A Treatise On the Analytic Geometry of Three Dimensions
A Treatise On the Analytic Geometry of Three Dimensions
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In our eReader you can find the full English version of the book. Read A Treatise On the Analytic Geometry of Three Dimensions Online - link to read the book on full screen. Our eReader also allows you to upload and read Pdf, Txt, ePub and fb2 books. In the Mini eReder on the page below you can quickly view all pages of the book - Read Book A Treatise On the Analytic Geometry of Three Dimensions
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288. It was proved (Art. 194) that if two surfaces u 1 +u i +&c. J «i + « 2 4- &c. touch, their curve of intersection has a double point, the two tangents at which are the intersections of the plane u x with the cone u 2 - t> 2 . When the plane touches the cone, the surfaces have what we have called stationary contact. It is also proved, as at Art. 196, that a sphere has stationary contact with a surface when the centre is on the normal and the radius equal to one of the principal radii of curv...ature. In fact, the condition for stationary contact between z + ax 2 + 2nxy f by* + &c, z + a'x* + 2n'xy + b'y 2 + &c.
is (a-a')(b-b') = (n-n'y, which when n and n both vanish implies either a=a' or b = b'.
The surface therefore z + Ax* + Cy* + &c. will have stationary contact with the sphere 2rz + x* -{- y* -{■ z* if r = — -. or — -,; but these are the values of the principal radii.
289. The principles laid down in the last article enable us to find an expression for the values of the principal radii at any point; the axes of co-ordinates having any position.
What to read after A Treatise On the Analytic Geometry of Three Dimensions?
You can find similar books in the "Read Also" column, or choose other free books by George Salmon to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
288. It was proved (Art. 194) that if two surfaces u 1 +u i +&c. J «i + « 2 4- &c. touch, their curve of intersection has a double point, the two tangents at which are the intersections of the plane u x with the cone u 2 - t> 2 . When the plane touches the cone, the surfaces have what we have called stationary contact. It is also proved, as at Art. 196, that a sphere has stationary contact with a surface when the centre is on the normal and the radius equal to one of the principal radii of curv...ature. In fact, the condition for stationary contact between z + ax 2 + 2nxy f by* + &c, z + a'x* + 2n'xy + b'y 2 + &c.
is (a-a')(b-b') = (n-n'y, which when n and n both vanish implies either a=a' or b = b'.
The surface therefore z + Ax* + Cy* + &c. will have stationary contact with the sphere 2rz + x* -{- y* -{■ z* if r = — -. or — -,; but these are the values of the principal radii.
289. The principles laid down in the last article enable us to find an expression for the values of the principal radii at any point; the axes of co-ordinates having any position.
What to read after A Treatise On the Analytic Geometry of Three Dimensions?
You can find similar books in the "Read Also" column, or choose other free books by George Salmon to read onlineMoreLess
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