A instead of V A or \/ JB for \f B, or by considering
either \f Aor *J B to have a negative value; and tan u will
A - A
become hence = -f- instead of \/ .
B li
Thus we see that, when C is negative, tan u Is positive and
u < and that, when C is positive, tun u is negative and
2
"." CO > .
2
The foregoing investigations lead immediately to the solutions
of the three following propositions :
70. To express the equations of the principal diameters
of a curve of the second order which is determined by the
general equation.
The co-ordinates of the centre, (55,) are
Cb ZBa _Ca
X'
4ABC* 4AB C*
Let u denote the inclination of one of the principal diameters
of the curve with the co-ordinate axis of x ; and, (54,)
C
from which
tan =
tan2u C
Ts T ow the diameter being inclined to the co-ordinate axis of x
at the angle a and also passing through the centre x'y' of the
curve, its equation, (22,) is
y y' -=.(x x') tan u.
Hence by substitution we have
Ca i
y T"7'i
for the equation of one of the principal diameters.
The other diameter passing through the centre x'y' perpen-
dicular to this, its equation, (36,) is
20
s \(A ) a -f C a (4 B}
or, which is the same,
? _
1ABC*
71. Cor. 1. If the origin of the ordinates be the centre of
the curve its equation, (56,) will be of the form
A x 1 -\- B 2/ 2 -f Cxy + c= o
and we shall have a =o, b = o. In this case therefore the
equations of the principal diameters are
and y =-
72. ^o/e. The equation
C
tan 2 u =
applies equally to both diameters. For, if 2 fulfil this equa-
tion, it will also hold good when 2 u ^T<n is substituted ; and,
u denoting the inclination of one of the diameters, u^r -~-
will evidently be that of the other.
From this equation we derive generally
tan * = * ec2 "~ 1 = V \(A
the upper sign appertaining to one of the axes and the under
sign to the other.
* By uniting these equations of the principal diameters wilh
the given equation of the curve we may thence find the positions
of the vertices.
,30
Thus, by making; use of the under sign, the equation (x) will
become the ame as the equation {*/), ami vice versa - ,
because
y | ( A B) 2 +C 2 } - (AB) V{(A BY-\-C*}-(A B)
~~C~ ~C~
1
When 4 AB & = o, see article 67.
73. The equation of a curve of the second order being
given to find the values of its principal semi-diameters.
The squares of the semi-diameters are, (53,) equal to
+ G G
C 2 ) B (A All C*
wherein, (54,)
A ' r= A cos 2 u -j- B sin 2 u -|- Ccos a sin u,
B" A sin 2 u -|- B cos 2 u C cos u sin u,
C
and tan 2 w = - .
A B
From the last we deduce
"If, 1 \_ 1 /,
=z VI 1 =: I 1
o > .*v-2/v,/ o v
cos u sn u r=
and hence we get
2
V M
anul tho foregoini;; value of G substituted, the squares
31
of the principal semi-diameters of the curve are found
equal to
2 {Cab Ab* Bat + c^AB C 2 )
(4 AB C*)A + B V (A
the under sign being- for the ellipse and either sign for the
hyperbola, (53.)
74. When the origin is at the centre of the curve, (61,)
a = o, b = Q ; and therefore in this case the squares of the
principal semi-diameters are equal to
+ 2c -j-2c
A+ B+V {(A -ff) 2 -^ ' ^+-B \/ { (A
75. To determine the particular description of a curve
of the second order from the immediate relative values of the
constants which belong to its equation.
In (51), (52) and the subsequent articles, the different cases
are severally stated, throughout the various relations of
A",B", G, 4 AB C 2 , &c., where A", B" are, (54,) expressed
in terms of the coefficients A, B,C, by means of the arc a as
a subsidiary. It is hence only necessary to transfer the
relations of A", B' to those of the immediate coefficients
A,B,C, which may be easily effected from their values which
have already been found, (73,) viz :
A+B+V
A" zzi
A+B-V
B =
2
Thus it is evident that, when (A-\-B) 2 is greater than (A B)*
-j-C 2 the sign of A-\-B cannot be affected with either the
32
addition or subtraction of \/ | (A U) 2 -f- <7 a , and conse-*
quently that the values of A", B" will both have the same
sign with A-\-B. But, when (J-j-.fi) 2 is greater than
(A B^+C 2 , we shall have (J-j-jB) 2 ^(AB)*-}- C 2 ^ =
4.4U C 2 positive. Hence, when &AB C 3 is positive
A" and 5" will both of them have the same sign with A -}- B,
that is, they will both be positive when A-\-B is positive and
both negative when A-\-B is so.
