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

346 HISTORY OF THE THEORY OF DETERMINANTS

we find the condensation-results

(ao,..., a.)1
(ao,.., a5)
7 (bo,... b4)3
(a0, I.., a )3
(bo..,b4)4

- a S

aos1 as82 + als, cta3 +... (S4 +..
aoS2 caoS+als9 as4+ -.... t8+...

s0 81 a0s2 CtS03 + a1s2 acs4 +as3 + a2s2
= ao4 81 S2 a083 CO4 + a1ls3 CaSg + a184 + a2S3
82 83 a(o4 aC0S5 + a14 Ct0s6 + a-1 + (t2-84

ISI
8- 3 S
i 2, S3

S1 82 C083 084 + a9183
82 83 a084 0t85 + a,84
83 84 COS85 a086 + a185
S4 S5 C0O86 (to87 +al86 1

((a..., a55)4
I (bo,..., b4)5

g8 Si
81 82
= ao9 S2 83
S3 84
84 85

82 S3 84
83 84 85
S4 85 S6
85 86 87
86 87 SS

all in agreement with Cayley's original result of 1846 (Hist., ii.
pp. 162-164). Taking the second of these for proof, we multiply
unity columnwise by the given bigradient array, obtaining

1

1. -so -81 i.
'. 1 o.... ' 1 I. I1

a1 a2
a0 a1
bo
bo b,
bl b2

Ct3
a2
bl
b2
b3

C4
a3
b2
b3

a5
a4
b3
b4

ao t1 a2 (a3 4 Ct5, ao a a2 Ct3 a
=.. bo 6bl b2 b3
*. aoS aoS2 + a acOs3+al 2s a cs4...
~., %s2 aoS3~ als,, a084 + Cals3 + a282 %OS9 +
ao90 aoS1 + also a82 + + a1,s + a2s, c.3 +.
= ao2 a0o1 -2 + a11 083 +- als2 + 2s1 aOS4 +
ao0.2 ao03 + a- 2 a4O' + ass + a2s a0s5 +.
and thence the final form desired.

ct5
a5
t4
aC +
^4 |
a0 +...
o84 +...
(to8 + * * ~ I
5 086 +...