387] 
LONDON MATHEMATICAL SOCIETY. 
23 
Analytically, Cremona’s transformation is obtained by assuming the reciprocals of 
#2* ])■!■> z -2 t° be proportional to linear functions of the reciprocals of x x , y x , z x —(of course, 
this being so, the reciprocals of x x , y x , z x will be proportional to linear functions of 
the reciprocals of x 2> y-2, z.f). Solving this under the theory as above explained, write 
11 
( 
a 
b 
c 
— 
— 
H— 
+ - 
0C2 
Vi 
z x 
l 
d 
, e 
f 
— 
- = . 
— 
_] 
+ y - 
- = ■[ 
1/2 
X x 
2/i 
1 
. /7 
h 
i 
— 
4— 
+ - 
l 
z. l} 
’ X x 
Vi 
ZiJ 
if 
P x = ay x z x + bz x x x + cx x y x , 
Qi = dy x z x + ez x x x +fx x y X) 
Pi = gyiZi + hz x x x + ix x y x . 
Hence 
x 2 : y, : z, = Q x R x : R X P X : P X Q X . 
Q x R x = 0, &c., are quartics, or generally uQ x R x + ¡3R X P X + yP x Q x = 0 is a quartic, having 
three double points (y x = 0, z x — 0), (z x = 0, x x — 0), (x x = 0, y x = 0), and having besides the 
three points which are the remaining points of intersection of the conics (Q x = 0, R x = 0), 
(R x = 0, P x = 0), (Pj =0, Q x = 0) respectively; viz., these last are the points 
— : — : — = ei — hf : fa — id : dli — qe, &c. &c. 
X x y x Z x J ja v 
The double and simple points are fixed points (that is, independent of a, ¡3, y), and the 
formulae come under Cremona’s theory. It is, however, necessary to show that if the 
points 4', o', 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, G, and let the points of the first two correspond 
by ordinary triangular inversion in respect of the triangle a x on the plane A, and /3 X 
on the plane B. Let also the planes B, C correspond by ordinary triangular inversion 
in respect of the triangle ¡3. 2 on the plane B, and y 2 on the plane C. Then the corre 
spondence between A and C is the one considered, the points 12 3 forming the 
triangle ol x and the points 4 5 6 forming the triangle y 2 . The points 4'5 / 6' and 1'2'3' 
in the planes A, G respectively correspond to the triangles /S 1} ¡3,; 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 ¡3 X , /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 “ Ueber- 
tragungsprincip,” Grelle, tom. lxvi. p. 15, which establishes a correspondence between 
the points of a plane and the point-pairs on a line. If Ax 2 + ZBxy + Cy 2 = 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, G.