100 
ON WRONSKIS THEOREM. 
[574 
with the determinants formed out of the matrix 
Ф / »// »/// #'//' 
(4> 2 )', (* 2 )"> (4>T> (4> 2 )"" 
(tf> 3 )', (tf> 3 )", (4> 3 )"'> W"" 
The series of equalities may be presented as follows, writing as above A to denote the 
function 
*'+f 
02 
+ 273 $ + • • • > 
1 1 n 
H 
P-e- 
II 
N 
1 " 1 I 
e, l 
1 
A 2 ~ f 8 
1’ 
1 +1 
hj\ \e , i 
i 
A 3 ' <f)' 6 
</>', <!>", </>'" 
‘1.1.2’ 
m, w, w" 
1 -1 
if, if, i# . 
1 
1 
ll 
-eq 
© | 
<f>' , 
»// AJ” 
> 9 y 
iff// 
9 
■ i 
m, (<t>r, w, 
wr 
m, (•#•■)"> (■#>’)"', 
m"' 
&c., 
where in each case the function on the left hand is to be expanded only as far as 
the power of 6 which is contained in the determinant : the numerical coefficients in 
the top-lines of the several determinants are the reciprocals of 
n(n— 1) ... 2.1, n(n— 1) ... 2, n(n — 1), n, 1, 
where n is the index of the highest power of 0. The demonstration of Wronski’s 
theorem therefore ultimately depends on the establishment of the foregoing equalities 
As a verification, in the fourth formula, write <£ = e a (a = 0), we have 
(e*-l) ° r (l + ^ + i<P+ *№+...)• * 
where the 
right hand is 
№ 
1, 1, 
2, 4, 
3, 9, 
№ 1 
1, 1 
8, 16 
27, 81 
= - ^ (- 1.12 + 16.72 - ¿0M32 + ¿0 3 • 72) 
= 1 _ 20 + #0*- 6 3 ,