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Page 64 - The derivative * of a function is the limit of the ratio of the increment of the function to the increment of the independent variable, when the latter increment varies and approaches zero as a limit.
Page 566 - From eight times the chord of half the arc, subtract the chord of the whole arc, and divide the remainder by 3, and the quotient will be the length of the arc, nearly.
Page 55 - That is, the limit of the quotient of two functions is equal to the quotient of their limits, provided the limit of the divisor is not zero.
Page 397 - A circle of this radius, having the same tangent at P, and its concavity turned the same way, as in the given curve, is called the 'circle of curvature,' its radius is called the 'radius of curvature,' and its centre the 'centre of curvature.
Page 63 - A is the area of the triangle formed by the chord of the arc and the two tangents at the extremities, and A' the area of that formed by the three tangents.
Page 133 - If both members of the last equation be divided by h, we shall have u' — u •which expresses the ratio of the increment of the function to that of the variable.
Page 123 - From a given circular sheet of metal it is required to cut out a sector so that the remainder can be formed into a conical vessel of maximum capacity ; prove that the angle of the sector removed must be about 66�.
Page 80 - Bx' identically ; and therefore, since the limit of a product is the product of the limits, dy _ dy du, dx du' dx' A useful application of the formula (3) occurs in the theory of rectilinear motion.
Page 471 - Let us suppose that, in the equation M, N are homogeneous functions of x and y, of the same degree. In this case the fraction M/N is a function of y/x only, and we may write ?-/(*) (1) dx J \x) If we put y = xv this becomes 0) The variables x, v are now separable, viz. we have C&F dv •a /(„)_�' • whence log.T = I . - + C (4) After the integration has been effected we must write v = y/x.
Page 331 - ... perpendicular to the plane of the arc, is given by And the radius of gyration (�) about a parallel axis through the middle point of the arc is given by 11.