Cayley A. Забранные математические бумаги, издание 5 (1892), Cayley A. Collected mathematical papers, vol. 5 (CUP, 1892)(800dpi)(T)(640s).djvu:

Cayley A. Забранные математические бумаги, издание 5 (1892)

Cayley A. Collected mathematical papers, vol. 5 (CUP, 1892)(800dpi)(T)(640s).djvu

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Date Oct 27, 2004

Cites: But in the special case, which, as already noticed, is here
considered irrespectively of the general one, the equation Disc*...
The equation of the satellite line giving a twofold and one-with-twofold centre is
# V z
a
the coordinates of the point of contact with the envelope are x : y : z = a4 :
The equation in 8 gives 81 = 82 = -r- for the values corresponding to the twofold
2
oentre; and 03 — —^- for the value corresponding to the one-with-twofold centre...
In the formulae for these transformations, and indeed
throughout the memoir, the three roots of the equation in 6 are represented by
i> 02, @3, and I write also
t/3, &2 == — "it
17...
It is easy to show that the curve has three nodes the coordinates whereof
are (—4, 1, 1), A, —4, 1), A, 1, —4); and this being known, the equation may be
transformed so fcas to put the nodes in evidence...
The equation
x3 + y3 + zs — (yz2 + y2z + zx2 + z2x + xy2 + x2y) + 3xyz = 0,
of the one-with-twofold centre locus may be transformed as follows, viz...
In fact, since the point @, v, — fju) is an arbitrary point on the line x = 0, a
line Xx + fjuy + vz — 0 passing through the point in question is an absolutely arbitrary
line, and the corresponding critic centres therefore do not lie on the line x — 0; that
is, they lie on the conic
2 (y-
y z x+y+z '
and it may also be remarked that the elimination of X, 6, from the system
0 + X : 6 + p : 6 + v : 0 = - : - : - :
x ' y ' z ' x + y + z}
or, what is the same thing, the elimination of 0 from the system
11 2
fjL : 0 + v : 0 =
y 'z 'x+y+z'
gives the last-mentioned equation, unencumbered by the factor x = 0...
We have identically
— 6fgh (x2 + y2 + z2 — 2yz — 2zx — 2xy)
= 2fgh (x + y + zf + ? Bfyz + 2gzx + 2hxy) — 8 (fx + gy + hz) (ghx + hfy -\-fgz),
so that the tangents in question meet the twofold centre conic
x2 + y2 + z2 — 2yz — 2zx — 2xy = 0,
at its intersections with the lines fx + gy+ hz = 0, and ghx + hfy -\-fgz — 0 : the latter
of these is in fact the tangent of the conic at the point (/2, g2, h2) of intersection
of the two tangents...
Starting with a critic centre on the conic x (x + y + z) — tyz = 0, the other
two critic centres lie one on the conic, and the other on the line y + z=0; viz...
For the asymptotic cubic, in order that it may
have a cusp, must satisfy two conditions; it may therefore be made to satisfy seven
more conditions, or to have a seven-pointic intersection; and then the original curve
having an inflexion or threefold point at infinity, the asymptotic cubic will ipso facto
have the same point as an inflexion, or threefold point at infinity, and be thus a
cuspidal divergent parabola...
And writing z = 0 for the equation of the line infinity, the equation of the cubic is
of the form U = V + fjuz2s — 0...
For the Divergent Parabolas: the asymptotic aggregate is a semicubical
parabola; let q — 0 be the equation of the cuspidal tangent, p = 0 the equation of the
line joining the cusp with the inflexion at infinity, then the equation is pz + \q2z — 0...
the satellite line may disappear altogether (^ = 0), and the curve
thus coincide with the asymptotic semicubical parabola...
The Trident Curve, the
Divergent Parabolas, and the Cubical Parabola, form each a single genus...
The only other case where the satellite line 5 = 0 is an arbitrary line, admitting
therefore of a double series of positions, is that of the Hyperbolas; and the division
into groups constitutes an extensive and interesting theory, which is insufficiently
discussed by Pliicker; and it was with a view to the development of this theory
that my Memoir, On a Case of the Involution of two Cubic Curves, (ante, pp...