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

RECURRENTS (CROCCHI, 1879)

247

and that the alephs are the same functions of the c's as the c's are of
the alephs; for example,
C1 -= C, C = 1,
N2 = CI C2= 1 1
C2 C1, N2 1I'
3- = C]1 1. C3= - 1
C2 Ci 1, 2 2 1
C3 C2 C1 N 3 2 1 N
As has already been pointed out, however (Hist., ii. p. 218), all
these results are of less importance than the three sets of equations
from which they are derived, namely,
SI-C- = 0 i-1-C1 = 0~ s21-i1 = 0~
s2-s1C1+2c2= 0 2 2-C 1+C2= ~ 2+5s122 =2- 0
They are but illustrations of the general proposition that the set
of recurrence-formulae
a1P~ + a2Q1 = 00
blP2+ b2P1Q1 + b3Q2 = 0
c1P3 + c2P2Q1 +c3P1Q2 +C 4Q3= O
gives Pr as a recurrent in terms of Q,., Q._,,..., and at the
same time Q,. as a recurrent in terms of P,., P,-.1....; for
example,
P3 = - a2Ql a1 * a 1bc1, Q, = - ajP, a2 - a2b3c4.
|b3Q2 b2Q1 b1 bQP2 b2Pl b3
C43 C3Q2 2Q1 I 1P3 C2P2 C3P1