A Treatise on Infinitesimal Calculus ...

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Contents

A general definite integral expressed in its complete form
91
Examples of definite integrals determined by indefinite
98
Cauchys principal value of a definite integral
101
The correction for infinity and discontinuity
105
Application of the theorem to particular forms of element functions
106
Simplification of definite integrals by means of transformation
107
The values of the preceding when the limits are changed
108
Another form of function treated on Cauchys method
109
Remarks on the method SECTION 6 Methods of Approximating to the value of a Definite Integral
110
Evaluation by direct summation
111
Evaluation by summation of terms at finite intervals
112
Geometrical interpretation of the process
113
Application of the process to Mensuration
114
Integral of a definite integral with respect to a parameter
115
Definite Integrals involving Impossible Quantities
121
142
126
147
128
гn+12 +e when ʼn 174
138
Approximate values given by the forms of definite integrals
147
Bernoullis series for approximate value 118 Approximate value deduced from Taylors series 119 The theory of integration by series 120 121 Maclau...
150
244
154
Definition of the gammafunction various forms of it 123 The gammafunction is determinate and continuous 124 Particular values of the gammafunc...
157
Equivalent forms of the betafunction
159
Examples of rectification
162
sin n
163
Another proof of the same for positive and integral values of
166
log г x dx
167
The third fundamental theorem of the gammafunction
169
Eulers constant Gauss definition of the gammafunction
170
The numerical calculation of г n 135 The minimum value of г
171
The deduction of the fundamental theorems of the gamma function from Gauss definition of the function
173
344
177
Similar application of the betafunction
183
Examples of the criterion given in the preceding article 243
184
Another form of definite integral determined by means
188
Various applications of the logarithmintegral
189
Criterion of convergence of these series
190
THE APPLICATION OF SINGLE INTEGRATION TO QUESTIONS
197
Fagnanis theorem
206
Quadrature of a plane surface contained between two given
222
Involutes of curves referred to polar coordinates and
226
THE APPLICATION OF SINGLE DEFINITE INTEGRATION TO
235
The geometrical interpretation of the general formula
249
Other series derived by definite integration from the pre
256
The problem of probabilities for which infinite summation
265
The calculus of variations is a calculus of continuous func
290
Geometrical interpretation of fundamental operations
296
to a Variable Parameter
297
Quadrature determined when the axes are oblique
311
Examples of these theorems
376
The probability of an event and of the precedent cause
380
The variables separated by means of a substitution
382
Examples of integration of the same
388
REDUCTION OF MULTIPLE INTEGRALS
389
Mode of integrating P dx + q dy 0 when Pa Qy 0
395
Further extension by Liouville
396
Discontinuous functions exhibited as periodic series
403
Criteria of singular solutions and examples
405
General method of integration and examples
411
The variation of a product of differentials
426
Geometrical problems involving total differential equations
430
Variation
435
The variation of a double integral
446
The number of arbitrary constants contained in the final
452
Problems of relative maxima and minima
465
The same method applied to simultaneous linear partial
482
Investigation of Critical Values of a Definite Integral
491
The integral can always be found
504
Investigation of the Critical Values of a Double
511
INTEGRATION OF DIFFERENTIAL FUNCTIONS OF
513
Definition of general integral particular integral singular
520
number
528
Integrating factors of equations of three variables and
556
Singular Solutions of Differential Equations
569
415
589
Substitution by means of polar coordinates
597
Solution of Geometrical Problems dependent
606
Variation of Fx y ý y
608
Application of Fouriers theorem to definite integrals
614
Geometrical interpretation of the result of Article 303
615
Linear Differential Equations
622
A differential equation linear in at least the first
629
A problem solved on the principle of Art 314
640
Examples in illustration of the process
643
Particular forms of Linear Differential Equations
651
Integration of ƒ x Fƒ¹x ƒ2x
663
Monges method
666
THE SOLUTION OF DIFFERENTIAL EQUATIONS BY THE CALCULUS
673
tion of this process
682
The Solution of Partial Differential Equations
683
Integration of linear simultaneous equations
689
differential equations
694
INTEGRATION OF DIFFERENTIAL EQUATIONS BY SERIES 484 Application of Taylors theorem
699
Application of Maclaurins theorem
700
The method of undetermined coefficients
701
The solution of Riccatis equation affected by the method
703
A particular case of this equation
704
The general solution of particular forms of Riccatis equa tion expressed in terms of a definite integral
705

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