182 
Proceedings of Royal Society of Edinburgh. [sess. 
and consequently that/,, involves only/, j\ , ... , f n _ x ; that is to 
say, that /, f , ... , f n are not independent. This result being 
obtained, it only needs to be noted that the proposition being 
manifestly true in the case where the number of functions is one , 
must be true generally. 
As an addendum, it is noted that since the vanishing of the 
m + 1 determinants 
'V + ^ .... ^ n ~i d/ w 
^ ~ dx dx x dx n _ 1 dx n ’ 
'V + *!L . .... d/tt-i s/n 
^ “ dx dx 1 0cc„_ 1 0£ n+1 ’ 
y + ^ .... e /»-i . 
^ ~dx ' 0a5 x 0B n _ 1 * 0aw m 
when the determinant, B, and the n 2 elements common to all 
these does not vanish, implies that/,, is a function of /, f\ , . . . , / w _ x : 
and since this mutual dependence of /, / x , ... f n _ l , /„ implies 
the vanishing of all the functional determinants formed with 
respect to any n + 1 of the n + m + 1 independent variables, we 
are led to the conclusion that, provided B does not vanish, the 
vanishing of all these functional determinants of which the 
number is 
(?z + m+ l)(n + m) • • • (m+\) _ (n + m + \)(n + m) • • • (n+1) 
1-2-3 (» + l) 01 1-2-3 • • • (m+1) 
is a consequence of the vanishing of a certain m + 1 of them. 
In order that the connection between the members of this 
group of functional determinants formed from the differential- 
quotients of /, f\ , . . . , f n with respect to any n + 1 of the 
variables x , x x , ... , x n+m may he better looked into, several 
identities regarding square arrays of functional determinants are 
next given (§ 16). 
Taking in addition to /, f , ... , f n the m arbitrary functions 
fn+l 5 / n+2 j • • • J f n+m 
of the same n + m+1 variables, and denoting the determinant 
y\± d l. d A 
^ dx dx x 
bfn-l_ 
bx n _i 
0/„ 
dXy 
by b k 
