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

Portada
Clarondon Press, 1852
 

Comentarios de usuarios - Escribir una reseña

No hemos encontrado ninguna reseña en los sitios habituales.

Páginas seleccionadas

Índice

The proof that
96
On impossible logarithms
102
Taylors Series
108
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
Laplaces Theorem
140
0 X oo oo
143
0
146
Extension of Maclaurins Theorem and an explanation of the method of Derivation
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
Certain Corollaries of the theorem of Art 99
174
The cause of quantities assuming the forms j and 55
181
Evaluation ofquantities of the forms 0 oo l00 0
190
Evaluation of indeterminate forms of Functions of Two Variables 114 Examples of such evaluations
191
On Functions of One Variable 115 An accurate proof of Taylors Series
194
The imperfect form of it given in Art 66
196
Deduction of Maclaurins Theorem from Taylors
197
On the limits of Taylors and Maclaurins Series
198
On the failure of Taylors and Maclaurins Series
200
Expansion of p x + A y + k
204
Expansion of f + A y + z + I
208
Expansion of rx y in ascending powers of x and y
209
ON MAXIMA AND MINIMA
211
Geometrical representation of the criteria
213
The values of ds and of lines and quantities connected
220
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
ON SOME QUESTIONS OF PURE ALGEBRA
244
Sturms Theorem
252
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
Definition of and equation to a tangent
285
Discussion of the equations to the tangent and the normal
291
Illustrative examples on differentiation 38
296
On Asymptotes to Plane Curves referred
299
Examples of applying the method
303
Another proof of the same theorem by means of an expansion
309
On Multiple Points
315
An explicit function is explained which well exhibits some
321
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 1 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
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 0 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
Mode of measuring torsion radius of torsion
482
r 326 Radius of absolute curvature
483
Angle of curvature
486
Geometrical illustrations
487
Torsion
488
Singular values of curvature and torsion
490
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
506
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

Otras ediciones - Ver todo

Términos y frases comunes

Información bibliográfica