The Elements of Analytical Geometry: Comprehending the Doctrine of the Conic Sections, and the General Theory of Curves and Surfaces of the Second Order

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E. H. Butler, 1848 - 288 من الصفحات
 

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Miscellaneous problems
25
If pairs of tangents intersect at right angles the locus of the inter
68
Construction of some simple algebraical expressions
81
Of rendering algebraical expressions homogeneous
82
Construction of irrational expressions
83
On Analytical Geometry e
85
Situation of a point fixed by the signs of its coordinates
86
When the axes of coordinates are rectangular
87
When they are oblique
88
Equation of a straight line passing through a given poin
90
Equation of a straight line through two given points
91
To find the expression for the angle of intersection of two straight
92
To determine whether lines from the vertices bisecting the opposite
98
To draw a tangent from a point without the circle
104
The base of a triangle and the sum of the squares of the sides
110
Modifications to be introduced when the axis of a is below
117
Afrticle Page
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
On the hyperbola and mode of describing it
141
Determination of its equation
142
Expression for the distance of any point from the centre
143
Different forms of the equation of the hyperbola
144
Squares of the ordinates as the products of the parts into which they divide the transverse diameter parameter a third propor tional to the transverse a...
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
On tangents normals c to the hyperbola
151
The equations and lengths of these lines
152
Properties analogousto those in the ellipse 153
153
Equation of the hyperbola when referred to its asymptotes
155
On polar coordinates
164
Polar equation of the ellipse when the focus is pole
165
Polar equation when the centre is the pole
166
Determination of the polar subtangent in each curve
167
Properties of focal chords
168
General examination of indeterminate equations of the Second Degree 118 Preliminary observations
169
determination of the locus of the equation My + Na + P
170
ā y Qa
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
Means of constructing the equation
176
Table of formulas to be employed in constructing central curves
177
Examples of their application
178
Means of judging when the equation represents a system of parallels
180
Determination of formulas for constructing parabolas
182
The preceding formulas apply only when the locus is referred to rectangular axes
183
Examination of the irrational part of the expression for either variable
185
Bº4ACN 0
187
137a y are absent
188
Construction of asymptotes
189
Discussion of the equation when B4AC0
190
Examples on the preceding discussion
191
Conditions which exist when the locus meets the axes of coor dinates
193
Given the base of a triangle and the sum of the tangents of the base angles to find the locus of the vertex
195
Given the base and difference of the tangents to find the locus of the vertex
196
To find the locus of a given point in a straight line of given length
197
If through any point chords are drawn to a line of the second
211
To determine a curve which shall pass through any proposed
214
PART II
221
Conditions of intersection of two straight lines
228
211
243
221
253
231
260

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الصفحة 67 - 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.
الصفحة 160 - In article (100), it was found that the principal parameter is equal to 4m, the coefficient of x, in the equation of the curve, when referred to rectangular conjugates. By analogy, the coefficient 4r, when any system of conjugates are employed, is called the parameter of that diameter, which is taken for the axis of x, so that generally the parameter of any diameter is equal to four times the distance of its vertex from the focus. By equation (5) the parameter is always equal to the double ordinate,...
الصفحة 30 - To find the side of a regular octagon inscribed in a circle whose radius is known.
الصفحة 200 - Given the base and the sum of the sides of a triangle to find the locus of the centre of the circle touching the base, and the prolongation of the other two sides.
الصفحة 62 - Given the hypothenuse (10) of a right angled triangle, and the difference of two lines drawn from its extremities to the centre of the inscribed circle (2), to determine the base and perpendicular. Ans. 8.08004 and 5.87447 PROBLEM XIX.
الصفحة 104 - To draw a tangent to a circle from a given point without it. Let (a...
الصفحة 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.
الصفحة 201 - Given the base, an angle adjacent to the base, and the difference of the sides of a triangle, to construct it.
الصفحة 120 - From these expressions for y and z, 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.
الصفحة 142 - A, which intimates that the hyperbola, like the ellipse, intersects the axis of x in two points, B, A, equidistant from O, the one to the right, and the other to the left, and that A expresses this distance. If, in the same equation, we suppose x = 0, we have for the corresponding value of y the expression y = V JA2 — c2^.

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