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absolute value absolutely convergent angle approaches a limit approaches zero asymptotic lines axes axis becomes infinite center of curvature circle coefficients condition constant continuous function contour convergent series coordinates corresponding defined definite integral denote determine developable surface diverges double integral double series dx dy envelope equal evident example expression finite follows formula function f(x given curve given series Hence increases indefinitely independent variables infinite number infinitesimal intersection interval Let us consider Let us suppose lies line integral lines of curvature locus necessary and sufficient obtained osculating plane parallel partial derivatives perpendicular plane curve polynomial positive number positive terms power series preassigned preceding principal normal radius of curvature ratio region replaced respectively root satisfy the equation sequence series converges skew curve straight line tangent plane theorem tion transformation vanish whence written xy plane
Page 2 - ... the second series. 2. Functions. When two variable quantities are so related that the value of one of them depends upon the value of the other, they are said to be functions of each other. If one of them be supposed to vary arbitrarily, it is called the independent variable. Let this variable be denoted by x, and let us suppose, for example, that it can assume all values between two given numbers a and If (a < 5).
Page 352 - ... of the sums of the two given series. This theorem, which is due to Cauchy, was generalized by Mertens,* who showed that it still holds if only one of the series (23) and (24) is absolutely convergent and the other is merely convergent. Let us suppose for definiteness that the series (23) converges absolutely, and let wn be the general term of the series (25): «>
Page 436 - Before entering upon a more precise discussion of the relations between a given curve and its evolute, we shall explain certain conventions. Counting the length of the arc of the given curve in a definite sense from a fixed point as origin, and denoting by a the angle between the positive direction of the x...
Page 277 - The geometrical interpretation of this result is easy : 2-np ds is the lateral area of a frustum of a cone whose slant height is ds and whose mean radius is p. Replacing the area between two sections whose distance from each other is infinitesimal by the lateral area of such a frustum of a cone, we should obtain precisely the above formula for A. For example, on the paraboloid of revolution generated by revolving the parabola...
Page 540 - C is a parabola whose vertex is at the origin and whose axis is the z axis. [Licence, Paris, July, 1880.] 11. Determine the asymptotic lines on a ruled surface which is tangent to another ruled surface at every point of a generator A of the second surface, every generator of the first surface meeting A at some point. 12. Determine the curves on a rectilinear helicoid...
Page 215 - We shall commence with the case in which the integrand is a rational function of x and the square root of a polynomial of the second degree.
Page 408 - ... introduce in this section, x, y, P(x), and Q(x) may be either real or complex. In the first place, both P(x) and Q(x) may be analytic at x = a; that is, they may possess Taylor expansions around the point x = a. When this happens, x = a is said to be an ordinary point of the differential equation. A point which is not an ordinary point is called a singular point. At a singular point, although P(x) and Q(x) do not both possess Taylor expansions, it may be that the products (x - a)P(x) and (x -...
Page 49 - It follows that the derivative ^*/2«, does not vanish for the initial values, and hence the general theorem is proved. The successive derivatives of implicit functions defined by several equations may be calculated in a manner analogous to that used in the case of a single equation. When there are several independent variables it is advantageous to form the total differentials, from which the partial derivatives of the same order may be found. Consider the case of two functions u and v of the three...
Page 524 - The study of triply orthogonal systems is one of the most interesting and one of the most difficult problems of differential geometry.