It is also pretty obvious that, when (A-\-B}* is less than
(A BY + * the va]ues of A " B " wil l have different signs,
that is, the one will be positive and the other negative,
In this case we shall have ( A -f- 5) 2 ^ (A B} z + C 2 1 =
&AB C 2 negative. Thus we see, when 4AB C 2 is
negative, that A", B" are of different signs.*
Again, under the class 4 AB C 2 = o, when the value of
o, we shall have 2VA(a^B b\fA)
or Ca 2 Ab = o.
Hence also, when a\/ B b^ A not =20, we shall have
Ca 2Ab not = o.
By carefully comparing these relations with the articles
(51), (52), (58), (59), (60), (66), and (68), we find the different
descriptions of the curve to be as in the following arrange-
ment, wherein
G = Cab- Ab* Ba* + c (4^B C 2 ).
* These relatious are also pretty evident from the equations
A" + B'=:A+ B,
4A'B' = 4AB C.
-'
in
fc
W
J
-
O
OS
H
E
2
!J
O
-
b
C
2
m
Z ~
^5
ff
^
H>
V)
*- ^
-
OS
(5*
W
O
-
?
s?
H
H
05
8
.2
(A
H
a e
8
8
u
O
^
^
05
s .
II II II II II
, r I I
o
rs
C
-
03
Sf> 4.
C
_=
^
34
SECTION VI.
FORMULA FOR CURVES, &c. INVOLVING THE DIF-
FERENTIAL AND INTEGRAL CALCULUS.
76. The equation of' a variable straight line being given
to find the point of intersection of two of its positions which
are indijinitely near to each other,
Let y = m x -j- h be the equation of the line.
Jt is plain that when m, h are given constants the straight
line is given and fixed thus, in order that the line may assume
another position it is necessary that one or both of the char-
acters i, h shall become of different values. Hence, under
the circumstances of the proposition one or both of the values
m,h must be subject to variation,
Let x'y' be the co-ordinates of the required point which will,
of course, fulfil the equation
y' = moo' -j- h.
If we now suppose the line to vary to an indefinitely near
consecutive position, the point pf intersection x'y' being fixed
during the change its ordinates op' and y' will hence remain
invariable. Hence differentiating, and considering x'y' as
constant, we get
o-=.x' dm -j- dh,
from which and the above equation we find the required
point to be
, dh , , dh
x =. ,y'=zh m .
dm dm
To compute these values it is necessary for A to be a function
of m; this is, in fact, necessary to impose a law on the varia-
tion of the line, the position of which would otherwise be
absolutely arbitrary.
77. Cor. 1. If the line be supposed in motion this point
will obviously be the centre of instantaneous rotation ; and
its locus will evidently be that curve to which the line is
,35
always a tangent* (see 81) - hence the nature of this curve
may be found by eliminating the introduced variable from
the values of cc' and y'.
78. Cor. 2. Similarly to the foregoing may we find the
point of intersection of two indefinitely near positions of a
variable curve by differentiating its equation and considering
x',y' as constant; the values of a?', y' being determined from
the given and the resulting equation.
79. To find the area of a curve comprehended between two
given values of y and the axis of x.
By taking two values of y indefinitely near to each other,
the space included between them may obviously be con-
sidered as rectangular and consequently as having a value
^=ydx.
Thus we have
The area = jydv.
This integral, between the limits x'y', xy will give the area
contained between the ordinates y' and y.
80. To find the length of any portion of a curve from the
equation between its rectangular co-oi dinates.
