301 
1910-11.] 
The Theory of Wronskians. 
given earlier in the paper (p. 293), namely, by evolving the left-hand side 
from the right-hand side after substituting for the differences in the latter 
their equivalents obtainable from the identity 
^%(m) = + - if k (m + i- 1) + ^ ~ } ] f k (m + i - 2) - . . . . 
1 . 2 
There is next derived the theorem 
y, ± vu.A(vu')./\ 2 (vu ") . . . A n (vu {n) ) 
= V 1 UV 0 . . . v n 
y. ± u.Au'.A 2 u". ... A 
n iM 
( 2 ) 
and from this again by putting v r =l /u r there is obtained 
y, ± u.Au. \ 2 u ' . . . A n u {n) = uu Y u 2 . . . u n . y 
A 2 
. . . A’ 
,(n) 
Then, by using this last theorem on itself, and continuing in like manner, 
there is finally reached the result 
2 ± w.Am'AV'. . . A n u (n) 
= (uu 1 . . . u n )(u l0 u\° . . . . . . m®L„) .... n ~ 2 u±~ 1, n ~ 2 )u n ' " _1 (3) 
where 
u^° = A 
ww, 0 
i = A-h_ } vr, 2 = a 
1 
In order to pass from differences to differentials Christoffel then puts 
vW = f v (me + fxe) , Vfj, — <f>{me + /xe) , me = X , e = 0 a? , 
the result obtained from (2) being 
Y + Af. . . . 5 ’W„ = <A»+i Y + / d A d ^x 
^ * 0a; 0x 2 0a; n ^ ‘ ' dx’ dx 2 ‘ ‘ ' dx n 
and from (3) being 
•“ ' 8*' 8a: 2 8a:” 7 •/i,o-/s.i • • •/«.»- 1 
where 
•••• 
There is next given a result dealing with change of variable, namely, 
y +/ % 3 = /^V n(n+1) y +f . . % % 
^ 0a; 0a; 2 0a; n \0ay ^ * dt df 2 dt n 
which is proved like (1) and (l 7 ), the identity used for substitution purposes 
being now 
( 4 ) 
(5) 
*f _ 
= — — + or — + os 
+ 
0.r r dt r \dxj dt r ~ l 1 dt r ~ 2 
where a (r \ at ] , ... do not involve differential- quotients of /. 
