 | Charles Davies, William Guy Peck - Mathematics - 1855 - 628 pages
...denote what R becomes by R', we have — 2j' cos <f> sin £ + q" cos*^ ; -5Г and by addition that is, the sum of the reciprocals of the radii of curvature...two normal sections at right angles to each other, when taken at the point of contact, is constant. We see from this result, that the section of least... | |
 | Charles Davies, William Guy Peck - Mathematics - 1859 - 622 pages
...aOb will be similar, and we shall have aO : ab : : be : Ac, or К : ds : . ds : Ac; whence, that is, the sum of the reciprocals of the radii of curvature of two normal sections at ri^ht angles to each other, when taken at the point of contact, is constant. Vi'e see from this result,... | |
 | George Salmon - Geometry, Analytic - 1862 - 490 pages
...equation then is z + A (** + y*) f &c. = 0, or the indicatrix is a circle. The origin is then an umbttic. From the expressions in this article we deduce at...This formula (with the inferences drawn from it) is duo to Euler. 261. It will be observed that th*e radius of curvature, being proportional to tbe square... | |
 | Théodore Strong - Calculus, Integral - 1869 - 640 pages
...Consequently, since the position of the axes of x and y in the plane of x, y, is arbitrary, it clearly follows that the sum of the reciprocals of the radii of curvature of any two vertical planes through the axis of z, which cut each other perpendicularly, is always equal... | |
 | Charles Smith - Geometry, Analytic - 1884 - 256 pages
...reciprocals of any two perpendicular semi-diameters of a conic is constant, it follows from the last article that the sum of the reciprocals of the radii of curvature of any two perpendicular normal sections is constant. 241. Since the semi-diameter of a conic has a maximum... | |
 | Charles Smith - Geometry, Analytic - 1886 - 268 pages
...reciprocals of any two perpendicular semi-diameters of a conic is constant, it follows from the last article that the sum of the reciprocals of the radii of curvature of any two perpendicular normal sections through a given point of a surface is constant. 244. Since the... | |
 | George Shoobridge Carr - Mathematics - 1886 - 1036 pages
...в, у = r sin в, and r* = 2pz, which substitute in (5793). 5807 COR. — The sum of the curvatures of two normal sections at right angles to each other is constant; or, if p, p' be the radii of curvature for those sections, and pa, pb the radii for the principal sections,... | |
 | Charles Sheldon Hastings, Frederick Elijah Beach - Physics - 1898 - 788 pages
...through a liquid surface of double curvature, the pressure changes by an amount, where r and r' are the radii of curvature of two normal sections at right angles to each other. It should be noted that in applying this equation to a soap film, since there are two liquid surfaces... | |
 | University of Sydney - 1912 - 1050 pages
...is meant by the indicatrix at any point of a surface ? By means of the indicatrix or otherwise prove that the sum of the reciprocals of the radii of curvature of any two perpendicular normal sections is constant. X _ 3! The catenary y='lec+tc\ revolves round the... | |
 | Great Britain. Parliament. House of Commons - Great Britain - 1876 - 570 pages
...the angle of torsion of a curve of double curvature. Find the angle of torsion of the helix. G. Prove that the sum of the reciprocals of the radii of curvature...sections at right angles to each other is constant. 7. Give a short account of the solution of differential equations : — (a.) by the method of the separation... | |
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From the expressions in this article we deduce at once, as in the theory of central conies, that the sum of the reciprocals of the radii of curvature of two normal sections at right angles to each other is constant ; and again, if normal sections be made... 