A treatise on infinitesimal calculus, Volume 2

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Contents

The theory and its application
103
The correction for infinity and discontinuity
105
Application of the theorem to particular forms of element functions
106
Cases where the correction for infinity vanishes
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
110
Interchange of this process with that of the definite integra
111
Further application of the preceding processes
117
133
133
rn+1 27 nn+1e when n
138
Methods of Approximating to the value of a Definite Integral 111 Evaluation by direct summation
139
Evaluation by summation of terms at finite intervals
141
Geometrical interpretation of the process
142
Application of the process to Mensuration
144
Approximation by means of known integrals 116 Approximate values given by the forms of definite integrals
147
Bernoullis series for approximate value
150
Approximate value deduced from Taylors series
151
Maclaurins series applied to integration
153
Definition of the gammafunction various forms of it 155
155
The gammafunction is determinate and continuous
156
Particular values of the gammafunction for particular values of the argument
157
Equivalent forms of the betafunction
159
The proof of the theorem fn+1 nrn
160
Another proof of the same for positive and integral values of n
161
Examples of rectification
162
The proof of the theorem r n T 1n inne
163
Evaluation of log x dx
165
Determination of the xdifferential of log rx
166
The third fundamental theorem of the gammafunction
167
Eulers constant Gauss definition of the gammafunction
170
The numerical calculation of r n 135 The minimum value of r n 136 The deduction of the fundamental theorems of the gamma function from Gauss ...
173
344
177
Similar application of the betafunction
181
The LogarithmIntegral
187
THE APPLICATION OF SINGLE INTEGRATION TO QUESTIONS
197
Fagnanis theorem
206
The Rectification of Curves in Space
212
Involutes of Plane Curves
222
173
228
THE APPLICATION OF SINGLE DEFINITE INTEGRATION TO
235
fx is convergent or divergent according as f x dx
241
Directions as to the application of the preceding tests
249
Other series derived by definite integration from the pre
256
The geometrical interpretation of the discontinuity of
264
The problem of probabilities for which infinite summation
265
Discontinuous functions exhibited as periodic series
273
Cauchys Method of Evaluation
279
Examples of definite multiple integrals
283
An extension of the theorem of the preceding article
287
Examples of transformation of definite multiple integrals
293
The differentiation of the same when one definite integra
299
The order of integrations changed and examples
306
curv
307
Examples in illustration
308
Quadrature determined by means of substitution
310
Quadrature determined when the axes are oblique
311
Quadrature of Plane Surfaces Polar Coordinates 226 The differential expression of a surfaceelement
312
Examples illustrative of it
313
The order of integrations inverted
315
Investigation of the surfaceelement in terms of r and p
316
Quadrature of a surface between a curve its evolute and two bounding radii of curvature
317
Quadrature of Surfaces of Revolution
319
Investigation of the surfaceelement 233 The area of a surface when the generating plane curve revolves about the axis of y
321
The area of a surface of revolution when the axis of revolu tion is parallel to the axis of x
322
The area of a surface of revolution when the generating curve is referred to polar coordinates i
323
Quadrature of Curved Surfaces 236 Investigation of the surfaceelement
324
Examples of quadrature of curved surfaces
326
The quadrature of the surface of the ellipsoid
327
The same expressed in terms of certain subsidiary angles
328
Catalans interpretation of the same
330
CHAP X
342
Examples of geodesics on a sphere and on a cylinder
343
Geodesies on the surface of an ellipsoid Joachimsthals theorem
344
Theorems concerning geodesics on an ellipsoid
345
The equations of geodesics in terms of curvilinear co ordinates
347
The cubature of a solid of revolution when the generating
350
Examples of cubature
356
Other transformations of the integral expressing the volume
362
The value of a definite integral when the elementfunction
372
The curve of probability
379
The variables separated by means of a substitution and examples
382
Bernoullis equation
383
REDUCTION OF MULTIPLE INTEGRALS
389
Further extension by Liouville
396
Application of Fouriers theorem
403
Criteria of singular solutions and examples
405
Cauchys method of evaluation and examples
406
CALCULUS OF VARIATIONS
411
Geometrical interpretation of fundamental operations
419
The variation of a product of differentials
426
Geometrical problems involving total differential equations
430
Variation of Fx dx dx y dy dạy 2 dz dz
435
Calculation of a variation of a variation
442
The calculus of variations considers a function of an infinite
448
Particular cases
454
Investigation of Critical Values of a Definite Integral whose ElementFunction involves DerivedFunctions 349 Determination of the necessary conditi...
491
Investigation of particular forms
492
Solution of various problems
493
Discriminating Conditions of Maxima and Minima 352 General considerations as to the required conditions
495
Statement of the requisites and Jacobis mode of satisfying them
498
Proof that 8 Hn dx is an exact differential
501
The form of its integral
503
The integral can always be found
504
Two particular cases wherein the criteria are applied
505
Application of the criterion to the general case
509
The criterion applied to a case of relative critical value
510
Investigation of the Critical Values of a Double Definite Integral 361 Determination of the necessary criteria
511
CHAPTER XV
513
A problem solved on the principle of Art 314
519
Definition of general integral particular integral singular
520
The definite integral of a total differential equation of three
527
Another form reducible to an homogeneous equation
533
Partial Differential Equations of the First Order and First Degree 384 Method of integrating partial differential equations and of introducing an arbitr...
536
Examples of such integration
539
Geometrical illustration of the process 387 Partial differential equations of any number of variables
543
Examples of integration of the same
545
Every differential equation of the first degree has an inte grating factor
546
And the number of these integrating factors is infinite
547
Mode of determining these integrating factors
548
Examples in which the integrating factor is a function of one variable
550
Mode of integrating P dx + q dy 0 when Px + Qy 0
551
Integrating factors of equations of three variables and ex amples in illustration
556
Application of the method to homogeneous equations
560
Another method of integrating differential equations of three variables
562
Geometrical interpretation of the criterion of integrability
563
A method of integration when the condition of integrability is not satisfied
567
Singular Solutions of Differential Equations 402 The general value of the integral of a differential equation 403 Conditions which the general integra...
569
Particular forms Clairauts form
584
The case where the coefficients of the powers of ı
591
Various Theorems and Applications
597
Solution of Geometrical Problems dependent
606
Geometrical problems involving partial differential equa
612
433
615
An a posteriori proof of the theorem
618
Examples of application of the process
626
A differential equation linear in at least the first
629
The determination of the constants
635
Examples in illustration of the process
643
Particular forms of Linear Differential Equations
651
Integration of fx y 0 and of fyy y 0
659
472
677

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