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

RECURRENTS (GUNTHER, 1873, 187 6)23

231

a result previously reached by Schlbmilch *through the form
STUDNICKA, F. J. (1874).
[Ueber die independente Darstellung der n-ten Derivation von
gebrochenen Functionen einer Verdinderlichen. Sitzungsb....
bohrn. Ges. d. Wiss. (Prag.), Jahrg. 1874, pp. 1-2.]
The expression obtai ned for the nth derivative of u/v is not
essentially different from Spottiswoode's of 1853, but is much less
simple, being in fact Spottiswoode's expression for the (n-i )t1
derivative of (VU' - UV') 1/V2.
WEIHRAUCH, K. (1874).
[Zur Determinantenlehre. Zeitschrift f. Math. U. Phys., xix. pp.
354-360.
When dealing with the number of terms in the final expansion of
zero-axial determinant (see below in chap xxi.),' Weihrauch
proves in two ways (pp. 357-358) that
(rn2 (rn-i)1 1I
(rn)3 (ni9 (m-2)1 m,
(m),. rn1, (rn- 2),,-2 (m-3),,- 3., 1n
His initial form of expression for the said number of terms is also
a recurrent.
We may note in passing that Weihrauch's result is readily
*,See Zeitschrift f. Math. u. Phys., iii. (1 858), Lit eraturs,., pp. 63-614; and for the
subject generally Giinther's Vernsischte Untersuchungen zur Ce~schichte d. math.
Wiss. (1876), pp. 93-135.