A Treatise on the Differential Calculus: And Its Applications to Algebra and Geometry, Founded on the Method of Infinitesimals

Front Cover
University Press, 1852 - Differential calculus - 540 pages
0 Reviews
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Maclaurins Theorem
87
60 fil On the roots of + 1 and of 1
98
Theory of the Equicrescent Variable
104
Taylors Series
108
67 Limits of Taylors Series
109
Examples of Taylors Series
110
Different forms of the problem
112
Requisite formulae for a function of two variables
113
Examples of transformations
114
Successive Differentiation of Functions of many Independent Variables 72 Explanation of the symbols
117
The order of successive differentiations with respect to many variables is indifferent
118
Application of the principles of the preceding Articles to functions of two and more variables
120
Eulers Theorems of homogeneous functions
123
77 Extension of the preceding principles to other and similar cases
126
Examples of preceding formulae
127
Expansion of one of the variables of an implicit function in terms of the others by means of Maclaurins Theorem
128
Calculations and properties of Bernoullis numbers
130
Lagranges Theorem
133
Laplaces Theorem
140
Extension of Maclaurins Theorem and an explanation of the method of Derivation
147
Elimination of constants from an implicit function
148
87 Elimination of given functions
151
8891 Elimination of arbitrary functions
152
Transformation of expressions involving partial derived functions into their equivalents in terms of other variables
160
Examples illustrative of the preceding principles
163
and Corollaries are deduced therefrom
172
Definition of order of infinitesimals
178
The cause of quantities assuming the forms r and f
181
Evaluation of quantities of the form 0 x oo
189
Evaluation of quantities of t lie forms 0 00 1 0
190
Evaluation of indeterminate forms of Functions of Two Variables 114 Examples of such evaluations
191
EXPANSION OF FUNCTIONS SECTION 1 On Functions of One Variable 115 An accurate proof of Taylors Series
194
The imperfect form of it given in Art 66
196
Deduction of Maclaurins Theorem from Taylors
197
On the limits of Taylors and Maclaurins Series
198
On the failure of Taylors and Maclaurins Series
200
Expansion of vx + h y + k
204
Expansion of vx + h y + k z + I
208
Expansion of FJT y in ascending powers of x and y
209
ON MAXIMA AND MINIMA
211
Geometrical representation of the criteria
213
The values of ds and of lines and quantities connected
220
Maxima and Minima of Implicit Functions
223
quisite conditions are not fulfilled
230
Discussion of the case of two variables which are connected
237
Mode of generating an evolute and formulae for determin
239
Object of the Chapter stated to be the discussion of
244
Ifjr has m equal roots fx has m1 roots equal
251
Des Cartes rule of signs
258
Cases considered of curves involving two and three
260
Corroboration of the preceding modes of interpretation
264
Necessity of symbols of direction
267
On the Generation of some Plane Curves of higher orders
274
ON PLANE CURVES REFERRED TO RECTANGULAR COORDINATES
285
Discussion of the equations to the tangent and the normal
291
On Asymptotes to Plane Curves referred
299
tance
302
On Direction of Curvature and Points of Inflexion
308
On Multiple Points
315
An explicit function is explained which well exhibits some
321
The normal to the curve passes through two consecutive
385
On the chord of curvature its definition and value
394
Two curves which have a common tangent intersect or
401
arbitrary constants Examples
402
Conditions under which a circle can have contact of the third order
405
Contact of curves with given curves
406
Theory of Envelopes
408
Explanation of the subject of envelopes families of curves variable parameters
409
Examples of envelopes
411
General case of n parameters and n 1 conditions
413
Examples in illustration
414
On Caustics 269 On the formation of caustics
419
General properties of such caustics
424
Particular case of the caustics by reflexion at a circular cylindrical surface
425
Caustic by reflexion on a logarithmic spiral
428
General properties of caustics by refraction
429
Caustic by refraction at a plane surface
431
277 On the equations to a straight line and to a plane
432
The equation to a tangent plane to a curved surface
434
The directioncosines of the tangent plane
435
Modified forms of the equation to the tangent plane when the equation to the surface is a explicit fi ho mogeneous and algebraical
436
The equations to a normal of a curved surface
438
Examples in illustration of the preceding
439
Singular forms of tangent planes Cones of the se cond and third orders
441
287 On the equations of curves in space
444
Examples of the preceding formula
450
Ruled surfaces
456
Examples of developable surfaces
469
On Surfaces generated by Circles
475
CURVATURE OF CURVES IN SPACE 323 Mode of measuring absolute curvature angle of contingence
481
Mode of measuring torsion radius of torsion
482
Radius of absolute curvature
483
Angle of curvature
486
Geometrical illustrations
487
Torsion
488
Singular values of curvature and torsion
490
Equation to the polar surface
491
The polar line and locus of polar lines
492
The osculating sphere
493
Evolutes of nonplane curve
494
Geometrical illustrations
497
Complex flexure and its measure
500
The osculating surface 201
501
Application to the helix
503
CURVATURE OF CURVED SURFACES
506
Normal sections
507
Perpendicularity of normal sections
511
Normal sections of maximum and minimum curvature
512
Eulers theorem of the curvature of normal sections
513
Application to the ellipsoid
515
Singular values of radii of curvature
516
Umbilics
519
357 Lines of curvature
520
Locussurface of centres of principal curvature
521
Modification of the conditions when the equation is explicit
523
Meuniers theorem of oblique sections
525
Explanation of properties by means of the indicatrix
526
Osculating surfaces
529
Differentiation satisfies the laws
533

Other editions - View all

Common terms and phrases

Popular passages

Page 429 - but varies for different media; that is, the sine of the angle of incidence bears a constant ratio to the sine of the angle of refraction. The envelope of all the refracted rays is called the caustic by refraction of the given surface*.
Page 283 - cycloid is the curve traced out by a point in the circumference of a circle, as the circle rolls along a fixed straight line. (a) Let the given straight line (fig. 39) be taken as the axis of x, and the radius of the
Page 276 - and y have already preoccupied the two directions at right angles to each other in the plane of the paper, which is (and conveniently so) called the plane of reference, we must seek for some other course by which a line,
Page 303 - when x = a, a line parallel to the axis of y, at a distance a from it, is an asymptote ; and if x = oo , when y = b, a line parallel to the axis of x, at a distance
Page 44 - (fig. 4) be the circle, of which let the radius be a; take o the centre, and let the angle AOC be the rath part of four right angles, and AC be the side of a regular polygon of n sides inscribed in the circle ; and make
Page 335 - it is plain that the curve is symmetrical with respect to the axis of x, and since the curve passes through the origin, the
Page 283 - a curve, such that FT, the length of the tangent intercepted between the point of contact and the axis of x, is always equal to OA, then the locus of P is the equitangential curve. Let OM = x, MP = y, OA = PT = a; then the definition of the curve above given leads, as will be seen in the next Chapter, to an equation of the form
Page 246 - Ex. 7. To find a point within a triangle, such that the sum of the lines drawn from it to the angular points may be a minimum.
Page 390 - the length of the evolute is equal to the difference of the radii of curvature of the involute corresponding to its two extremities. Of this we subjoin some examples
Page 358 - the equation represents an ellipse, parabola, or hyperbola, according as e is less than, equal to, or greater than unity. Hence the equation to the parabola

Bibliographic information