This action may take several minutes for large corpora, please wait.
doc#129 | image of the point of tangency of the given | line | . Thus pencils of tangents to Q are transformed |
doc#129 | <formul> are two planes intersecting in a | line | l, tangent to Q at a point P, the two free |
doc#129 | tangent l<prime>. Hence, thought of as a | line | in a particular plane <pgr>, any tangent |
doc#129 | <formul> since it meets <lgr>, and hence every | line | of <formul> in the <formul> invariant points |
doc#129 | <lgr> and since it obviously meets every | line | of <formul> in a single point. The congruence |
doc#129 | <formul> which are tangent to g. Clearly, any | line | , l, of any bundle having one of these points |
doc#129 | of <formul> which is tangent to g at T. A | line | through two of these points, <formul> and |
doc#129 | (1,1) curve on Q, meets the image of any | line | of <formul>, which we have already found |
doc#129 | points. Hence its image, C<prime>, meets any | line | of <formul> in <formul> points. Moreover |
doc#129 | . Moreover, C<prime> obviously meets any | line | <formul> in a single point. Hence C<prime> |
doc#129 | observations make it clear that there exist | line | involutions of all orders greater than |
doc#129 | To do this we must first show that every | line | which meets g in a point P meets its image |
doc#129 | <formul> points cut from C by a general | line | , l, of the pencil correspond to the point |
doc#129 | k coincidences, each of which implies a | line | of the pencil which meets its image. However |
doc#129 | necessarily a cone, it follows finally that every | line | through a point, P, of g meets its image |
doc#129 | the invariant locus must have a multiple | line | of multiplicity either <formul> or <formul> |
doc#129 | first possibility requires that there be a | line | through P which meets g in <formul> points |
doc#129 | points; the second requires that there be a | line | through P which meets g in <formul> points |
doc#129 | each pencil are the multiple secant and the | line | joining the vertex, P, to the intersection |
doc#129 | secants. </p><p> Now consider an arbitrary | line | , l, meeting Q in two points, <formul> and |