. The principles of projective geometry applied to the straight line and conic . oints ofthe ranges be a, a, b, b. Let r and t be the lines ah. aband ab. ab. Let the tangents from rt tothe circle or conic be I and m. Self-corresponding Elements 237 Then since L and M are thedouble points of the involution onthe circle or conic (Art. 75), SLand SM are the double rays of theinvolution pencil. (ii) Take any circle and let sany tangent to this circle cut theinvolution in AABB. Construct by the right-handside of (i) the double elements Land M of this involution range. The rays joining these pointst
. The principles of projective geometry applied to the straight line and conic . oints ofthe ranges be a, a, b, b. Let r and t be the lines ah. aband ab. ab. Let the tangents from rt tothe circle or conic be I and m. Self-corresponding Elements 237 Then since L and M are thedouble points of the involution onthe circle or conic (Art. 75), SLand SM are the double rays of theinvolution pencil. (ii) Take any circle and let sany tangent to this circle cut theinvolution in AABB. Construct by the right-handside of (i) the double elements Land M of this involution range. The rays joining these pointsto S will be the double rays of theinvolution pencil. Then since I and m are thedouble rays of the involutionsystem of tangents to the circle orconic (Art. 75), si and sm are thedouble points of the involution. Take any circle and projectfrom S any point on the circle therange into a pencil {S. aabh). Construct by the left-handside of (i) the double elements Iand m of this involution pencil. The points in which these raysmeet s will be the double pointsof the involution 111. To find the common pair of conjugates—when real-of two involutions situated on a conic (ct. Art. 55). Let the involutions be deter-mined by the pairs of conjugatepoints AA, BB and KK, ^QiAA. BB be 0 and R Join OR. Then the points P, F, ifreal, in which OR meets theconic are the common pair ofconjugates of the two involutions.(Art. 95 (c).) By the correlative method the two tangents, which are a pair of conjugates intwo invohitions of tangents to the conic, may be constructed. 112. Method of false positions. In attempting to solve ageometrical problem it frecjuently happens that the solution of theproblem is found to depend on finding the configuration for which twopoints coincide. Such points are frequently situated on some givenline or conic and in this case it often happens that for differentpositions of one of the points it may be proved that the second pointdescribes a range project
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