Let denote the length of the curve corresponding with the
ordinates coy and reckoned from any given point, and we
shall evidently have
ds 2 = dx 2 -f- df
and s =
* This is rendered evident by considering it inversly ; thus,
by supposing a tangent to move over a curve line its successive
indefinite intersections will obviously coincide with the points of
contact and therefore trace out the same curve.
36
By taking this integral between the limfts x'y' and yy we
find the length of the portion of the curve intercepted by
those points.
81. D<f. Let BPC be
any curve.
Then a straight line /?, drawn
*o as to touch it at any point P
is called a tangent; and the
point P is called the point of
contact.
Another straight line PIT, drawn perpendicular to the curve
at the point P and consequently also perpendicular to the
tangent RS, is called a normal at that point,
By the length of a tangent we generally understand that
portion which is limited by its intersection with the axis of x
and the point of contact ; the value of the normal is similarly
understood to be that portion of it which is intercepted by
the axis of x and the point P. Thus, P T is the length of
the tangent and PJVthat of the normal.
OD and DP being the ordinates of P, TD is called the
subtangent and DA* the subnormal.
82. The equation of a curve being given to Jind the equa-
tion of the tangent at any point.
Let x'y' be the co-ordinates of the point of contact and ca
the inclination of the tangent with the axis of x then, (22,)
the equation is
y y'-=. (x x') tan u ;
but we obviously have
dtf
tan u n:
dx'
'.' the required equation is
, dy f . ,,
"dx'
or (x x'} -~ (y y') = o,
dy'
37
wherein x'y' is the point of contact and xy any point what-
ever in the tangent.
Or it may be expressed differentially
dy' (or so'} dx' (y y'} = o.
83. Cor. 1. Hence the equation to the normal through
the same point, (36,) is
y y' = (x x'\
dy' v
or (**') + W (y _y,) 0m
dor
This may also be expressed differentially
dx' (x x'} -\- dy' (y y') = o.
84. Cor. 2. The equation of the tangent at the point
x'y' being
dy' (x x'} dx' (y y'}^=o
or dy' . oo dx' .y-\- (y' dx' x' dy'} =. o,
the perpendicular drawn to it from the origin, (38,) is equal to
x 1 dy' y' dx' _ , x' dy' ?/' dx'
/2 ~
^ (dx'*-\-dy'*) ds'
85. Cor. 3. Similarly, the perpendicular from the origin
upon the normal is equal to
x' dx' -\- y' dy' , x' dx' -\- y' dy'
86. Cor. 4. By taking y = o in the equation of the tan-
gent, x will determine its intersection with the axis of x and
hence in this case x' x is the subtangent at the point x'y'.
'.' Subtanqent -=.x' x = ^ .
dy'
By similarly taking y=o, in the equation of the normal,
we find
The Subnormal = x f ^x= * .
dx'
Hence also
the Tangent = V ( y* + ^- } = t** ,
V dy' 2 J dx'
and f te formal =: V ( y'* +
\ y
*' being the arc of the curve corresponding with the point x'y'.
ASYMPTOTES.
87. Two curves or a curve and straight line are said to
be asymptotic when they continually approach nearer and
nearer to each other but do not meet at any finite distance.
By an asymptote to a curve we generally understand a
straight line such that if it and the curve be indefinitely con-
tinued they will continually approach each other but never
meet; or it may be considered as a tangent to the curve when
the point of contact is at an infinite distance.
In the equation of the tangent let y = o and we shall find
the intercept of the axis of #, between the origin and the
langent at the point x'y ', to be
3/' d$' _ x' dy' - y' dx'
& if> " - -- - -- .
dy' dy'
Again by taking x = o we similarly find the intercept of the
axis of y, between the prigin and the tangent, to be
dy' ' dx'
If, when x' DC 00 or y' = oo , either of these values of x and y
are finite, the curve has asymptotes which will thence be de-
termined.
When x is Jinite, but y infinite, the asymptote is parallel to
the axis of y.
When y isjinite, but x infinite, the asymptote is parallel to
the axis of a-.
But when the values of x and y are loth of them infinite,
ihe asymtote is at an infinite distance from the origin. In
this case the curve is said to have no asymptote.
