42 E. B. Wilson, 
The ratios +1:%2...:%, of these coordinates may be regarded as 
as homogeneous coordinates in a space of ~—1 dimensions. In 
particular that space may be taken as the space at infinity in the 
original space of 2 dimensions. As the coefficients in (82) can be 
arbitrary, the transformation (82’) is the general projective trans- 
formation in 2—1 dimensions. The correspondence between the 
two is not one to one: for all the equations in (82’) may be multi- 
plied by a constant. In particular the constant may be so chosen 
that the determinant of (82’), which is supposed not to vanish, may 
be + 1.1. Thus the correspondence may be considered to be be- 
tween unimodular strains and the collineations. 
In this correspondence any projective reflection in the (7—1)- 
dimensional space at infinity becomes an oblique reflection of the 
types here considered by merely passing spaces through the fixed 
spaces of the projective reflection and through the origin, and con- 
versely, any reflection in the spaces /t;, Rn; of the v-dirnensional 
space becomes a projective reflection in the plane at infinity and 
with the intersections of that plane and Ay, Ay—x as its fixed spaces. 
In the projective reflection the distinction between the reflections 
of types 0, ” or 1, w—1 or 2, m—2 or ... entirely disappears: there 
is nothing corresponding to reversal of direction, as only the ratios 
of the coordinates are considered. Moreover the Hamilton-Cayley 
equation of the matrix of the coefficients in a projective trans- 
formation may be written 
(83) 22 — 2, Q2-1 + Qo, Qn-~2 —... + Qn—2, 5 2? + Qn-1,52 
ait iy Hl 0, 
without any factors arising from the factor of proportionality which 
may effect the coordinates: for that factor enters into Q;; to the 
power & and into 2,y-; to the power n—&, and hence may be can- 
celed out. If the projective transformation may be resolved into 
— 

the product of two projective reflections the equation for 
” 
V | Qn 
must be reciprocal and the invariant numbers corresponding to a 
pair of reciprocal roots of the scalar equation must be equal. 
The connection with Smith’s work already referred to is inter- 
esting. If a quadratic form in the homogeneous variables in the 
plane at infinity is invariant, under any projective transformation 
of the variables, the same quadratic form must be invariant under 


1 The distinction between +1 and —1 may be disregarded except for 
questions of reality. 
