. The principles of projective geometry applied to the straight line and conic . stems of tangents and 236 Princl2)les of Projective Geometry SM and SL arc the self-corre- to the conic (Art. 105(6)),sponding rays of the pencils. sm and si are the self-corresponding points of the ranges. Take any circle and projectfrom any point S on the circle theranges into pencils () and(S . ). Construct by theleft-hand of (i) the self-corre-sponding rays I and m of thesepencils. These rays will meet s in theself-corresponding points of theranges, viz. L and M. 110. To construct the
. The principles of projective geometry applied to the straight line and conic . stems of tangents and 236 Princl2)les of Projective Geometry SM and SL arc the self-corre- to the conic (Art. 105(6)),sponding rays of the pencils. sm and si are the self-corresponding points of the ranges. Take any circle and projectfrom any point S on the circle theranges into pencils () and(S . ). Construct by theleft-hand of (i) the self-corre-sponding rays I and m of thesepencils. These rays will meet s in theself-corresponding points of theranges, viz. L and M. 110. To construct the double elements—when real—of aninvolution form. To construct tJie double elements To construct the double elements of an involution pencil of an involution range {). (). (ii) Take any circle and letany tangent s to the circle cut thepencils in A, B, C and A, B, by the right-hand sideof (i) the self-corresponding pointsL and M of these ranges. The rays joining these pointsto ;S^ will be the self-correspondingrays of the pencils, viz. I and (i) Through S describe anycircle or conic to meet two pairs ofcorresponding rays in A, A, B, B. Let R and T be the and Let the line RT meet thecircle or conic in L and M. Describe any circle or conic totouch s and let the tangents fromtwo pairs of corresponding points 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
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