A Treatise on Infinitesimal Calculus: Containing Differential and Integral Calculus, Calculus of Variations, Applications to Algebra and Geometry, and Analytical Mechanics, Volumen 1

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The University Press, 1852 - 540 páginas
 

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ON SUCCESSIVE DIFFERENTIATION
80
Expansion of an Explicit Function of One Variable
87
97 The proof that under stated conditions
97
On impossible logarithms
102
Taylors Series
108
67 Limits of Taylors Series
109
Examples of Taylors Series
110
Different forms of the problem
112
Requisite formulae for a function of two variables
113
Examples of transformations
114
Successive Differentiation of Functions of many Independent Variables 72 Explanation of the symbols
117
The order of successive differentiations with respect to many variables is indifferent
118
Application of the principles of the preceding Articles to functions of two and more variables
120
Eulers Theorems of homogeneous functions
123
77 Extension of the preceding principles to other and similar cases
126
Examples of preceding formulae
127
Expansion of one of the variables of an implicit function in terms of the others by means of Maclaurins Theorem
128
Calculations and properties of Bernoullis numbers
130
Lagranges Theorem
133
Extension of Maclaurins Theorem and an explanation of the method of Derivation
141
Elimination of constants from an explicit function
147
Elimination of constants from an implicit function
148
87 Elimination of given functions
151
8891 Elimination of arbitrary functions
152
Transformation of expressions involving partial derived functions into their equivalents in terms of other variables
160
Particular cases of the theorem of Art 89
175
Mode of evaluating and examples of such quantities
190
Asymptotes are also tangents to a curve at an infinite dis
194
The imperfect form of it given in Art 66
196
Expansion of rx + h y + k
204
ON MAXIMA AND MINIMA
211
Geometrical representation of the criteria
213
Maxima and Minima of Implicit Functions
223
Conditions of such singular values of a function of three
232
Mode of generating an evolute and formulae for determin
239
Object of the Chapter stated to be the discussion of
244
If fx has to equal roots fx has to roots equal
251
Des Cartes rule of signs
258
Cases considered of curves involving two and three
260
Corroboration of the preceding modes of interpretation
264
Seeewíy of ттсЬс4 oí recaen
267
Modification of the equation in case the function be
286
18? Explanation of the course of a curve at points where
292
The values of ds and of lines and quantities connected
350
Examples in illustration
356
Curves referred to Rectangular Coordinates
368
Other values of p and of dr
374
ing its equation
377
On the circle of curvature
384
Singular values of the radius of curvature at a point of
390
CONTACT OF CURVES AND ENVELOPES
398
arbitrary constants Examples
402
Conditions under which a circle can have contact of the third order
405
Contact of curves with given curves
406
Theory of Envelopes
408
Explanation of the subject of envelopes families of curves variable parameters
409
Examples of envelopes
411
267 General case of n parameters and n 1 conditions
413
Examples in illustration
414
On Caustics 209 On the formation of caustics
419
General properties of such caustics
424
Particular case of the caustics by reflexion at a circular cylindrical surface
425
Caustic by reflexion on a logarithmic spiral
428
General properties of caustics by refraction
429
Caustic by refraction at a plane surface
431
277 On the equations to a straight line and to a plane
432
The equation to a tangent plane to a curved surface
434
The directioncosines of the tangent plane
435
Modified forms of the equation to the tangent plane when the equation to the surface is a explicit 9 ho mogeneous and algebraical
436
The equations to a normal of a curved surface
438
Examples in illustration of the preceding
439
Singular forms of tangent planes Cones of the se cond and third orders
441
287 On the equations of curves in space
444
Examples of the preceding formulae
450
Ruled surfaces
456
Examples of developable surfaces
469
On Surfaces generated by Circles
475
Mode of measuring absolute curvature angle of contingence
481
Torsion
488
Evolutes of nonplane curve
494
Complex flexure and its measure
500
CURVATURE OF CURVED SURFACES
506
Normal sections of maximum and minimum curvature
512
Singular values of radii of curvature
519
Meuniers theorem of oblique sections
525
On the relation between у and its equivalentт of the equa
533

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Página 289 - Find its equation. Show that the radius of curvature at each point of the curve is inversely proportional to the length of the normal intercepted between the point on the curve and the ?/-axis.
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Página 289 - The Cycloid. The cycloid is traced out by a point in the circumference of a circle as the circle rolls along a straight line.
Página 252 - Find a point within a triangle such that the sum of the square of its distances from the three angular points is a minimum.
Página 423 - From a fixed point on the circumference of a circle chords are drawn, and on these as diameters circles are drawn.
Página 360 - Now the conic section is an ellipse, parabola, or hyperbola according as e is less than, equal to, or greater than unity.
Página 392 - As shown on p. 84 for the cycloid, the arc of the evolute is equal to the difference of the radii of curvature at its end-points.

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