A Treatise on Infinitesimal Calculus: Differential calculus. 1857 (Google eBook)

Front Cover
University Press, 1857 - Calculus of variations
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Contents

Differentiation of functions of many variables all of which
79
CHAPTER III
89
The nth differential of x in terms of successive values
97
7 Imaginary logarithms
113
If in the theorem of the last Article fx x xo then
114
Taylors Series
119
77 Transformations in terms of a new variable
125
137 The theorem for functions of two variables analogous
137
Eulers Theorems of homogeneous functions
138
Calculation and properties of Bernoullis numbers
145
Examples of Lagranges Theorem
153
Expansion of fx in ascending powers of fx
155
Extension to implicit functions
162
Elimination of given functions
169
Transformation of partial differential expressions
180
CHAPTER IV
187
Particular cases of the theorem of Art 114
196
Evaluation of quantities of the forms
202
Mode of evaluating and examples of such quantities
210
The number of given point? through which a curve of
211
The failure of Taylors and Maclaurins Series
217
A particular form of the preceding
224
Expansion of p x + h y + k z +1
230
Method of determining asymptotes by means of expansion
232
Method of determining such singular values of fx
236
Examples of maxima and minima
243
Maxima and minima of implicit functions of
252
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 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 fx fx is divisible by x a
284
The roots of fx are intermediate to those of fx
285
177 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 Cartesrule of signs
295
The application to space of the law of continuity
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
326
nth degree may pass
327
The number of these points which may be on a curve of the mth degree
330
CHAPTER X
331
Definition of and equation to a tangent
332
Modification of the equation in case the function is a implicit 0 homogeneous
333
Equation to the tangent when the equation to the curve is homogeneous and of three variables
334
Values of ds and of sin t cos r sin cos il
336
Discussion of the equations to the tangent and the normal and of lines and quantities connected with them
338
Explanation of the course of a curve at points where 0 and 00
340
Examples illustrative of the preceding articles
341
On rectilinear asymptotes
358
Examples of the method
359
Direction of curvature and points of inflexion
367
Criterion of points of inflexion when the equation of
373
At a multiple point jj double points and their
379
The number of double points of a curve of the nth degree
385
On tracing curves by means of their equations
393
Interpretation of and 6 when affected with negative signs
411
Other values of p
417
Direction of curvature and points of inflexion
423
CURVATURE OF PLANE CURVES
432
Value of the radius of curvature when the equation is
439
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
The theory of envelopes
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 3 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 formula?
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 55
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
417 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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