A treatise on infinitesimal calculus, Volume 1

Front Cover
University Press, 1852 - Calculus
 

Contents

On functions of many variables
69
ON SUCCESSIVE DIFFERENTIATION
80
Maclaurins Theorem
87
Extension of preceding principles to the consideration
94
On impossible logarithms
103
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
If in the theorem of the last Article ƒ x xx then
172
Definition of order of infinitesimals
180
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 Fx+h y + k
204
Definition of maximum and minimum
210
Geometrical representation of the criteria
213
The values of ds and of lines and quantities connected
220
Maxima and Minima of Implicit Functions
223
Asymptotic circles
226
Conditions of such singular values of a function of three
232
Mode of generating an evolute and formulæ for determin
239
ON SOME QUESTIONS OF PURE ALGEBRA
244
PART II
260
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
Tan x x and sin x are equal when x is infinitesimal
296
On Asymptotes to Plane Curves referred
299
On Multiple Points
315
An explicit function is explained which well exhibits some
321
Examples illustrative of the preceding principles 163
331
Interpretation of r and 0 when affected with negative signs
345
Explanation of curvature definition of curvature of a circle
368
Other values of p and of
374
Examples of evolutes
379
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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