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

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Clarondon Press, 1852
 

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Fundamental theorems on infinities and infinitesimals
9
Examples on the above theorems
10
On the relation of the finite to the infinite and the infinitesimal
11
On functions on dependent and independent variables
12
1315 Functions are implicit and explicit algebraical and tran
13
scendental simple and compound continuous and dis continuous
15
On the generation of continuous quantity
16
On the particular mode of generating continuous number as it is considered in the Differential Calculus
17
On derivation and derivedfunctions
18
Description of the Differential Calculus 1
20
Evaluation of 1 + x when x is infinitesimal
21
Tan x x and sin are equal when is infinitesimal
22
versin x 2 when a is infinitesimal
23
Differentiation of
30
Differentiation of a product of many functions
37
On functions of many variables
69
ON SUCCESSIVE DIFFERENTIATION
80
Expansion of an Explicit Function of One Variable
87
On impossible logarithms
103
The cause of quantities assuming the forms
106
Taylors Series
108
On the relation between y and its equivalent fx of the equa
109
Examples of Taylors Series
110
Different forms of the problem
112
Requisite formulæ 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
Extension of the preceding principles to other and similar cases
126
Examples of preceding formulæ
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
Laplaces Theorem
140
Extension of Maclaurins Theorem and an explanation of the method of Derivation
147
Elimination of constants from an implicit function
148
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
Examples illustrative of the preceding principles
163
If in the theorem of the last Article fx xx then
172
Definition of order of infinitesimals
178
Evaluation of quantities of the form 0 x
181
Asymptotes are also tangents to a curve at an infinite dis tance
194
Examples of applying the method
195
The imperfect form of it given in Art 66
196
Curvilinear asymptotes
197
Direct proof that a curve is convex or concave downwards
198
SECTION 2
211
Geometrical representation of the criteria
213
d2y
215
The values of ds and of lines and quantities connected
220
Maxima and Minima of Implicit Functions
223
according as is positive or negative
224
Asymptotic circles
226
Conditions of such singular values of a function of three
232
Examples illustrative of the preceding methods
238
Mode of generating an evolute and formulæ for determin
239
ON SOME QUESTIONS OF PURE ALGEBRA
244
If fx has m equal roots fx has m 1 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
Necessity of symbols of direction
267
On the Generation of some Plane Curves of higher orders
274
ON PLANE CURVES REFERRED TO RECTANGULAR COORDINATES
285
Discussion of the equations to the tangent and the normal
291
Illustrative examples on differentiation
296
On Asymptotes to Plane Curves referred
299
Method of determining asymptotes by means of expansion
307
Examples in illustration
310
Form of the criterion when the equation to the curve is an implicit function
312
9
333
12
343
Interpretation of r and when affected with negative signs
345
217219 Equations in terms of r and p of the involute of the circle the circle and the epicycloid
347
18
351
24
352
25
353
26
354
Explanation of curvature definition of curvature of a circle
368
Other values of p and of
374
29
376
Examples of evolutes
379
The normal to the curve passes through two consecutive
385
On the chord of curvature its definition and value
394
Two curves which have a common tangent intersect or
401
arbitrary constants Examples
402
Conditions under which a circle can have contact of the third order
405
32
406
35
407
37
408
Form of the problem when only one parameter is involved
409
Examples of envelopes
411
General case of n parameters and n1 conditions
413
Examples in illustration
414
On Caustics 269 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
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 8 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
On the equations of curves in space
444
Examples of the preceding formulæ
450
Ruled surfaces
456
Examples of developable surfaces
469
On Surfaces generated by Circles
475
CURVATURE OF CURVES IN SPACE 323 Mode of measuring absolute curvature angle of contingence
481
Mode of measuring torsion radius of torsion
482
Radius of absolute curvature
483
Angle of curvature
486
Geometrical illustrations
487
Torsion
488
Singular values of curvature and torsion
490
Equation to the polar surface
491
The polar line and locus of polar lines
492
The osculating sphere
493
Evolutes of nonplane curve
494
Geometrical illustrations
497
Complex flexure and its measure
500
The osculating surface 201
501
Application to the helix
503
CURVATURE OF CURVED SURFACES 345 Normal sections
506
Curvature of principal normal sections
507
Perpendicularity of normal sections
511
Normal sections of maximum and minimum curvature
512
Eulers theorem of the curvature of normal sections
513
Application to the ellipsoid
515
Singular values of radii of curvature
516
Umbilics
519
Lines of curvature
520
Locussurface of centres of principal curvature
521
Modification of the conditions when the equation is explicit
523
Meuniers theorem of oblique sections
525
Explanation of properties by means of the indicatrix
526
Osculating surfaces
529
Examples of the method 301
532
Symbolical form of Taylors Series
538

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Términos y frases comunes

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Información bibliográfica

TítuloA Treatise on Infinitesimal Calculus: Containing Differential and Integral Calculus, Calculus of Variations; Applications to Algebra and Geometry, and Analytical Mechanics. I, Volumen 1
AutorBartholomew Price
EditorClarondon Press, 1852
Procedencia del originalBiblioteca Estatal de Baviera
Digitalizado26 Nov 2008
  
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