The Differential and Integral Calculus: Containing Differentiation, Integration, Development, Series, Differential Equations, Differences, Summation, Equations of Differences, Calculus of Variations, Definite Integrals,--with Applications to Algebra, Plane Geometry, Solid Geometry, and Mechanics. Also, Elementary Illustrations of the Differential and Integral Calculus

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R. Baldwin & Cradock, 1842 - Calculus - 849 pages
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culculus book by Augustus

Contents

general and singular solutions 77 78 Examples 7980 81 Detached artifices
82
General term of a series 83 Differences of a rational and integral function 83
83
when ra 80 to 90 Reduction of fractions into sums of more simple fractions
90
Other instances 8990 Linear Equations 91 Arbitrary functions 9293 9495
92
tions subsequently introduced 87 88 Mode of tabulating the manner in which
95
1+3 98 ſlog rPrdr 99 ſar a++ 100 Reduction
97
gives ſprair conducted by unequal increments 102 Connexion of ſſr dt
104
a++ 101 to 103 Reductions of ſaAr+Brdr 104 Reduction
111
Consequences of neglect of the constant 106 Instances of formula in p
112
TRIGONOMETRICAL ANALYSIS
118
tan pk tan 4 124 Verification of the preceding 125 De Moivres Theorem
127
Linear equations of one variable 126 Partial differential equations
129
and minima 131133 Common theory 133 134 The diff co considered
135
Discussion of cos rHasin r 139 and 140 Discussion of cos aaHb
138
General method and instances 143 144 Isolated equations which admit
142
Length arc of a curve 110 141 Area of a curve 141 142 Solidity or volume
143
cases of motion 150 151
150
as f 5 156 gration o 2 vonnexion of giana as lar as n 5 156
156
theorem 153 Expansion of r in powers of tr 154 Deduction of Lagranges
159
Simple examples of the calculus of operations 160 Herschels theorem Expansion
168
I 2
169
CHAPTER X
172
Consequences of Herschels theorem 169 Expression of a linear function
177
xx andž 174 0 oc 1+ 175 Failure of Taylors theorem 176 First
182
Iog r and more correct approximation to 1 2 3 e 178 at Au as Au +
189
Branches of a curve 347 348 General theory of contact 349 350 Tangent
358
sions applied to these diff co 372374 Transformation of equations intersec
376
point 379382 Conjugate point 382 Pointed branch 382384 Area
387
Geometrical illustration 390 391 Extension of the method to the case in which
395
Classes of surfaces 399401 Characteristic of a surface 401 Connecting
430
Lines of curvature 434440 Various problems proposed 441 Shortest line on
443
e CHAPTER XVII
475
formation of the expression of rotations 482 483 Geometrical consideration
511
Lagranges general method 518522 Variation of parameters 523530
523
Lagranges general forms 530535 The fundamental equations connected with
535
ChapTER XVIII
542
Chapter XIX
560
1+3 576 577 Definition of Tr 577 Series for log Tz+1
579
n to the smallest number of distinct
587
TABLE of Tr for the twelfths of a unit 590 Determination of Tr by continued
601
tion of their discontinuity 608 Expansion of functions into periodic series 609
609
tinuity 614618 Fouriers theorem and remarks on proposed verifications
621
tion of the case in which the subject of integration becomes infinite between
644
approximating to Fresnels integrals 647649 On the celebrated property
656
log r 660662 Table of Soldners integral 662663 Integrals depending
664
Method of using A ſt wav 665666 Miscellaneous integrations from
673
Elem Illust pp 3135 and 4345
767
subject of diff equ are made in works on gravitation heat
771
1+rH 223 Combinatorial analysis 224 Development of aoHairH c
774
Elem Illust pp 2225 Appendir p 776
776

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Page 472 - ... connexion of the parts of the system one with another. Suppose perpendiculars drawn from the new positions of the points upon the directions of the forces acting at the points in their positions of equilibrium. The distance of any perpendicular from the original point of application of the corresponding force is called the virtual velocity of the point with respect to that force, and is estimated positive or negative, according as the perpendicular falls on the side of the point towards which...
Page 495 - ACD is a semicircular conducting loop of radius r with centre at O, the plane of the loop being in the plane of the paper. The loop is now made to rotate with a constant angular velocity co about an axis passing through O and perpendicular to the plane of the paper.
Page 560 - The history of algebra shows us that nothing is more unsound than the rejection of any method which naturally arises, on account of one or more apparently valid cases in which such a method leads to erroneous results. Such cases should indeed teach caution, but not rejection; if the latter had been preferred to the former, negative quantities, and still more their square roots, would have been an effectual bar to the progress of algebra . . . and those immense fields of analysis over which even the...
Page 519 - Ex. 36. A particle on the concave surface of a sphere is repelled from the lowest point by a force which varies inversely as the square of the distance to the particle. Find the position of rest.
Page 394 - This plane, which passes through the centre of the sphere and is perpendicular to the chords it bisects, is called the diametral plane of the sphere for chords parallel to b.
Page 126 - English ; a maximum is not necessarily the greatest possible value of a function, nor a minimum the least. The greatest value of the function is the greatest of all its maxima, and the least value is the least of all the minima. A maximum may even be less than a minimum ; or the value of a function where its increase stops in one state may be less than that where its decrease stops in another state.
Page 104 - An interesting example for following out the processes of the engine would be such a form as x" dx or any other cases of integration by successive reductions, where an integral which contains an operation repeated n times can be made to depend upon another which contains the same n — 1 or n — 2 times, and so on until by continued reduction we arrive at a certain ultimate form, whose value has then to be determined. The methods in Arbogast's Calcul des Derivations are peculiarly fitted for the...
Page 565 - OA = -m, 0 = + m, (fig. 33), we find the areas PAOY and QOBY both positive and infinite, which agrees with all our notions derived from the theory of curves. Again, if we attempt to find the area PYQB by summing PA 0 Y and YOQB, we find an infinite and positive result, which still is strictly intelligible.
Page 624 - In fact it has become usual to regard it as a method of oaaooa suggestion of new integrals to be verified by other methods rather than as a mode of investigation. For instance, De Morgan remarks : " It is a matter of some difficulty to say how far this practice may be carried, it being most certain that there is an extensive class of cases in which it is allowable, and as extensive a class in which either the transformation, or neglect of some essential modification incident to the manner of doing...
Page 470 - Now, the product, mv, is called the momentum of the mass m, moving with the velocity v.

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