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

ORTHOGONANTS (CAYLEY, 1846)

303

considered without any regard to the mode in which they may
have originated.
CAYLEY, A. (1846).
[Sur quelques proprietes des determinants gauches. Crelle's
Journ., xxxii. pp. 119-123; or Collected Math. Papers, i.
pp. 332-336.]
There is clear evidence that Rodrigues' paper of 1840 made
a strong impression upon Cayley. In a paper published in 1843*
he introduces his subject by speaking of Rodrigues as having
"given some very elegant formulae for determining the position
of two sets of rectangular axes with respect to each other,
employing rational functions of three quantities only"; and he
proceeds at once to demonstrate these formulae as a necessary
preliminary to the essential part of his paper. In another
paper published in 1845,t the first part of which deals with a
quaternion identity, he makes the important observation that a
set of nine coefficients which occur in the identity is precisely
the same as the set of nine given in Rodrigues' transformation;
and he adds, " It would be an interesting question to account a
priori for the appearance of these coefficients here." We are
thus not wholly unprepared for a communication from Cayley
himself on the subject of the construction of a linear substitution
for the transformation of X2+ x+... into C+ 2 +.... The
following is his procedure, four variables being used in place of
his n.
With unity and any six quantities whatever there is first
formed the square array
1 l12 13 14 11 112 '113 114
-112 1 123 124 or say 121 2 123 124
12 23 24 1 22 23 24
113 "-123 1 134 131 132 133 134
-1l4 -124 -134 1, 141 142 143 144
* Cayley, A., "On the motion of rotation of a solid body." Cambridge Math.
Journ., iii. pp. 224-232; or Collected Math. Papers, i. pp. 28-35.
t Cayley, A., "On certain results relating to quaternions." Philos. Magazine,
xxvi. pp. 141-145; or Collected Mlath. Papers, i. pp. 123-126.