A Treatise On the Analytic Geometry of Three Dimensions

George Salmon

The book A Treatise On the Analytic Geometry of Three Dimensions was written by author George Salmon Here you can read free online of A Treatise On the Analytic Geometry of Three Dimensions book, rate and share your impressions in comments. If you don't know what to write, just answer the question: Why is A Treatise On the Analytic Geometry of Three Dimensions a good or bad book?

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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.

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