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

RECUIRRENTS (LUCAS, 1876, 187 7) 2

239

and thence obtain
f(n-4l)-f(1) = aoSl.+aISr.-i+ +a-O
As a matter of fact, however. the forms of f(n) which he finds useful
are not very varied, being, in addition to Wr,
the second of which leads to the recurrence-formula
(2vn+1)2 —1 = (2S+1)r-(2S —1)`,
and thence to
22r+1. 35... (2r-+-)S.,.,

2,I-I?-+1
( - W

(2ii+1)1
(2n-j1)
(2+1)5

1
1
1

('3)2
(5)2

(5)4....

(2n +1)21'1l 1 (2?-1')2 (2ir-l1)4. 2..2.'(2nb+ 1)2~'l 1 (2r+1)2 (2r+1)4....(i+)1 -Since with Lucas the Bernoulli numbers, Pc, /31 132....are
our B,. being thus equal to (- 1 )_1321., hie is enabled to write
Bernoulli's relation in the form
r (r-1 -1"')= (O3+V)Y" - Or,
and thus obtain such recurrence-formulae as
O + 0 = q( r-1

STUDNICKA, F. J. (1877).
[Ueber -die independente Darstellung der n-ten Derivation einer
Potenz, deren Basis und Exponent verschiedene Funktionen
einer Variablen bilden. Sitzungsb.... Ges. d. Wiss. (Prag),
pp. 3068-373.]
[Weitere Beitriige zur Differentialrechnung. Sitzungsb.... Ges.
d. Wiss. (Prag), pp. 393-399.]
In the first of these papers the n'11 differential-quotient of Vi
obtained in the form
U V. 71',