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

192 HISTORY OF THE THEORY OF DETERMINANTS

it may be written
I Yz451 I z12X45 I X12Y45
Y I Z346 I j and be denoted by A.
I Y56Z23 1 I Z56X23I I X56Y231
Multiplying rowwise by
IX12 Y72 Z12
X34 Y34 Z34
IIX45 Y45 Z45
we obtain
I. 1~~Ixi2Y45z341
A* I X12Y34Z45 I= I X34I61Z12, I IX34y6Z451
I x56Y23z121 I X56Y23Z341 I I I2345
whence
= I Xi f X34Y6ZI j
I x56Y23zJ21 I X56Y23Z451
X1Y2Z6 I x~y3Z4 I -IX3Y4/Z.5 X1Y4Z6 I
I -X I x XyYz3 I* x5y~z6 X 4Yz,56 I1 X2Y3z6 iX I
Y lx1yY.x y3z 4 I.1X2Y3Z51 1X4y5z6 1
- I 1y2z3 I X1y4z6 I I X2Y5z6 I xI X3Y4z I5
The vanishing of the left-hand member is in co-ordinate geometry
the expression of Pascal's theorem, and the vanishing of the righthand member is the analogous equivalent of a theorem of Pappus
or I esargues. The proof of the identity of the two members may
be interestingly compared with Cayley's procedure of 1843 (Hist.,
ii. pp. 10-13).
Using other multipliers than I j12Y34Z5, for example
Ix12Y34z611 I X12Y2 I.. 5
Hunyady obtains the requisite variety of similar expressions for A.
it has also to be noted that by taking a multiplier in which no
suffix is repeated an equally interesting result is reached: for
example
A I X12Y314Z561 = 125 * 126d- 134 ~ 234V- 356 ~ 456
- 123 ~ 124 1 56 *256 *345 *346,
where 125, 126.... stand for I x1y2z5 1' IY2Z