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

CHAPTER VI.
RECURRENTS, FROM 1841 TO 1860.
LIKE Wronskians, and for the same reason, recurrents were dealt
with in our first volume among " Miscellaneous Special Forms":
their previous history is thus to be found under Wronski 1812,
Scherk 1825, and Schweins 1825 in the chapter so entitled.
(History, i. pp. 472-474, 478-481.)
The name is quite recent, having apparently been first used by
E. Pascal in 1907 in a paper published in the Rendiconti...
1st. Lombardo (2), xl. pp. 293-305.
SPOTTISWOODE, W. (1853, August).
[Elementary theorems relating to determinants. Second edition,... Crelle's Journ., li. pp. 209-271, 328-381.]
In the last chapter or section (~ xi.), which is headed
"Miscellaneous Instances of Determinants," Spottiswoode gives
(pp. 373-374) an expression for the lth differential-quotient of
u/v in terms of the nth and lower differential-quotients of ui
and of v. The first four cases are
=. v v' v-2, v v' v-,
(V).=. V 3?V-4, --.,. 4v' Vv~'3,
\\Zt?/ /'w/
v Uv V VI
v 2v' 3". v 3v' 6v"
v v' v" v'/. v 2v' 3v" 4v"'
U i' Ui" U'" V V' V/ " V iv
Uq U/' '," ui/n" iv"