172 Proceedings of Royal Society of Edinburgh. [sess. 
expressible as the sum of (n+ l) 2 terms of the form 
d~ 
0R dx k 
M da 
x 
This, however, by substitution of the expression just referred to, 
becomes 
R 
dx k 
& 
or 
so that we have 
dfj da 
dx k da m 
\ ’ Sar, ’ 
*=» Jdfj 
2R _ -p VI da dx k 
da dx k 'Wi' 
i = 0 k = 0 * 
i=n 
a f * 9 
r, as Jacobi puts it, 
a/. 
2/ 
aiogR aa °3a 
da df h df 1 + ’ ' + df n 
da 
With this theorem is associated another having no connection 
with it save the fact that the proof of it is dependent on the use 
of the same expression for the co-factor of an element of the 
functional determinant. Recalling the general theorem proved 
in his memoir De formatione . . . viz., 
2, ± a a; . . . A.™ = (2 ±ad 1 . . . a™ y n . ^ ± • * • a n ] , 
he applies it to the functional determinant, viz., to the case where 
a d) _ dp 
* ~ dx k ’ 
i dx k 
and where the cofactor 