When the values of <r and y are both p: o the asymptote
39
passes through the origin and its position must be determined
,/../
from the val ue of - , when x' = oo or y' = x> .
dx'
88. The equation of a curve being given to find whether,
at a particular point, it is convex or concave to the axis of x.
As in article 82, let x'y' denote the ordinates of the pro-
posed point in the curve and u the inclination of the tangent
at that point: with the axis of x, and
du'
J! = tan u.
&*
Now when the curve at the point x'y' is concave to the axis
of x the angle ca will evidently, if x increase, decrease when
y is positive and in crease when y is negative ; and therefore
</, --L will have a sign contrary to that of y. But when the
dx'
curve is convex to the axis of x the inclination of the tangent
will obviously, when x increases, increase or decrease accord-
ingly as y is positive or negative and consequently d. -^L.
dx'
will have the same sign with y'.
Hence, taking dx' constant, the curve at the point x'y' will
be concave towards the axis ofci' when d* y' has a contrary
sign withy'; and it will present a convex side towards the
axis of x when d 2 y' has the same sign withy'. Or, which
amounts to the same, the curve is convex or concave to the
axis of x accordingly as y' d' l y' is positive or negative.
89. Cor. 1. Hence also, by supposing dy' constant, the
curve at the point x'y' will be convex or concave to the axis
of y accordingly asd' 2 x' has the same or a different sign
v ith x' ; of it will be convex or concave towards the axis of y
accordingly as x' d^x' is positive or negative,
90. Cor. 2. When a curve is first convex and becomes
afterwards concave to the axis of x it must have passed a
40
point of contrary flexure -- in this case, supposing 1
dx' constant, rf 2 y' will hence experience a change of sign ;
and the point of contrary flexure will evidently be where
91. The equation of a curve being given to find the radius
of curvature, or the radius of that circle which touches it
most intimately at any given point.
A tangent to any curve may be conceived to be a straight
line drawn through two of its points which are indefinitely
near to each other ; and hence the first differentials of the
ordinates which appertain to the tangent must correspond
with those of the curve at the point of contact.
Similarly may we conceive the osculating circle or the circle
of curvature to be that circle which passes through three suc-
cessive points of the curve which are indefinitely near to each
other ; in this case, therefore, both the first and second differ-
entials of the ordinates which belong to the circle and curve
must correspond at the point of contact.
Let a:" y" be the co-ordinates of the centre of the circle, and
we shall have ac a?", y y" for the two lines drawn from it
respectively parallel to x and y and terminating in the cir-
cumference at the point of contact ; hence, denoting its radius
by , its equation, (25,) is
(* *")+(? y")==i*.
Now since, as has been observed, this circle corresponds with
the curve at two other points contiguous to the point, of con-
tact, we may differentiate twice and consider the first and
second differentials of the ordinates xy as agreeing with
those of the curve. Hence differentiating, observing that x"y'
are invariable, we get
dx (x x ")-{-dy ( y y"} = o,
d\x [x at") -\- d*y (y y <") -\- </s 2 = o ;
where
rf* = dx* + dif,
s being the length of the curve.
41
From these two equations we deduce
_ d x ds*
dx d*
Hence we find
_ dy ds 1 _ _ d x ds
" dyd^x dxd^y^ y ~ dy d*x
x xyY
dtr*
- dy d*x dx d' 2 y
In this expression for the radius of curvature we may assume
an independent variable at pleasure.
92. It may be otherwise very clearly determined by con-
ceiving the centre of the circle of curvature to be the inter-
section of two normals drawn from two points of the curve
which are indefinitely near to each other. Let a normal be
drawn from the point xy and also another normal from a
contiguous point, the intercept of the curve between these
points being ds. Let also two tangents be drawn at these
points, the former making an angle u with the axis of x.
Then the angle IE du included by the tangents will evidently
be the same as that included by the normals; and as the
normals, which are radii of the osculating circle, subtend the
arc ds of the curve, we obviously have
du = ds
ds
du
But
,
and . . $ = 4.
