A Treatise on Infinitesimal Calculus: Containing Differential and Integral Calculus, Calculus of Variations, Applications to Algebra and Geometry, and Analytical Mechanics, Volumen 2

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The University Press, 1854
 

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Various examples of integration of exponential functions
69
Integration of sinxndx and of coaxndx
78
Integration of fx sin1 xdx fx tan1 dt c
85
Further theorems of definite integrals
91
Examples of Definite Integrals
98
A similar proof of Maclaurins Series
105
Successive integration
111
integral calculus applied to the rectification
118
Discussion of properties of the arc of an ellipse
127
Examples in illustration
132
Value of lengthelement in terms of r and p
133
Investigation of the general equation and examples
135
Involutes of curves referred to polar coordinates and
141
The order of integrations changed and examples
149
Remarks on elimination by means of a system of linear
150
Examples illustrative of it
156
Quadrature of Surfaces of Revolution
162
Quadrature of Curved Surfaces
166
Investigation of the surfaceelement
169
Investigation of volumeelement when axis of y is that
173
Variation of a definite integral due to the integration of
176
Necessity of caution as to the order of integrations
181
GENEKAL PROPERTIES OF MULTIPLE INTEGRALS AND THEIR
188
General result derived from explicit functions
196
Modification of the result when an equation of condition
198
Investigation of a method for determining the new limits
202
The three confocal surfaces of the second order intersect
212
An integral involving an irrational function transformed
223
The radius of absolute curvature of a geodesic is equal
229
Variation of a definite integral due to that of a constant
230
Difference as to operations and symbols between the dif
236
Geometrical interpretation of fundamental operations
242
Geometrical interpretation of the result of the last Article
248
Calculation of a variation of a variation
257
Examination of the several parts of the result
263
Modification of the result when derivedfunctions and
269
275 The substitution required by separation of the variables shewn to be equivalent to multiplication by an inte grating factor 355
275
An a posteriori proof that an homogeneous equation when
276
On Geodesic Lines
293
The radius of torsion of a geodesic
296
Only one general form of function satisfies a differential
305
Length of a geodesic on an ellipsoid
307
Solution of various problems
313
Proof that 8 H u dx is an exact differential
322
Examples of integration
359
Bernoullis equation
360
Partial Differential Equations of the first order and degree 281 Method of integrating partial differential equations and of introducing an arbitrary fu...
361
Examples of such integration
363
Geometrical illustration of the process
367
Partial differential equations of any number of variables
368
Integrating Factors of differential Equations 285 Every differential equation of the first degree has an inte grating factor
371
And the number of such integrating factors is infinite
372
Mode of determining integrating factors
373
Integrating factor of a homogeneous equation of n dimen sions and two variables
374
To find the surface every point of which is an utnbilic 518
375
Examples in illustration
376
290291 Integrating factor of the linear equation of the first order
377
Examples of other forms wherein the integrating factor can be found
380
Integrating factors of equations of three variables
381
Examples in illustration
383
Application of the method to homogeneous equations
386
Another method of integrating differential equations of three variables
388
297 Geometrical interpretation of the criterion of integrability
390
Firstly by Monges theorem
392
299301 Secondly by Bertrands theorem
395
A method of integration when the condition of integrability is not satisfied
397
Criteria of singular solutions and examples
402
Examples of the process
408
Particular forms Clairaults form
414
The case where the coefficients of the powers of y1 are
422
Eulers differential equation
429
INTEGRATION OF DIFFERENTIAL EQUATIONS OF ORDERS HIGHER
439
Similar conditions that it should be integrable m times
442
Application of the process to an equation of the third order
448
Construction of a linear differential equation when particu
455
Modification if the roots are impossible
462
Examples in illustration of the process
469
Examples
475
Other modes of employing the operative symbols
477
Integration of a linear differential equation whose coeffi
484
Integration of fx y y 0 and offy y y 0
492
Examples in illustration
498
367 Trajectories of plane curves referred to rectangular coor
504
Geometrical problems involving partial differential equations
511
INTEGRATION OF SIMULTANEOUS DIFFERENTIAL EQUATIONS
521
Linear simultaneous equations of higher orders and of con
528
Application of Maclaurins theorem
534

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Página 164 - ... is equal to the product of the length of the curve and the length of the path described by the centroid of the curve.
Página 284 - ... of any form and size. When stretched the rod becomes thinner, so that the several particles undergo lateral as well as longitudinal displacements. There is one fibre or line of particles which is undisturbed by the lateral contraction. Let this straight line, which we may regard as the central line, be taken as the axis of x, and let the origin be at the fixed extremity of the rod. We suppose that the stretching forces at the two ends are distributed over the extreme cross sections in such a...
Página 502 - Find the curves in which the perpendicular from the origin on the tangent is equal to the abscissa of the point of contact. [The circles r - 2a cos в.] 13.
Página 292 - F3 are proportional to the directioncosines of the normal to the surface at the point (x, y, z), that the points determined through these equations are such that lines joining them to the point (a, b, c) stand normal to the surface. If...
Página 510 - By making *' = 0, we find the co-ordinates of the point where the normal meets the plane of xy, also, the length of the normal, intercepted between the surface and the plane of wy, is РнoB.
Página 168 - ... and one of whose edges passes through the centre of the sphere. Find the area of the surface of the sphere intercepted by the cylinder. .Let the cylinder be perpendicular to the plane of xy ; then the equations of the cylinder and the sphere are respectively yz = ax — x2 and я2 -f y2 + z2 = a*.
Página 346 - R ' ( } where P, Q, and R are functions of x, y, and z.
Página 306 - It is not possible to have a series of geodesic lines passing through a point and touching a given line of curvature. In fact the words "originating at a point" should be omitted, as they are not required or used. On page 308 we read " it may be proved in the same way as the analogous theorem in plane geometry, that the geodesic radii vectores make equal angles with the curve of curvature.
Página 502 - A. + u for the rectification of a plane curve, where p is the perpendicular from any assumed point called the pole on a tangent to the curve, A the angle between this perpendicular and any fixed line drawn through the pole, u the portion of the tangent intercepted between the point of contact and the foot of this perpendicular. Let Q be the centre of curvature of the arc at A, AB a tangent at A...
Página 116 - K is also the rate of change of the angle between the tangent to the curve and a fixed direction.

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