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 26, 2004

Cites: In virtue of the general property, the first-mentioned two centres must be considered as
lying on the line which is the polar of the onefold critic centre in regard to either of
the conies...
But this system of four equations contains not only the cusp system, but the
system made of the three linear equations and the equation x2 = 0...
In the case in question of a twofold and one-with-twofold value of k, the line
\x + fiy + vz = 0, or say the satellite line, envelopes a curve which might be termed the
twofold and one-with-twofold envelope, but which is spoken of simply as the envelope...
if X = 0, Y = 0, Z = 0 are the equations of the sides of the triangle formed by
the critic centres, then the equations of the tangents at the three critic centres
respectively are of the form
AX"
...
Writing the equations under the form
_ — xJryJrz_x—
xyz Xx fiy vz 0'
where 8 is an auxiliary parameter to be determined, we find
349] ON A CASE OF THE INVOLUTION OF CUBTC CURVES...
y — 0, z = 0
of the triangle are inflexions on the curve; and that the tangents at these points are
respectively
= 0, — 4# 4- 5y — 4# = 0, — 4<x — 4<y 4- 5z = 0...
the twofold centre or cusp, and
the point where the tangent at the cusp is met by the other tangent (that is the
tangent not passing through the cusp) at the one-with-twofold centre...
considering the pencil of lines through A, the locus
of the fourth harmonic of the point in which a line of the pencil meets T, in
regard to the two points in which the same line meets the conic ©, is a conic
which is the harmoconic in question...
Let wly ylt zx be the coordinates of a critic centre, then the equation of the
polar in regard to the twofold centre conic is
and the equation of the conic through the five points is
y
z
and these equations together determine the remaining two critic centres...
the
polar of the first centre in regard to the twofold centre conic meets the line in one
point, and the conic in two points; of these one is the harmonic of the point on
the line in regard to the twofold centre conic; this point on the conic, and the
point on the line, are the other two centres...
* 4 * 4?
which points are therefore the critic centres for the line 3x + y + z = 0>
The last-mentioned line, it is clear, is one of the system of three lines
82...
Consider for a moment the case v = /ju-\-e, where e is ultimately =0, the
equation in 6 is
1 1 1 2 A
j — o •
then if a root is 6 = — /jl + A e, we have
"T"
so that, e being indefinitely small, we have
= 0 or J...
The two additional species are, I believe, first mentioned in Murdoch's Genesis Gurvarum per Umbras
A746), but one of them is there ascribed to Cramer...
For a Divergent Parabola there is not any asymptote or asymptotic conic ; but
we may consider an asymptotic cubic, viz...
Finally, when as in the
Trident Curve the intersection is nine-pointic, the line s — 0 has with the curve a
;iven threefold intersection at infinity ; that is, it coincides with the line infinity, z = 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...
And, conversely, when the satellite line is parallel to an
asymptote such asymptote is an osculating one, and when the satellite line is at infinity
the three asymptotes are osculating...