296 Proceedings of the Koyal Society of Edinburgh. [Sess. 
XVI. — The Theory of Wronskians in the Historical Order of 
Development up to 1860. By Thomas Muir, LL.D. 
(MS. received June 13, 1910. Read July 4, 1910.) 
The previous history of Wronskians being not at all lengthy, was in- 
cluded in the chapter on “ Miscellaneous Special Forms ” {History, i., 
chap, xvi.), and is to be found there under Wronski 1812, Wronski 1815, 
Wronski 1816-17, and Schweins 1825 (pp. 472-478, 482-485). 
The name dates only from 1882, being first suggested on p. 224 of my 
Text-book on Determinants.* 
Liouville, J. (1838). 
[Note sur la theorie de la variation des constantes arbitrages. Journ. 
{de Liouville) de Math., iii. pp. 342-347.] 
The Wronskian which incidentally appears here is of a special kind, 
namely, that in which the originating functions are in the so-called relation 
of being first differential-quotients of one and the same function, for 
example, in later notation, 
dx dx 
05 0C 
d 2 x d 2 x 
dbdt debt 
d 3 x d 3 x 
dbdt 2 dedt 2 . 
dx 
da 
d 2 x 
dadt 
d 3 x 
dadt 2 
It is worthy of note also that the expression for the differential-quotient 
of this with respect to t is obtained in the form which accords with the case 
of Schweins’ theorem of 1825 {Hist., i. p. 484). 
Z d || |zd)“Aj . {Zdf +1 A i .... (Z dY +n - 1 k,^j 
= || (Z dfA l . (Zrf)“+'A 2 .... (Z^)“+”- 2 A„_ 1 . (Ztf)»+”A„) 
where Z = 1 and a = 0. 
* Muir, Th. A Treatise on the Theory of Determinants, . . . viii + 240 pp., London. 
