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arbitrary constant arbitrary function asymptote axis becomes infinite binomial theorem calculus co-ordinates condition considered cos.x curve deduced degree denominator derived determine differential calculus eliminated equal equation becomes exact differential exponent expressed factor ferential equation finite follows formula fraction fudx Geometry given point gives homogeneous functions hyperbolic imaginary independent variable integration of differentials l)th Let the equation Let the proposed Let udx limits logarithms maxima and minima maximum or minimum method Multiplying negative obtained osculating osculating circle osculating curve parabola partial differential coefficients partial differential equations particular values plane primitive equation PROP proposed differential equation proposed equation quantities radius rational function reduced render result roots second differential coefficient second order SECTION sin.x singular solution substituted successive differential coefficients supposed surface system of values Taylor's theorem tegral theorem tion vanish variation
Page 109 - To divide a given number a, into two parts, such that the product of the mth power of the one and the nth power of the other shall be the greatest possible. Let x be one part, then a — * is the other, and y = a...
Page 176 - Equation (4) states that a constant factor may be taken outside the sign of integration; (5) states that the definite integral of the sum of two functions is equal to the sum of the integrals of the separate functions. Equations (4) and (5) permit us to work with linear combinations of functions. Thus, / [cifi(x) +••••+ cnfn(x)]dx = ct / fi(x)dx + • • • + cn I fn(x)dx.
Page 14 - ... by multiplying the differential of each function by the product of the others.
Page 14 - For instance, the product formula is d(uv) = udv + v du: in words, the differential of the product of two functions is equal to the first function times the differential of the second plus the second times the differential of the first.
Page 9 - ... quantities. Again ; we know by the theory of equations, that every radical has as many different values as there are units in its exponent; and that, consequently, every...
Page 466 - E^, are in the plane This is the equation of a plane passing through the centre of the indicatrix and parallel to the planes which touch the indicatrix at either of the points where the line X /2 v intersects it.
Page 176 - Passing to the limit we have cue whence [duv = vPdx + uQdx — vdu + udv, that is : The differential of the product of two functions of the same variable, is equal to the sum of the products obtained by multiplying each function by the differential of the other.
Page ii - AN ANALYTICAL TREATISE ON PLANE AND SPHERICAL TRIGONOMETRY. By the Rev. DIONYSIUS LARDNER, LL.D. Second Edition, Corrected and improved. 8vo. 12s. cloth. VII. AN ELEMENTARY TREATISE ON THE DIFFERENTIAL AND INTEGRAL CALCULUS. By the Rev. DIONYSIUS LAHDNER, LL.D. 8vo. 21».
Page 143 - ... case <£'(#i) cannot be either positive or negative (Art. 48) and must therefore vanish, since it is by hypothesis finite. The geometrical statement of this theorem is that if a curve meets the axis of x at two points, and if the gradient is everywhere finite, there must be at least one intervening point at which the tangent is parallel to the axis of x. See, for example, the graph of sm x on p.