[574 
574] ON WRONSKl’s THEOREM. 99 
• &c., 
This must be equivalent to Wronski’s theorem; it is in a very different, and, I think, 
a preferable form; but the results obtained from the comparison are very interesting, 
and I proceed to make this comparison. 
Taking the foregoing coefficient (f/ 3 this should be equal to Wronski’s term 
1 1_ 0', (0 2 )' , f 3 F' ; 
1.1.2 ^ ^ s y/ > (pFJ 
<T> W> (PF'T 
or, what is the same thing, the determinant should be 
= i-i.2 
3) " 
j +&C. 
= 1 • 1 • \pr (¿)" + 2 (pry (¿)' + (fly ¿}, 
that is, the values of 
should be 
-VW-PW* PW"-<r(Pr 
ng ultimately put = a; 
respectively. Or, what is the same thing, if 
/00 2 \3 A 0 + AJ + j A.6 + ..., 
lis being ultimately put 
then the last mentioned functions should be 
1.1.2<£'«A, 1.1.20' 6 2A lt 1.1.2(f)' 6 A 2 . 
We have 
„ i , 3 <f>" A </>'" , 30" 2 
A °~ 2 <*>'<’ s f 4 f ’ 
or the identities are 
as a function of a only 
iifferentiations in respect 
differentiation in regard 
20' 3 = 0' (0 2 )" - 0" (0 2 )' , = 0' (200" + 20' 2 ) - 0". 200', 
- 60V 2 = 0'" (0 2 )' - 0' (0 2 )'", = 0"'. 200' - 0' (200" + 60'0"), 
+ 60" 2 0' - 20'" 0' 2 = 0" (0 2 )'" - 0"' (0 2 )", = 0" (200'" + 60'0") - 0'" (200" + 20' 2 ), 
which is right. And in like manner to verify the coefficient of X 4 , we should have 
to compare the first four terms of the expansion of 
1 
• 4-J +&c. 
13—2