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. 1861 Excerpt: ...of the reciprocal curves. Hence any two curves will have as many points of intersection as their reciprocals have common tangents, and as many common tangents as their reciprocals have points of intersection. If the curves touch one another, then two of their points of intersection coincide; and consequently the two ...
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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. 1861 Excerpt: ...of the reciprocal curves. Hence any two curves will have as many points of intersection as their reciprocals have common tangents, and as many common tangents as their reciprocals have points of intersection. If the curves touch one another, then two of their points of intersection coincide; and consequently the two corresponding tangents to the reciprocal curves will coincide, and therefore the reciprocal curves will also touch one another. 5. From what has been said above, it will be seen that the total number of tangents, real or imaginary, which can be drawn to E or L from any point (not on the curve itself) is equal to the total number of points, real or imaginary, in which L or E is cut by any straight line, not a tangent to it. A curve, to which n tangents can be drawn through the same point, is said to be of the nth class, and we may therefore express the above proposition by saying that the degree of a curve is the same as the class of its reciprocal, and the class of a curve the same as the degree of its reciprocal. 6. The number of tangents, real or imaginary, which can be drawn to a conic from a given point is known to be two. Hence, the reciprocal of a conic is intersected by a given straight line in two points, real or imaginary, and is therefore of the second degree, that is, it is itself a conic. 7. This proposition may also be proved analytically as follows. Def. The conic with respect to which the poles and polars are taken is called the auxiliary conic. We have seen (Art. 15, Chap, n.) that any two conics may be expressed by equations involving the squares of the variables only. Let then the auxiliary conic be denoted by Li, + M? + Nf=0 (1), and the conic to be reciprocated by la2+mp2+nrf = 0 (2). If (f, g, h) be any point on the required...
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