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

CONTINUANTS (WORPITZKY, 1865)

441~

shown to be equal to N,81.N8~51, and -N1;,-2N~2,,n respectively,
and he thus reaches the result
N N1,,:- N8~1,n + a Nk,s2N,~2,
already obtained in a different way by SchMifii.
Lastly, taking a determinant of the same form as N.,, but
having
-a,-,ICsl '1-CC;1, -a -aI CL -a7,+,+1,..., - a,-,, -a
for its minor diagonal of a's, he obtains for it by isolating the
first a5 the expression
NS,'+1 Nk,j + a1,Ns,,~2Nk+ln,
and by isolating the second a,
Ns,kNNk+l, + akN5,,~1 N7,~2,;
and thus deduces
NTk,n Nk5+,s - Nks NkS-I,n - a(N5~+l,nNN5+2,s - N7,.+lS Nkf2,n),
It is then noted that the bracketed expression on the right differs
from the expression on the left merely in having k +1 in place of
k; so that there results
Nk~nNkSl,s - N5,sN,+1,,, = (- 1)2aaaa+l(Nk+2, nTk+,s - Nk+2,s Nk+3,n)
(1S-k+1 akak,,,, a,+, N,+3,,
This also, it will be seen, is connected with a result of Schlafii's:
for putting s = - I we have *
N5 -,n-l = Yk ( a -l) aa, a,
Nk+1,wlL Nk,n 7,azfl ''al
which becomes identical with Schhifii's last proposition on transposing the two rows of the determinant and (what is equally
immaterial) putting lo = 1.
* In giving to N+f,,s, Ns+2,s, Ns+3,s the values 1, 1, 0 which are necessitated by
assuming the generality of the recursion-formula
Nk,qz = N71~s,?z ~ akNs+s,n,
Worpitzky forgets to note that in these eases the proposition Nk,,, = N,,,k, used
by him in the demonstration, does not hold.