A Treatise on Infinitesimal Calculus: Differential calculus. 1857

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University Press, 1857 - Calculus of variations - 26 pages

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Contents

CHAPTER III
89
Limits of Maclaurins Theorem
99
CO63 Certain trigonometrical expressions and theorems
103
Imaginary logarithms
113
If in the theorem of the last Article Jx x xi then
114
Taylors Series
119
77 Transformations in terms of a new variable
125
Expansion of fx in ascending powers of fx
135
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
Extension to implicit functions
162
Elimination of given functions
169
Transformation of partial differential expressions
180
CHAPTER IV
187
CHAPTER V
198
127 Evaluation of quantities of the form 00 oo
209
The number of given points through which a curve of
211
The imperfect form of Maclaurins Theorem given in Art 57
215
A particular form of the preceding
224
Expansion of f x+h y + A z +I
230
Method of determiningasymptotes 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
256 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 offunctions 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 squareB
266
Examples of the method of least squares
270
Maxima and minima of functions when alt 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 offx fx is divisible by aa
284
The roots of fx are intermediate to those of fx
285
177 If fx has m equal roots f x 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
18fi The definitions of some geometrical terms founded on
298
Contact depends on the number of consecutive points which
304
Interpretation of + and of + 4
305
On the generation of some plane curves of higher orders
311
207 Various forms and the number of terms of an algebraical
321
Particular case of the caustics by reflexion at a spherical
326
The number of these points which maybe on a curve of
330
Values of as and of sin r cos t sin yfr cos
338
General properties of the tangent of a curve of the
344
The equation of the pencil of tangents
350
Other values of p
417
Direction of curvature and points of inflexion
423
Explanation of curvature definition of curvature of a circle
432
28 The number of points in a curve at which the curvature
440
On the circle of curvature
448
297 The order and the class of the evolute singular properties
454
Two curves which have a common tangent intersect or
464
Explanation of the subject of envelopes families of curves
471
The theory of reciprocation
480
Properties of reciprocal polars
486
surface
493
327 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 j8 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
347 Examples of the preceding formulae
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
397 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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INFINITESIMAL
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