1
tan u :=: -_;
dx
d tan u dx dx
Therefore by substitution
42
93. Since
we have, differentiating
.'. o=(dx d*x-\-dy d*y}* (dsd*sY.
Adding the square of dy d 2 x dx d*y to this, the result is
and hence
ds*
(d V) 2 j
By taking * for the independent variable, this gives
_J ds* _ _
V (d
94. By taking a? for the independent variable, or supposing
dx to'be constant,
^ vr
2 a; dx d
becomes simply
1 ' -de*
y
95. From the equations
__ dy ds* _ dx ds 3
__ fa fl'Zy '
we find the position of the centre of the circle of curvature
to be
*jc dxd*y'
dx ds 2
y z=.y-~- -- ;
dyd*x dxd^y
or, supposing dx constant,
' __ dy ds z
y =
43
By means of these and the equation of the curve, the ordi-
natesan/and their differentials may be eliminated; and an
equation will thence be found expressing the relation between
x" and y", which will hence define the locus of the centre of
the osculating circle for all points in the curve. This locus
is denominated the evolute of the curve ; and, on the contrary,
the curve is called its involute.
96. As the centre of the osculating circle may be conceived
to be the point of intersection of two normals which are in-
definitely near to each other, it is obvious that the normal
at 'any point of the curve must be a tangent to the evolute.
See article 77.
Thus we see that a tangent drawn to the evolute at any
point coincides with the radius of the osculating circle which
is drawn to the point of contact.
97. The equation of this tangent, (62,) gives
dy" (x x) dx (y y") = o.
Let us now differentiate the equation
*'* " 8 = '
supposing x"y" to vary, and we have
(dx dx ") (x x") -|- (dy dy") (y y") == ^ ;
but, x"y" appertaining to the normal of the curve at the point
xy, we have by its equation
dx (x -* a?") + dy(y y") = o,
which rejected and the signs changed, we get
dx' [a- at"} -\-dy" (y y"} = $e/$.
From this and the preceding equation of the tangent to the
volute we find
, dx" , du"
** --^.,y^y =- S </ S .^,
wherein
s" being the arc of the evoJute from any given point.
These values of a? x" and y y" substituted in the equation
44
give
The integral of this gives
*" = , ',
where ' is the radius of curvature corresponding with the
given point from which s" is estimated.
Hence we find the length of the arc of the e volute between
any two points to be equal to the difference between the radii
of the corresponding osculating circles.
98. By means of this principle we discover that the curve
may be described by the unwinding of an inextensible thread
from off the evolute. Thus let p be any point in the curve
and pp" the normal or radius of curvature touching the
evolute at the point />" ; then, this line pp" being conceived
to be a thread extending round the evolute, it is obvious,
from the above property, that by unwinding this thread,
keeping/?/)" always stretched, the point/) will trace out the
curve.
Considering the evolute as a curve, its involute is thus de-
scribed.
99. It appears, from the foregoing, that any given curve
can have but one evol ute, but may have an indefinite number of
involutes as the value ofpp" at any point p" is indeterminate.
Hence, for any particular involute, the value ofpp" must be
known at a given point p".
100. The evolute being given, the equation of its involutes
may be found by means of the values of tf",r/", (95,) in terms
of xy and their differentials.
45
SECTION VII.
FORMULAE APPERTAINING TO POLAR EQUATIONS.
101. Besides the application of co-ordinate axes there is
another method of rendering the relative positions of points
and the properties, &c. of curve lines, in the same plane, sub-
ject to the power of the algebraic analysis, by means of what
is usually called a polar equation.
Thus a given indefinite right line OJ1,
originating at O is denominated the axis ;
the fixed point O is denominated the
pole or origin ; any variable right line
OP drawn to a point P is called a radius
vector, to that point, and its angle of inclination POA, with
the axis, the polar angle.
The radius vector OP we shall denote by r, and the polar
angle POA by <p.
102. The polar equation of a curve is thus expressed
F(r,<f>)=o;
and, in most cases, r may be separated so as to give
F and f denoting given trigonometrical functions.
