Elements of the Differential and Integral CalculusA.S. Barnes & Company, 1838 - 283 σελίδες |
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angle asymptote axis of abscissas become binomial bx² circular segment circumference co-ordinate plane consequently curvature cycloid d²u d³u da² da³ denominator Differential Calculus differential equation dx dx dx dy dx² dx³ dy dx dy² equal example exponent expressed factor Find the differential find the value formula found Art fraction function Geom hyperbolic spiral increment h infinite integral involute Let us take limit logarithmic spiral logarithms maximum minimum multiplied Naperian negative obtain ordinate origin of co-ordinates osculatory circle parabola passing perpendicular quantity radius radius of curvature radius-vector ratio rectangle reducing render represent roots second differential coefficient second member sine solid spiral sub-tangent Substituting these values Substituting this value suppose supposition surface tang tangent line Taylor's theorem unity variable
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Σελίδα 162 - P', we shall have, for the first, R = s + a, and for the second R'-s' + a; hence, and hence, the difference between the. radii of curvature at any two points of the involute is equal to the part of the evolute curve intercepted between them. 170. The value of the constant a will depend on the position of the point from which the arc of the evolute curve is estimated. If, for example, we take the radius of curvature for lines of the second order, and estimate the arc of the evolute curve from the...
Σελίδα 59 - Art. 255); and the equation will become id \ i du a(lu) = le — , tv that is, the differential of the logarithm of a quantity is equal to the modulus of the system into the differential of the quantity divided by the quantity itself. 57. If we suppose a = e the base of the Naperian system, and employ the usual characteristic I...
Σελίδα 29 - or by reducing to a common denominator , , vdu — udv _ hence, the differential of a fraction is equal to the denominator into the differential of the numerator, minus the numerator into the differential of the denominator, divided by the square of the denominator.
Σελίδα 15 - The part 2 ax, which is independent of h, is therefore the limit of the ratio of ; the increment of the function to that of the variable.
Σελίδα 20 - The value of the ratio of the increment of the function to that of the variable is composed of two parts, 2ax and ah.
Σελίδα 122 - Of Asymptotes of Curves. 125. An asymptote of a curve is a line which continually approaches the curve, and becomes tangent to it at an infinite distance from the origin of co-ordinates. Let AX and AY be the co-ordinate axes, and the equation of any tangent line, as TP.
Σελίδα 28 - ... hence, the differential of the product of any number of functions, is equal to the sum of the products which arise by multiplying the differential of each function by the product of the others.
Σελίδα 243 - Substituting these values, we have (1 + z2)3' which may be integrated by the method of rational fractions^ Rectification of Plane Curves. 273. The rectification of a curve is the expression of its length. When this expression can be found in a finite number of algebraic terms, the curve is said to be rectifiable, and its length may be represented by a straight line. 274. The differential of the arc of a curve; referred to rectangular co-ordinates, is (Art. 128) dz = Hence, if it be required to rectify...
Σελίδα 31 - Hence, generally, the differential of any power of a function, is equal to the exponent multiplied by the function with its primitive exponent minus unity, into the differential of the function.
Σελίδα 166 - ... Curves may be divided into two general classes : 1st. Those whose equations are purely algebraic ; and 2dly. Those whose equations involve transcendental quantities. The first class are called algebraic curves, and the second, transcendental curves. The properties of the first class having been already examined, it only remains to discuss the properties of the transcendental curves. Of the Logarithmic Curve. 175. The logarithmic curve takes its name from the property that, when referred to rectangular...