THE ELEMENTS OF ANALYTICAL GEOMETRY; COMPREHENDING THE DOCTORINE OF THE CONIC SECTIONS AND THE GENERAL THEORY OF CURVES AND SURFCES OF THE SECOND ORDER

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Contents

Modifications to be introduced when the axis of x is below
117
Other modes of expressing the formulas of transformation
118
Inferences from this expression
124
The axes of an ellipse and the vertex of any diameter being
130
Other properties unfolded by the same equation
136
section will be a circle
140
On the hyperbola and mode of describing it
141
Determination of its equation
142
Determination of its vertices and asymptotes
143
Different forms of the equation of the hyperbola
144
tional to the transverse and second axes
145
Expression for the radius vector
146
Transformation of the equation of the hyperbola from rectangular to oblique conjugates
147
Properties deduced analogous to those in the ellipse
148
The same property true for any system of supplemental chords
149
The axis and vertex of a diameter being given to find the length of that diameter and of its conjugate
150
Situation of a point fixed by the signs of its coordinates
151
The equations and lengths of these lines
152
Properties analogous to those in the ellipse 153
153
Equation of the hyperbola when referred to its asymptotes
155
Properties of lines drawn between the asymptotes
156
To construct the curve when a point in it and the asymptotes are given
157
Its equation and vertex determined
158
Equation in terms of the parameter
159
Transformation of the equation
160
Equations of the tangent normal c the subtangeht double the abscissa and the subnormal constant
161
Properties of the focal tangent
162
To find the locus of the intersection of pairs of rectangular tangents
163
On polar coordinates
164
Polar equation of the ellipse when the focus is pole
165
Article Page 112 Polar equation when the centre is the pole
166
Determination of the polar subtangent in each curve
167
Properties of focal chords
168
SECTION IV
169
determination of the locus of the equation M zb N zb P
170
_ _i_ ys Qaf
171
Mode of obtaining the properties of the parabola from those of the ellipse
172
Examination of the equation when some of its terms are absent
174
Criteria for determining the nature of the curve represented by any equation of the second degree
175
Discussion of the equation when B4ACA0
186
B24ACV0
188
Construction of asymptotes
189
Discussion of the equation when B24AC0
190
Examples on the preceding discussion
191
Conditions which exist when the locus meets the axes of coor dinates
195
Given the base and difference of the tangents to find the locus of the vertex
196
Article Page
197
From two given points two straight lines are drawn so as to inter
203
If through any point chords are drawn to a line of the second
211
General Scholium with remarks on the higher curves
218
Equations of a straight line in space
224
Equation of the plane
230
Article Page 191 To draw a perpendicular from a given point to a plane and to determine its length
234
To determine the inclination of a straight line to a plane
235
The locus of every equation of the first degree containing three variables must be a plane
236
SECTION
237
Equation of a tangent plane
238
Conical surfaces described
239
Equation determined
240
Surface of revolution defined
241
an ellipse
242
Surfaces of the second order in general
243
Diametral planes of surfaces without a centre
244
The hyperboloid of a single sheet
245
The hyperboloid of two sheets
248
The hyperbolic paraboloid
249
Tangent planes to surfaces of the second order
251
On conjugate diametral planes
253
SECTION III
254
The square of a surface equal to the sum of the squares of its projections on three rectangular planes
255
If any number of areas be projected on different systems of rect
257
Article Page
260
Case in which the formulas become simplified
263
Final reduction of the equation
269
Position of a plane determined so that if a given triangle be pro
276
Three rectangular planes touch a central surface of the second
280
PROBLEM XXIV
286

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Page 50 - In a triangle, having given the ratio of the two sides, together with both the segments of the base, made by a perpendicular from the vertical angle, to determine the sides of the triangle.
Page 108 - In the first case, there must obviously subsist the conditions r -|- d~7 r, r -f- r' -7 d, r -|- d! ~7 r, which prove that if two circumferences cut, the distance of their centres must be less than the sum, and greater than the difference of the radii. In the second case where y becomes imaginary because of a negative factor, we must have one of the conditions d...
Page 201 - Given the base, an angle adjacent to the base, and the difference of the sides of a triangle, to construct it.
Page 30 - Having given the side of a regular decagon inscribed in a circle whose radius is known, to find the side of a regular pentagon inscribed in the same circle.
Page 163 - FP • FA = FR • FA. From this property we may derive an easy method of drawing a tangent to a parabola from a. point either within or without the curve. Thus, let P be a point either within or without the curve, through which it is required to draw a tangent. Draw PF, upon which describe a semicircle...
Page 196 - Given the base of a triangle and the difference of the angles at the base, to determine the locus of the vertex. Taking the same axes as before, and putting a, a', for the tangents of the angles at the base...
Page 153 - ... does. From the general expression for the subtangent just given, it follows that T,x' = (x' + A') (x' — , A'), that is, as in the ellipse, the rectangle of the \ subtangent and abscissa of the point of contact is equal to the rectangle of the sum and difference of, the same abscissa and semi-transverse axis Thus OM • MR = A'M • MB'.
Page 120 - From these expressions for y and x, it appears that for the same value of x, there are two values of y numerically equal, but having contrary signs ; hence the chord AB bisects all the chords drawn parallel to CD. In like manner with regard to x ; this also has two values numerically equal, but differing in sign for one value of y; therefore, the chord CD bisects all the chords drawn parallel to AB. Moreover, since, for x = A, or x — — A, the corresponding value of y is 0, it follows that parallels...
Page 163 - D' is a right angle, and the angle DPF = DPD', and PF = PD', .-. DFP is a right angle. In like manner, DFP' is a right angle ; hence, first, the part of the tangent intercepted between the point of contact and the directrix, subtends a right angle at the focus ; second, the line joining the points of contact of perpendicular tangents always passes through thefocut.
Page 37 - In an isosceles triangle, the square of a line drawn from the vertex to any point in the base, together with the rectangle of the segments of the base, is equal to the square of one of the equal sides of the triangle.

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