The characters r, <p thus rendered subject to an equation, we
shall evidently have particular values of r for each successive
value of <p ; and hence the point P will be limited to a par.
ticular curve, determined by the nature of the equation.
103. Particular values of r, <f>, or such as belong to given,
and, of course, invariable points are thus distinguished,
r'<p', r"<p", &c. and the points to which they appertain are
usually called the points r'<p',r"<p", &c.
104. It is often useful to transform expressions, involving
rectangular co-ordinates, into their equivalents in terms of the
H
46
radius vector mid the polar angle and conversly. This may
be effected by means of the two following' propositions.
105. To reduce any expression or formula, involving the
rectangular co-ordinates xy of any point and their differ-
entials, into one involving the radius vector and the polar
angle.
By taking the axis of x for the polar axis, and the origin for
the pole, we shall obviously have
x =. r eos $ t y = r sin q> ;
and heuce also, by differentiation,
dx =. dr cos <p rdty sin @,
dy = dr sin <p -\- rd<p cos <p ;
d~x rrr d 2 r co* <p 2 dr dq> sin <p rcty* cos <p rd' 2 $ sin <>,,
d z y = d' 2 r sin <p -[~ * dr dtp con <p rd$ l sin <p -j- rd-q> cos <p ;
See. &r. &c.
These values substituted in the given expression, will produce
its equivalent in terms of r, <p and their differentials.
By supposing f/<p to l?e con&tunt, the terms wherein d-q> occur
106. If it be required lo have the pole at a given point ^''//',
we may previously t'.ansfer ihe ovigin of the rectangular axes
to that point, by article 43. Or we may substitute
'j; =. r cos- <p -J- JL-', y = r sin <f> -j- y',
and x'y' being constant, the values of dee', d^ d'h', d*y, &c.
as above.
And, if the polar axis be required to make an angle o with
I he axis of a 1 , we must obviously substitute <p -j- &/ instead of i>.'
107. To reduce any expression, involving the radius vector
tiud the polar angle, into one involving rectangular co-ordi~
nates. . .
By taking the polar axis* for the axis of jr, and the pole for the
origin of .ry, we sjiall have, from the foregoing equatipns,
"'
47
also
' :r
cos <Z> r= __ r=
tan $
or
. - ' II
r=m - y- z=z
V (tf + tt*) V(x 2 -ry*) x
wherein cos - - - signifies the arc whose cosine
>
Tlie required tram-formation may be accomplished by the
pubstitution of these values; and the origin may afterwards
be transferred to any given point, (43).
108. The polar equation of a curve beiny yiven to Jind
the lenyth oj' any arc of it.
Ky referring- the points of the curve to rectangular co-ordinate
axes, we have, (HO,)
Hence, Kiibstituting the values of dx, dtj, (105,) we j^e
d*> = df 2 -\- M f
/ dsV (rf^ + r^rf^)
(dr 2 -^- r'W(p-)-f- const.,
(he value of the constant being such as to make the complete
integral vanish at the point whence the arc is estimated.
J09. To Jind the perpendicular from the oriyin on the
tangent at any given point.
The equation of the tangent at any given point X''y' t (82,) is
dj/' (a *'') d.v' (y y') = o,
or, dy'. x dx'. y (.*' dy' y' da;'} =. o.
\#i p be the required perpendicular, and, as in article 84,
we shall hence have
x r dy / y' dx 1 \tx' dy' y'
This reduced for the polar equation, (105,) gives
/> =
ds
1 10. To find the sectoreal area contained between the
curve and any two radii vectores.
Let us imagine two radii vectores infinitely near to each
other, containing the indefinitely small angle </<p, and sub-
tending the element ds of the curve. The sectoreal element
thus formed by these radii veetores and ds may obviously
be considered as a plane triangle ; and the perpendicular
from the origin on the opposite side ds will obviously cor-
respond with that on the tangent. Therefore, p denoting
this perpendicular, we shall hence have the area of this sec-
toral element = f- .
2
That is,
d. (Sectoreal Area) = ^.
But, (109,)
_ a>dy ydx _ r 2 d<p
ds ds
.V d. (Sectoreal Area) =
. = f r ^ + const.
