178 Proceedings of Royal Society of Edinburgh. [sess. 
and each of the F’s being a function not only opM , x 1 , ... , x n , 
f, fi, ... , f n hut also of 
fn + 1 i f n+2 J * ■ • 5 fn+m • 
If the last m of the equations were solved for the last m of the f s, 
and these f s thus eliminated from the other + 1 equations, the 
functional determinant desired would, by the proposition sought 
to be generalised, be equal to 
where the brackets are used to indicate that the enclosed F’s are 
in the] altered forms resulting from the substitution referred to. 
Multiplying enumerator and denominator by 
Z + w+ l 11+2 .... dl n+m 
~ ¥n + 1 ’ ¥n + 2 ”” ¥n+m 
we obtain a new enumerator which by a later proposition is 
equal to 
y + aF 8 F t dF n 8F„ +1 S 0F w+2 . . _ 8F n+m 
dx ' dx x dx n df n+1 ’ m +2 3 f n+m ’ 
and a new denominator which for a similar reason is equal to 
V+— —i dFn+m 
2j-df' <ff'";y n+m ' 
The]functional determinant desired is thus found to be equal to 
V + — . — . 5F ^+2 b¥ n+m 
/ -i \ ^ dX-j dx n fn+\ a/ n _)_o bf n+m 
" V , 0F , 8F 1 dF n +m ’ 
^ ~ a/ a/ x a/ fl+m 
where, it will be observed, all the F’s are in their original form. 
As usual, the^ extreme case is noted, viz., the case where a single 
function f of one variable x is given by means of m + 1 equations 
connecting x, /, f , ... , f m the result then being 
'V dFj 3F m 
df^f 2u — 0/^ 0/’ m 
dx v + ^_' aib’ TO ’ 
" ” a/ ’ a/j 0 /; n 
