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

338 HISTORY OF THE THEORY OF DETERMINANTS

a b c d ex a b c d Ax
a b c dx+e a. a c A
p q rx +s or.. p q B
p q r sx pq r Bx
p q 8 p r s Bx,
a b c d e Ax2
*a b c d e Ax
ac b e d A
_p*pqvs Bx
*p q r s. BX2
p q v s Bx3o
where, be it remarked, the ultimate forms, namely, those explicitly
involving A and B, are Cayley's of 1848.
Cayley's relation between any three consecutive 'simplified
remainders' is next given (pp. 140-142), the proof arising quite
naturally and being mainly dependent on the equality a2lca. = Dr-1 -Thus, taking the equations that indicate the nature of the divisionprocess, namely,
A = QjB -R
B Q21R,-R2
R1= Q3R.2-R3
R2 Q4R3-R4
and substituting pr/ar for R,., we obtain
a, A a1QiB - pi
a1a2 B = a2Q2 'P1- a1.p2
(IaasPi a C~la3Q3'p2 - '1a2p3
a3a4.p2 = a2a4Q4.p3 - a2a3.P4
In this way there results the general equation
ct,1-a1C Pr -2 a..2a.Q.. AP-i - ar.i,.a2 r. Pr'
and thence
D2 a,-aQ, 2 'i
Iri = ar.2r, 7Pr-i- D r9*Pr,
showing that Pr-2 and p,. have different signs for any value of x that,
makes pA-i vanish.