The theory of determinants in the historical order of development, by Sir Thomas Muir.

ORTHOGONANTS (CAYLEY, 1846)

305

the full substitution being
= 2L lll i 2Li~ 2L ~ 2L14
2L2 (2L22231)$~ 2L2 2L2
2L3l$ 2LL 2L2.= 3 + e2 + e)6+2I+
2L,, 2L,2 2L__ ___
X4 e41+ a42~4
We may add, that had the relation of the reverse substitution to
this not been already known it would have been evident from
the set of equations which here produce both. The result
reached is that the n2 coefficients a11,,, a,,,, for the transformation of rectangular co-ordinates can be expressed rationally in
terms of In(n-1) arbitrary quantities Ins satisfying the conditionls Irs = -s l) rr = 1 by forming the determinant 11,1122... L,,1,
or A say, and thereafter the adjugate determinant. IL2....
and takoing
2LIrg 2Lrj.
By way of illustration Cayley works out the cases where n= 3
and where n = 4. For n = 3 he begins with three quantities
v -/Lu
and obtains the substitution-coefficients
1~X2_ - 2 -_v2 2 (XxU + v) 2(vX- )
1~+ X2~+ IA2+ r2 1 + X2 + t2 -[- 12 1 + X2 + ~A2 ~ v2
2(Xx - v) 1+ -v2 X2 2 (MPv+X)
l~+X2-%s2+v2 l-+fX2 +jYv2 1 [-X2~+x2 +v2
2(vX + 0) 2(Jv - X) I + V2_-Xh2 2
1+X2 +A2 ~1v2 1+X2 + 2 v2 1~X2+ V 2+v2
remarking, in passing, on Rodrigues' introduction of them (but
on this point see Euler's memoir of 1770) and on their connection
M.D. I1. U