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edition:
Authors: Etienne Bezout
serie:
ISBN : 0691114323, 9780691114323
publisher: Princeton University Press
publish year: 2006
pages: 362
language: English
ebook format : PDF (It will be converted to PDF, EPUB OR AZW3 if requested by the user)
file size: 2 MB
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price : $9.57 11 With 13% OFF

Cover......Page 1

Title Page......Page 2

Table of Contents......Page 6

Translator’s Foreword......Page 12

Dedication from the 1779 edition......Page 14

Preface to the 1779 edition......Page 16

Definitions and preliminary notions......Page 26

About the way to determine the differences of quantities......Page 28

A general and fundamental remark......Page 32

Reductions that may apply to the general rule to differentiate quantities when several differentiations must be made.......Page 33

Remarks about the differences of decreasing quantities......Page 34

About sums of quantities......Page 35

Remarks......Page 36

About sums of rational quantities with no variable divider......Page 37

About complete polynomials and complete equations......Page 40

Problem I: Compute the value of N(u . . . n)T......Page 41

Problem II......Page 42

Problem III......Page 44

Remark......Page 45

Initial considerations about computing the degree of the final equation resulting from an arbitrary number of complete equations with the same number of unknown......Page 46

Determination of the degree of the final equation resulting from an arbitrary number of complete equations containing the same number of unknowns......Page 47

Remarks......Page 49

About incomplete polynomials and first-order incomplete equations......Page 51

Problem IV......Page 53

Problem V......Page 54

Problem VII: We ask for the degree of the final equation resulting from an arbitrary number n of equations of the form (ua . . . n)t = 0 in the same number of unknowns......Page 57

Remark......Page 59

Problem VIII......Page 60

Problem X......Page 61

Problem XI......Page 62

(3) The other unknowns do not exceed a given degree (different or the same for each), but, when combined groups of two or three among themselves as well as with the first two, they reach all possible dimensions until that of the polynomial or the equation......Page 63

Problem XII......Page 64

Problem XIII......Page 65

Problem XIV......Page 66

Problem XVI......Page 67

We further assume that the degrees of the n 3 other unknowns do not exceed given values we also assume that the combination of two, three, four, etc. of these variables among themselves or with the first three reaches all possible dimensions, up to the......Page 70

Problem XVII......Page 71

Problem XVIII......Page 72

Summary and table of the different values of the number of terms sought in the preceding polynomial and in related quantities......Page 81

Problem XIX......Page 86

Problem XX......Page 87

Problem XXII......Page 88

About the largest number of terms that can be cancelled in a given polynomial by using a given number of equations, without introducing new terms......Page 90

Determination of the symptoms indicating which value of the degree of the final equation must be chosen or rejected, among the different available expressions......Page 94

Expansion of the various values of the degree of the final equation, resulting from the general expression found in (104), and expansion of the set of conditions that justify these values......Page 95

Application of the preceding theory to equations in three unknowns......Page 96

General considerations about the degree of the final equation, when considering the other incomplete equations similar to those considered up until now......Page 110

Problem XXIII......Page 111

General method to determine the degree of the final equationfor all cases of equations of the form (ua . . . n)t = 0......Page 119

General considerations about the number of terms of other polynomials that are similar to those we have examined......Page 126

Conclusion about first-order incomplete equations......Page 137

About incomplete polynomials and second-, third-, fourth-, etc. order incomplete equations......Page 140

Problem XXIV......Page 143

About the form of the polynomial multiplier and of the polynomials whose number of terms impact the degree of the final equation resulting from a given number of incomplete equations with arbitrary order......Page 144

Useful notions for the reduction of differentials that enter in the expression of the number of terms of a polynomial with arbitrary order......Page 146

Problem XXV......Page 147

Table of all possible values of the degree of the final equations for all possible cases of incomplete, second-order equations in two unknowns......Page 152

Conclusion about incomplete equations of arbitrary order......Page 159

General observations......Page 162

A new elimination method for first-order equations with an arbitrary number of unknowns......Page 163

General rule to compute the values of the unknowns, altogether or separately, in first-order equations, whether these equations are symbolic or numerical......Page 164

A method to find functions of an arbitrary number of unknowns which are identically zero......Page 170

About the form of the polynomial multiplier, or the polynomial multipliers, leading to the final equation......Page 176

About the requirement not to use all coefficients of the polynomial multipliers toward elimination......Page 178

About the number of coefficients in each polynomial multiplier which are useful for the purpose of elimination......Page 180

About the terms that may or must be excluded in each polynomial multiplier......Page 181

About the best use that can be made of the coefficients of the terms that may be cancelled in each polynomial multiplier......Page 183

Other applications of the methods presented in this book for the General Theory of Equations......Page 185

Useful considerations to considerably shorten the computation of the coefficients useful for elimination.......Page 188

Applications of previous considerations to different examples interpretation and usage of various factors that are encountered in the computation of the coefficients in the final equation......Page 199

General remarks about the symptoms indicating the possibility of lowering the degree of the final equation, and about the way to determine these symptoms......Page 216

About means to considerably reduce the number of coefficients used for elimination. Resulting simplifications in the polynomial multipliers......Page 221

More applications, etc.......Page 230

About the care to be exercised when using simpler polynomial multipliers than their general form (231 and following), when dealing with incomplete equations......Page 234

More applications, etc.......Page 238

About equations where the number of unknowns is lower by one unit than the number of these equations. A fast process to find the final equation resulting from an arbitrary number of equations with the same number of unknowns......Page 246

About polynomial multipliers that are appropriate for elimination using this second method......Page 248

Details of the method......Page 250

First general example......Page 251

Second general example......Page 253

Third general example......Page 259

Fourth general example......Page 262

Observation......Page 266

Considerations about the factor in the final equation obtained by using the second method......Page 276

About the means to recognize which coefficients in the proposed equations can appear in the factor of the apparent final equation......Page 278

Determining the factor of the final equation: How to interpret its meaning......Page 294

About the factor that arises when going from the general final equation to final equations of lower degrees......Page 295

Determination of the factor mentioned above......Page 299

About equations where the number of unknowns is less than the number of equations by two units......Page 301

Form of the simplest polynomial multipliers used to reach the two condition equations resulting from n equations in n 2 unknowns......Page 303

About a much broader use of the arbitrary coefficients and their usefulness to reach the condition equations with lowest literal dimension......Page 326

About systems of n equations in p unknowns, where p < n......Page 332

When not all proposed equations are necessary to obtain the condition equation with lowest literal dimension......Page 339

About the way to find, given a set of equations, whether some of them necessarily follow from the others......Page 341

About equations that only partially follow from the others......Page 343

Reflexions on the successive elimination method......Page 344

About equations whose form is arbitrary, regular or irregular.Determination of the degree of the final equation in all cases......Page 345

Remark......Page 352

Follow-up on the same subject......Page 353

About equations whose number is smaller than the number of unknowns they contain. New observations about the factors of the final equation......Page 358

First 10 Pages Of the book