299 
1910-11.] The Theory of Wronskians. 
by considering the particular case where u = <J> = e x , The final result 
thus is 
U(f>° 
d(ucf>° ) . . . 
. d m (ucl> 0 ) 
u<h l 
d(u<fi l ) . . 
. d m (u<t> 1 ) 
= 1 ! 2 ! 3 ! . 
. . m ! u ni+1 (dcf))^ m+1) , 
u<f> m 
d(u<f> mS ) . . 
. d m (ucf> m ) 
which on putting u — 1 becomes Wronski’s result of the year 1816, and 
therefore a case of Schweins’ generalisation of 1825. 
A[badie, T.] (1852). 
[Sur la differentiation des fonctions de fonctions : series de Burmann, 
de Lagrange, de Wronski. Nouv. Annates de Math., xi. pp. 
376-383: or French translation of Brioschi’s Teorica dei Deter- 
minanti, pp. 182-193.] 
By a method similar to Prouhet’s, namely, by solving a set of equations 
in two different ways, Abadie obtains 
df~ 
dh 
”"1 V 
n + h 
1.2.3 ... n 
where 6 and D r (f> s stand for 
2, [ ± . B 2 <£ 2 . D n F] 
T [ ± D 1 ^ . D 2 <£ 2 .”.... D n "y _1 . D n (f> n ] 
tjx + h) -jfrfr) *_ U( x) \° 
h dx r I W j 
respectively. Thence, by equating coefficients of D n F, Wronski’s result 
2 [ ± D !<£ . D 2 <£ 2 D n cf> n ] = 1 ! 2 ! 3 ! . . . w !{ } in{n+1) 
is reached ; and this in its turn is then used to simplify the result which 
has just originated it, another of Wronski’s formulae being thus arrived at. 
Brioschi, F. (1855). 
[Sur une propriety d’un determinant fonctionnel. Quart. Journ. of 
Math., i. pp. 365-367 : or Opere Mat., v. pp. 389-392]. 
In later phraseology the property in question is that if the Wronskian, 
W say, of y v y 2 , . . . y n be a known function of the independent variable 
x, then any one of the y’s, y r say, can be expressed in terms of the others 
and W. Denoting the s th differential-quotient of y r by yf we have of 
course 
V\ 
y 2 
• • Vn 
W = 
y? 
y { f 
• • 2 /? 
yin-D 
yr 11 • 
v (n-l) 
• • i)n 
