This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1850 Excerpt: ...to, and therefore makes with the axes the angles a, /3, 7. Let a surface be described whose equation is H = c, and let a plane be drawn through the origin conjugate (with regard to this surface) to the line drawn from the origin to the point on the curve. Produce the normal to the. curve till it meet this plane, and ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1850 Excerpt: ...to, and therefore makes with the axes the angles a, /3, 7. Let a surface be described whose equation is H = c, and let a plane be drawn through the origin conjugate (with regard to this surface) to the line drawn from the origin to the point on the curve. Produce the normal to the. curve till it meet this plane, and let the line so produced be called n. We have then evidently x-x n v-V z-z cosa=, cosp=---, cosy =, n 'n 'n x'y'z being the point in which n cuts the conjugate plane. That is, parallel to the tangent plane to the surface M = c, at the point in which the line to which it is conjugate cuts that surface. It is easily seen that, as n is a homogeneous function, this plane is unique. Equation (F) may therefore be written nsinw 1 r / du du du I, da, du, du But since x'y'z is a point in the conjugate plane, we have, as in p. 152, du, du, du, . dx dy dz and since u is a homogeneous function, du du du dx J dy dz r m denoting the degree of the function. Hence n sin uj = mp. Hence, preserving the foregoing definitions, we have the following theorem: If u be a homogeneous function of the co-ordinates of a point upon a given surface, the curve which renders uds a maximum or minimum is such, that if the normal at any point be projected upon the osculating plane, its projection is equal to m times the radius of curvature. A similar substitution will reduce the first of the foregoing equations (F) to 1 L--p2 p'2 n2' Either of these theorems will furnish means for obtaining the osculating plane and the radius of curvature at any point, if the direction of the tangent be known. Example. 86. To find the shortest line which can be drawn between two points upon a given surface. This is the simplest case of the preceding proposition, from which it may be deduced by making p...
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