ORTHOGONAL GROUPS. 187 
These four sets of relations hold for the orthogonal group ; if 
we have given any one set of these relations we can deduce 
the other three. 
THEOREM 18. The orthogonal group is a special case of the 
eroup G;(K) when the equation of the conic K is in the canonical 
form 
De eye ae 0. 
222. The Second Canonical Form of G,(K). The quad- 
ratic function f can also be brought by linear transformation 
to the canonical form f=y*?— «wz. We may therefore set 
—i— anda ——gi— ws) Lhetrelations: A, (24) 
then reduce to 
a2 = a3, bs? = bibs, C,* =¢\C;, 2 dob. = a\b6; + 43b,, 
2 doc2=a)¢C;+ 43¢, — 1, 2 boto = b,c; + b3¢). 
ete usiaput | @)—a-s @,—97 ¢ — 0". se5—i0° >) we then: find 
b,=208, b,=2yd, b.=ad+Py, and (ad—@By)=1. The 
matrix M is now in terms of «, 3, y, 46, thus 
a2 2a;3 8? || 
M=l|ar as+ fy Bol ; 
l| 72 270 62 || 
and the determinant of Mis («3—y)’=1. The invariant 
conic has the equation y? = «2 and the invariant triangle of 7 
coincides with the triangle of reference. 
This form of 7 is connected in an interesting way with the 
one-dimensional transformation 
oy = aa! + 34! 
yi =a! + dy! ~ 
Taking the square of the first, the product of the first and 
second and the square of the second, we get after replacing 
CA DvAD Ny Dy, 2 .and ee yyy. 
Xv = a2u + 2ajsy + /32z, 
Yi =ara+ (40+ 87 )y + 752, (37) 
4 =u + 2yr dy + iz, 
with the condition y,? — 7,2, = y’ — xz. 
This shows that the conic 7’ = wz is invariant and that the 
transformations in the group G,(K) have a one-to-one corre- 
spondence with those of the group G, of one-dimensional 
