This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1910 edition. Excerpt: ... is the envelope of x. Now the pole of x lies on b;.'. the pole of b lies on x;.'. x envelopes the pole of b;.'. the reciprocal of a, viz. A, is the pole of b. q.e.d. THEOREM 101. (1) If P, Q are conjugate points w.r.t. Sv then p, q are conjugate lines w.r.t. S2; and conversely. (2) If PQR is a self ...
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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1910 edition. Excerpt: ... is the envelope of x. Now the pole of x lies on b;.'. the pole of b lies on x;.'. x envelopes the pole of b;.'. the reciprocal of a, viz. A, is the pole of b. q.e.d. THEOREM 101. (1) If P, Q are conjugate points w.r.t. Sv then p, q are conjugate lines w.r.t. S2; and conversely. (2) If PQR is a self-conjugate triangle w.r.t. Sx, then pqr is a self-conjugate triangle w.r.t. S2. The proof is left to the reader. THEOREM 102. (1) If O is the centre of the base-conic S, the centre of Sx reciprocates into the polar of O w.r.t. S2, and the polar of O w.r.t. Sx reciprocates into the centre of SQ. (2) The reciprocal of O is the line at infinity; and conversely. (3) The asymptotes of Sj reciprocate into the meets of S2 with the polar of O w.r.t. S2. (4) is an ellipse, parabola, or hyperbola, according as O lies inside, on, or outside Sx. The proof is left to the reader. 1. Prove Theorem 95. 2. Prove Theorem 96. 3. Prove Theorem 99. 4. Prove Theorem 101. 5. Prove Theorem 102. 6. If two conics touch each other, prove that their reciprocals touch each other. 7. What is the reciprocal of a common chord of two conics? 8. What is the reciprocal of a common tangent of two conics? 9. Two conics cut at P; what is the reciprocal of the tangents at P to the conics? 10. What is the reciprocal of two parallel tangents to a conic? 11. What is the reciprocal of a pair of conjugate diameters of a conic? 12. What is the reciprocal of two parallel chords of a conic? 13. P, Q are two points on a tangent // to a conic 5i; the other tangents from P, Q to Sx meet at R; construct the reciprocal figure. 14. PQR is a triangle, self-conjugate to a conic Su and circumscribing a conic S; what is the reciprocal figure? Attention has already been directed to the fact that all...
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