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

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

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Date Oct 27, 2004

That is, the system
= o, iif=o, jy=
is made up of the cusp system, and of the system (L = 0, itf = 0, JV = 0, A — 0, #?2 = 0);
or since A — 0 is a consequence of the other equations, the second system is
0, M=0, N=
Consider now the curve XTJ + fiU' + vJJ'f — 0, which will have a cusp if the ratios
X : fju : v are properly determined...
The investigation (which
is a development of two short papers already published in the Philosophical Magazine)^)
was undertaken in order to applying it to the explanation and discussion of Pliicker's
Classification of Curves of the Third Order; but such application will properly be made
in a separate memoir, On the Classification of Cubic Curves, and it has also appeared
to me convenient to give therein the discussion of the geometrical forms of certain
loci which present themselves in the present memoir...
I consider the involution
xyz + k (x + y + zf (Xx -f /juy + vz) = 0,
where x = 0, y=0, z = 0, x + y + z=0 may be considered as representing any four lines
no three of which meet in a point, and Xx+fjuy+vz — O, as representing any fifth line
whatever: k is a variable parameter...
We see that 8 is determined by a cubic equation, and that the ratios x : y : z
and the parameter h are rational functions of 8...
It is easy to see that the curve is touched by the lines x = 0, 2/ = 0, z =
at their intersections with the lines y — z = 0, z — x = 0, x — y = 0 respectively, or (what
is the same thing) in the points @, 1, 1), A, 0, 1), A, 1, 0) respectively...
We have
1 -2
Xx + iiy 4- vz = X + F +
and the equation therefore is
n l 4A( )
where II denotes the product of the three factors obtained by writing X, fi, v
successively in the place of X...
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...
The equation of the tangents to the harmonic conic at its intersection with
the line x + y + z = 0 (which tangents meet of course in the last-mentioned pole, that
is in a point of the twofold centre conic) is found to be
2fgh (x + y + zJ + ? Bfyz -\-2gzx + 2hxy) = 0 ;
if for shortness
2- 2gh - 2hf- 2fg,
or what is the same thing
64...
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 above-mentioned work of Newton contains, under the heading "Genesis Curva-
rum per Umbras," the remarkable theorem that the curves of the third order may all
of them be considered as the shadows of the five Divergent Parabolas; I reserve for
a separate Memoir the whole series of considerations to which this theorem gives rise...
For a Parabolic Hyperbola there is at the onefold point at infinity a tangent,
which is an asymptote...
[350
a twofold point at infinity it is at least four-pointic; and at a threefold point at
infinity it is at least six-pointic...
for a
given form of the asymptotic aggregate V = 0, and corresponding to each characteristically
distinct position in relation thereto of the satellite line s = 0, we have a Group...
I proceed now to explain the classification of Newton so far as relates to the
division into genera, and the classification of Pllicker so far as relates to the divisions
immediately superior to the groups...