The value of the constant must be determined from the
position of the radius vector from which the area is to be
computed.
111. To express the inclination of the tangent to a curve
with the radius vector.
49
Let the required angle under the tangent and radius vector
be TV, and p the perpendicular on the tangent T. Then is
V cos Tr = - v tan Tr = ... P
r
Therefore, substituting the value oi ' p 009^
sin Tr = r ^ =
ds V (tjn - -j- r'dip 1 )
m dr dr
cos Tr =. =.
ds V (dr* -f i*
dr
Any of these formulae w ill serve for the determination of the
required angle.
112. To express the polar subtangent in terms of r and <p.
The polar subtangent is the straight line drawn from the
pole perpendicular to the radius vector and terminating in
the tangent. Since
~~dr~'
we have
Subtangent = r tan Tr -? .
dr
113. Cor. Sometimes the equation of a curve is expressed
between the radius vector and the perpendicular on (he tan-
gent. In this case we may make use of
tan Tr =. - ^ _ ,
which gives
Subtanqent =. T P .
7 / / o _o\
50
114. To express the radius of curvature of a curve in terms
of the radius vector and the polar angle.
We have, (92,)
___, ds*
Now, pursuing the suppositions used in article 105, we get
dy d^x dx d 2 y =
dr (sin <f> d"*x cos q> d*y) -\- rdq> (cos p d\v -\- sin $ d 2 y)
= dr (2 dr dtp -j- rd 2 <p) -\- rdq> (d*r rdqfi)
__ rf<p (r 2 rf(p 2 -\- 2 dr 2 rdV) rdr d*q>.
Hence
I
"
By supposing if to vary independently
__ I ds 3 _ , (dr
S ~" ~ ~'
115. 7 T Ae equation of a curve between the radius vector
and the corresponding perpendicular on the tangent being
given, to Jind the angle contained by any two radii vectores.
By (109),
V(dr* + r
This solved for eftp gives
the integral of which between the proposed limits will give
the angle sought.
116. Given the equation between the radius vector and
the perpendicular on the tangent, F (*%/>) = > to fi na the
length of any arc of the curve t and also the sectoreal area
between any two radii vectores.
It has been found, (115), that
51
This substituted for </p in (108) and (110), we find
/ rdr
I -
jyflflpf
rdr / rdr
ds =. - , s = I - - 4- const. ;
' ^
and the
Sector eal Area = I - -- -|~
/ 2\/(r 2 p 2 )
117. TV) express the radius of curvature in terms of the
radius vector and the perpendicular on the tangent.
By (109),
"
Let </<p be supposed constant and
d = r</r <fy ( r W -f- 2 rfr a rrf 2 r)
But, (114,)
_ I
-
-\-2dr* rd*i
.'. 5 dp = -j- r</p
_i_ rrfr
118. T^e equation of a curve between the radius vector
and the perpendicular on the tangent being given, to Jind
the similar equation for its evolute.
The radius of curvature of the curve at any point rp coin-
cides with the normal and touches the evolute, (96). Let
R, P be the radius vector and the perpendicular on the
tangent which belong to the evolute at the point of contact.
By drawing the figure it will be at once perceived that/;
and P constitute a rectangle with the tangent and normal
of the curve; also that
which inverted, and the value of P ly the former substituted
in the latter, we get
r 2 p*=P 2 ,
(S />)* + ' 2 /> 2 = S 2
The value of ? being previously determined, (117,) we can
by means of these and the given equation of the curve,
f (r,p) o, eliminate r and p, which will produce the equa-
tion wanted.
119. Let R' and P' be the radius vector and the perpen-
dicular on the tangent which belong to an involute of the
curve ; as the curve is its evolute we have from the foregoing
, ... .. R'dR' f
equations, substituting - _ for ,
UMr
The values of pr by these equations substituted in the equa-
tion of the curve we shall find an equation involving R'P' and
their differentials. If it can be integrated the equation of the
involutes of the curve will thence be found.
END OF PART FIRST.
X
933 6
9
JUL 16 1947
JUN 2 1 19fi
IWY 1 2
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LIBRARY FACILITY
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