








44 E. B. Wilson, 
Yr 
ath Se = 
Lo s 0 
| Loe ate 
Ga 0 O 
Ys OHO 
3 0% 2/0 






as seen above are invariant. If a root is +1 without shearing terms 
the invariant terms of the second degree are 




8 
and ‘the determinant does not vanish; and similarly in case of the root 
—1. If there are shearing terms corresponding to +1, the trans- 
formation may be written 
Li Cee oe Koa hy os —— Ke a See 
and are invariant quadratic terms are 







|) Gai De | Dae Lee 
Nir ap ema Mle 0 
2 ee SC Sal POO 
in pease O | O 0 
0 QS? 0 0 


The determinant vanishes if 72> 3, and similarly for the case of 
a root —1. If these sets of quadratic terms corresponding to the 
various roots with or without shearing terms be arranged along 
the main diagonal of a matrix of order , and if all the other spaces 
be filled with zeros, the result is a quadratic form in variables 
which is invariant and which certainly has a non-vanishing deter- 
minant, unless +-1 or —1 is a root with as many as three consec- 
utive shearing terms. 
It therefore appears that there are linear transformations in more 
than three variables which are compounded of two reflections and 
which leave no quadratic surface (with non-vanishing determinant) 
invariant. In other words, the converse of Smith’s theorem is not 
always, although it is generally, true. The simplest example of the 
failure of the converse is in the collineations of three dimensions. 
oS eet Eee 
a tad 
