The collected mathematical papers of Arthur Cayley.

387]

LONDON MATHEMATICAL SOCIETY.

23

Analytically, Cremona's transformation is obtained by assuming the reciprocals of
X2, y2, Z2 to be proportional to linear functions of the reciprocals of xa, y,, z,-.(of course,
this being so, the reciprocals of x1, yi, z1 will be proportional to linear functions of
the reciprocals of x2, Y2, z2). Solving this under the theory as above explained, write
1 ( ac b cp
X2 | TV1 tl + l P
1 I d e f:-> =:.-+- +- + =: Q
y2 I 1?1 Z1 1
1 q h i I~:. + - i: R
Z2 ) x. X yi zy )
if
P1 = ayz- + bz6x1 + cxlyi,
Q1 = dy, z + ez xi +f-z'iyi,
R, = gylz + hzxi + i, y,.
Hence
x2: y2: Z=QBR,: RP1 P Ql.
QR, = 0, &c., are quartics, or generally aQRJ1.+/3RJP 7yPQi =0- is a quartic, having
three double points (y, = 0, z = 0), (z, =0, x = 0), (x, = 0, y, = 0), and having besides the
three points which are the remaining points of intersection of the conies (Q, =0, R1 = 0),
(RI =0, P = 0), (P1 =0, Q1 = 0) respectively; viz., these last are the points
-: -: ei - hf: fg - id: dh - ge, &c. &c.
XI Yl Zl
The double and simple points are fixed points (that is, independent of a, 13, y), and the
formulae come under Cremona's theory. It is, however, necessary to show that if the
points 4', 5', 6' are in a line, the points 1', 2', 3' are also in a line. This may be
done as follows:
Let there be three planes A, B, C, and let the points of the first two correspond
by ordinary triangular inversion in respect of the triangle a, on the plane A, and 813
on the plane B. Let also the planes B, C correspond by ordinary triangular inversion
in respect of the triangle /3 on the plane B, and 72 on the plane C. Then the correspondence between A and C is the one considered, the points 1 23 forming the
triangle a, and the points 4 5 6 forming the triangle 72. The points 4'5'6' and 1'2'3'
in the planes A, C respectively correspond to the triangles 1, 132; and the conditions
that 4', 5', 6' shall be in a line and that 1', 2', 3' shall be in a line, are the same
condition, namely, that the triangles /31, 3 shall be inscribed in the same conic.
Analogous properties must apparently belong to Cremona's other transformations, and
the investigation of them will form an interesting part of the theory.
It is important, also, to notice the relation of the transformation to Hesse's "Uebertragungsprincip," Crelle, tom. LXVI. p. 15, which establishes a correspondence between
the points of a plane and the point-pairs on a line. If Ax2 + 2Bxy + Cy2= 0 is the
equation of a point-pair the coordinates in the plane are taken by Hesse directly,
but in the present Paper inversely proportional to A, B, C.