216
ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES.
[447
where (P', Q', R', P/, Q x , P/) are linear functions of (x, y, z'), we have
that is to X', Y', Z', where X' = 0, Y' = 0, Z' = 0 are conics each passing through the
same three points in the second plane.
53. The lineo-linear transformation is thus the same thing as the quadric trans
formation. It is, moreover, clear that the equations must, by linear transformations on
the two sets of variables respectively, and by linear combination of the two equations,
be reducible into forms giving the before-mentioned values of x : y : z and x' : y' : z'
respectively. Thus, in the general case, where in each plane the three points are
distinct points, the lineo-linear equations will be reducible to
xx' — yy' = 0, XX — zz' — 0 ;
in the case where B, G in the first plane, and B', C' in the second plane respectively
coincide, the forms will be
XX - yy' = 0, yz -y'z = <0\
and in the case where A, B, C in the first plane, and A', B r , C' in the second plane
respectively coincide, the forms will be
xy — yx = 0, xz — yy + zx = 0.
The determination of the actual formulse for these reductions would, it is probable,
give rise to investigations of considerable interest.
The General Rational Transformation between Two Planes {resumed).
54. Consider, as above, the first plane or figure with a principal system (a 1} a... a n _ x ),
and the second plane or figure with a principal system (a/, a 2 "... aV-i). To a line in
the second plane there corresponds in the first plane a curve of the order n passing
1 time through each of the points a u 2 times through each of the points a 2 , 3 times
through each of the points a 3 , &c.; or, as we may write this:
First figure.
Second figure.
Points oíj a 2 a 3 ... a,
Points a/ a/ a 3 '... a
1 23 n-1
123 n-1 A
h curve order n
1 : 3 n-1
0 0 0 n — 1 \
000 n—1 ui
L curve order 1
0
viz., the l’s denote the number of times which the curve of the order n passes through
the several points vl x respectively; the 2’s the number of times which the curve passes
through the several points a 2 respectively; and so on.