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

,304 HISTORY OF THE THEORY OF DETERMINANTS

remembering that [,. = 1 and 4, = - [,. Then taking a new set of
four variables 01, 02, 03, 04, and using for their coefficients the
-quantities in the square array, firstly as disposed in rows, and
secondly as disposed in columns, he puts
11101 + 11202 + 11303 + 11404 = l
2101 + 12202 + 12303 + 12404 =
[3101 + 13202 + 13303 + 13404 =
[4101 + 14202 + 14303 + 14404 =.and
1101 + [2102 + [3103 + [4104 =
11201 + [2202 ~ [3203 + [4204 = $2
11301 + [2302 + [3303 + [4304 = $3
11401 + 12402 + [3403 + [a04 = $4
thereby ensuring that
xi 2+X 221X 32 =12~$22~$33...
Solving the two sets of eqnations separately for each of the 0's
and equating the results, he next obtains
L11x1 + L21x2 + L31xX + L4,x4 = L11 $ + L12C2 + LI3$ + L14&$
L12xj + L22x2 + L32x3 + L42x4 = L21$1 + L22$2 + L233 + L24$4
L13x1 + L23x2 + L 33x + L43x4 = L31$ + L32$2 + L33$3 + L 34
L,4x, + L24x2 + L34X3 + L44X4 = L41$el +L42$2 ~ L43$3 + L44$4
where L,., is nsed for the cofactor of i, in the determinant (A say)
of the initial array. It only then remains to obtain from this
the x's in terms of the e's, or the e's in terms of the x's. This
Cayley does by using as multipliers, in the former case the
elements of any row of the original array, and in the latter case
the elements of any column, Thus, multiplying by I11, I12' I13' t1
respectively and adding, he obtains
Ax1 = (211L11 - A)$e + 2[11L12$ + 2[11L1X3$ 211AL14$4