A Treatise on Infinitesimal Calculus: Differential calculus. 1857

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University Press, 1857 - Calculus

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Contents

Fundamental theorems on infinities and infinitesimals
9
Examples on the above theorems
10
The relation of the finite to the infinite and the infinitesimal
11
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
The generation of continuous quantity Page 7 8 9 11 12 15
16
Differentiation of
30
The particular mode of generating continuous number as it is considered in the Differential Calculus
31
Derivation and derivedfunctions
32
Description of the Differential Calculus SECTION 2 Fundamental Lemmas of the Infinitesimal Calculus
34
Differentiation of a product of many functions
37
32
57
34
58
Functions of many variables
71
Differentiation of functions of many variables all of which
85
Limits of Maclaurins Theorem
99
Imaginary logarithms
113
Taylors Series
119
Transformations in terms of a new variable
125
Eulers Theorems of homogeneous functions
137
Expansion of one of the variables of an implicit function
143
Lagranges Theorem
149
Examples of Lagranges Theorem
156
more variables
172
Transformation of partial differential expressions
180
CHAPTER V
198
CHAPTER VI
212
A particular form of the preceding
224
Expansion of F x+h y+k z + l
230
Method of determining asymptotes by means of expansion
232
Maxima and minima of explicit functions of one variable
236
Examples of maxima and minima
243
Maxima and minima of implicit functions of
250
Quadruple points 392
257
Application of the method to total minima
258
The sufficiency of the process
259
Examples of the process
260
A consideration of a case wherein the requisite conditions are not fulfilled
262
Maxima and minima of functions of three and more independent variables 163 Conditions of such singular values of a function of three independent...
263
The requisite conditions in the most general case
264
The method of least squares
266
Examples of the method of least squares
270
Maxima and minima of functions when all the variables are not independent 167 Investigation of the most general case of many variables
271
Discussion of the case of two variables which are connected by a given equation
273
Examples illustrative of the preceding methods
274
CHAPTER VIII
279
The continuity of algebraical expressions
280
Proof that every equation has a root
282
If a is a root of ƒ x ƒx is divisible by xa
284
The roots of fx are intermediate to those of ƒ x
285
If fx has m equal roots fx has m1 roots equal to them
287
Sturms Theorem
288
Examples in which Sturms Theorem is applied
291
The criteria of the number of impossible roots of an equation
292
Fouriers Theorem
293
Des Cartes rule of signs
295
GEOMETRICAL APPLICATIONS
297
fxo + h fx0
301
Necessity of symbols of direction
303
191
307
On the generation of some plane curves of higher orders
311
Various forms and the number of terms of an algebraical
321
Particular case of the caustics by reflexion at a spherical
326
Criterion of points of inflexion when the equation of
373
Examples of double points
381
The relation of a curve to its Hessian
388
On tracing curves by means of their equations
393
CHAPTER XI
411
Other values of p
417
Direction of curvature and points of inflexion
423
Definition of radius of curvature and of circle of curvature
433
Value of the radius of curvature when the equation is
439
On the circle of curvature
448
The order and the class of the evolute singular properties
454
Contact depends on the number of consecutive points which
461
Conditions under which a circle can have contact of
468
General case of n parameters and n1 conditions
475
First polar envelope
481
General properties of such caustics
491
surface
493
Caustic by reflexion on a logarithmic spiral
494
General properties of caustics by refraction
495
All caustics are rectifiable
496
Caustic by refraction at a plane surface
497
CHAPTER XIV
498
The equation to a tangent plane to a curved surface
500
The directioncosines of the tangentplane
501
Modified forms of the equation to the tangent plane when the equation to the surface is a explicit B homo geneous and algebraical
502
The equations to a normal of a curved surface
503
The equations to a perpendicular through the origin on a tangent plane
504
Singular forms of tangent planes Cones of the second and third orders
506
CHAPTER XV
509
The equation to the normal plane
511
The equations to the binormal
513
Examples of the preceding formulæ
514
The distinguishing criterion of plane and nonplane curves
516
CHAPTER XVI
518
Ruled surfaces
520
Developable surfaces
521
Examples of developable surfaces
534
CHAPTER XVII
547
Torsion
553
Evolutes of nonplane curves
559
The osculating surface
565
Normal sections
571
Curvature of any normal section
575
Normal sections of maximum and minimum curvature
576
Application to the ellipsoid
579
Umbilics
582
Lines of curvature
584
The Theorem of Dupin
586
Three confocal surfaces of the second order
589
Modification of the conditions when the equation is explicit
591
Meuniers theorem of oblique sections
593
Explanation of properties by means of the indicatrix
594
Osculating surfaces
597
Measure of curvature
598
CHAPTER XIX
600
The laws of commutation distribution and iteration
601
The extension of the same to algebraical functions
603
The law of total differentiation
604
Three fundamental theorems
606
Illustrative examples
607
Leibnitzs Theorem and particular forms
608
Another form of Leibnitzs Theorem
609
Extension of Eulers Theorem
610
INFINITESIMAL

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