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. 1917 Excerpt: ...either regulus, in a conic. For the cutting plane meets either regulus in a conic (Art. 47). If the cutting plane contains a ray of either regulus, we have the following theorem: Any plane which passes through a ray of either regulus contains a ray of the other regulus and does not meet the surface outside these rays. ...
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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. 1917 Excerpt: ...either regulus, in a conic. For the cutting plane meets either regulus in a conic (Art. 47). If the cutting plane contains a ray of either regulus, we have the following theorem: Any plane which passes through a ray of either regulus contains a ray of the other regulus and does not meet the surface outside these rays. For, suppose the plane a passes through the ray - of the regulus V and meets two other rays of the same regulus in the points M'and N (Fig. 67). The straight line MN meets three rays of the regulus V and must, therefore, meet all of them. It is then a ray of the other regulus U, say the ray Hence a has the two lines f, -and in common with the surface That it can have no point outside these lines in common with the surface follows in this way; suppose there is a point P, not lying on either vi or uj, but which is common to the plane and to the surface. We can then draw any number of lines through P meeting both vi and uj and therefore, having three points in common with the surface. But this is impossible unless all these lines lie entirely on the surface (Art. 80), in which case the entire plane must form a part of the surface. But a plane cannot form part of the surface since no two rays of either regulus lie in any plane. We conclude, therefore, that a cannot meet the surface outside the two lines vi and uj. The plane a can be any plane through any ray of either regulus. 82. Tangent Lines and Tangent Planes.--A plane which contains a ray of either regulus on a ruled surface of second order is a tangent plane to the surface, the point of contact being the intersection of the two rays lying in the plane. For all the lines which pass through the intersection of the two rays and lie in the plane cannot meet the surface outside this point of ..
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