A Treatise on Infinitesimal Calculus: Differential calculus. 1857

Front Cover
University Press, 1857 - Calculus
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Contents

Fundamental theorems on infinities and infinitesimals
22
The relation of the finite to the infinite and the infinitesimal 12 Functions on dependent and independent variables 1315 Functions are implicit and e...
25
The generation of continuous quantity
29
Derivation and derivedfunctions considered in the Differential Calculus
32
Description of the Differential Calculus
34
CHAPTER II
51
Differentiation of a product of many functions
60
The differentiation of functions of many variables
70
CHAPTER III
77
The nth differential of f x in terms of successive values
96
Imaginary logarithms
113
Certain corollaries of the theorem of Art
114
Taylors Series
119
Examples wherein orders of infinitesimals are determined
120
Transformations in terms of a new variable
125
The order of successive differentiations with respect to many
131
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
The formation of differential equations by means
165
Transformation of partial differential expressions
180
Evaluation of quantities of the forms
184
Evaluation of 1+x when x is infinitesimal 22 Tan a a and sin are equal when a is infinitesimal 23 2 versin a r when a is infinitesimal Page
186
11
193
15
194
18
195
23
196
Evaluation of quantities of the form
202
The number of given points through which a curve of
211
CHAPTER VI
212
25
217
that of Art 11I
222
Fundamental Lemmas of the Infinitesimal Calculus
225
Expansion of x+h y+k
228
Method of determining asymptotes by means of expansion
232
CHAPTER VII
235
27
237
Direct proof that a curve is convex or concave downwards
240
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
31
259
Examples of the process
260
A consideration of a case wherein the requisite conditions are not fulfilled
262
34
263
37
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 ƒx are intermediate to those of ƒx
285
If fx has m equal roots fx has m 1 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
Necessity of symbols of direction
303
191
307
The normal is the longest or shortest line that can be drawn
356
If the curve is of the nth degree the number of points
374
Examples of double points
381
The number of double points of a curve of the nth degree
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
Explanation of curvature definition of curvature of a circle
432
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 OF CURVES AND ENVELOPES
461
Conditions under which a circle can have contact of
468
General case of n parameters and n 1 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
2
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
384
555
Evolutes of nonplane curves
559
Double points and the Hessian when the equation of
561
The osculating surface
565
Perpendicularity of 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

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