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I.—On Polyzomal Curves, otherwise the Curves √U + √V + &c. = 0

Published online by Cambridge University Press:  17 January 2013

Cayley
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If U, V, &c, are rational and integral functions (٭)(x, y, z)r, all of the same degree r, in regard to the co-ordinates (x, y, z), then √U + √V + &c. is a polyzome, and the curve √U + √V + &c. = 0 a polyzomal curve. Each of the curves √U = 0, √V = 0, &c. (or say the curves U = 0, V = 0, &c.) is, on account of its relation of circumscription to the curve √U + √V + &c. = 0, considered as a girdle thereto (ζῶμα) and we have thence the term “zome” and the derived expressions “polyzome,” “zomal,” &c. If the number of the zomes √U, √V, &c. be = ν, then we have a ν-zome, and corresponding thereto a ν-zomal curve; the curves U = 0, V = 0, &c., are the zomal curves or zomals thereof.

Type
Transactions
Information
Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 25 , Issue 1 , 1868 , pp. 1 - 110
Copyright
Copyright © Royal Society of Edinburgh 1868

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References

page 29 note * Phil. Transactions, vol. cxlix. (1859), pp. 61–90. See p. 86.

page 54 note * It may be remarked that if the equation of the first pencil of lines be

(x − ay)(x − by) (x − cy) (x − dy) = 0,

and that of the second pencil

(z − aw)(z − bw)(z − cw)(z − dw) = 0,

then the equations of four conics are

xw − yz = 0,

(a + d − b − c) xz + (bc − ad)(xw + yz) + (ad(b + c) − bc(a + d))yw = 0,

(b + d − c − a)xz + (ca − bd)(xw + yz) + (bd(c + a) − ca(b + d))yw = 0,

(c + d − a − b) xz + (ab − cd)(xw + yz) + (cd(a + b) − ab(c + d))yw = 0.

page 56 note * It will appear, post Nos. 161–164, that if starting with three given points as the foci of a bicircular quartic, we impose the condition that the nodes at I, J shall be each of them a cusp, then either the quartic will he the circle through the three points taken twice, in which case the assumed focal property of the given three points disappears altogether, or else the three points must be in lineâ, or the curve be symmetrical, that is, a Cartesian.

page 63 note * This investigation is similar to that in Salmon's Higher Plane Curves, p. 196, in regard to the double tangents of a quartic curve.