[447
447] ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES.
203
26
■2
that
ining
f the
lively.
cients
f the
as if,
on of
point constituted by an intersection of the curves r, n — r. {Observe that the two curves
have only this single intersection; viz., the remaining r (n — r) — 1 intersections are
at points a/ + a 3 '... + a'of the principal system of the second plane.} We have thus,
in the second plane, a series of curves, each of them having a new double point;
viz., these are the several curves which correspond to the lines through a r in the first
figure. Each of the curves is a fixed curve r together with a variable curve n — r.
The new double point is an intersection of the two curves; that is, it is a variable
point on the curve r. The locus of the new double point is thus the curve r; therefore
the curve r is part of the Jacobian of the reseau of the second plane. Since each
point a r gives a curve r, the curves in question form an aggregate curve of the order
ofj + 2a 2 ...+ (n — 1) a M _ 1} = 3n — 3 ; viz., this is the order of the Jacobian; or, as stated,
the curves r (that is, the principal counter-system of the second plane) constitute the
Jacobian of the reseau of this plane.
41. The numerical systems (a 1} c 2 ...a n ^ 1 ) and (a/, a/... are each of them a
solution of the same two indeterminate equations
Xr 2 a r — n 2 — 1, Xra r = 3 n — 3,
but not every solution .of these equations is admissible; for instance, if r > \n, then
a. r is = 0 or 1, for a r =2 would imply a curve of the order n with two ?’-tuple points,
and the line joining these would meet the curve in more than r points; similarly,
r>\n, Or is =4 at most, for a r — o would imply a curve of the order n with five
r-tuple points, and the conic through these would meet the curve in more than 2n
points; and there are of course other like restrictions. The different admissible systems
up to n = 10 are tabulated in Cremona’s Memoir; and he has also given systems
belonging to certain specified forms of n : these results are as follows:
it the
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ion is
, new
igular
being
urves.
lecond
it a r ,
cursal
igular
;s are
louble
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r-2) +