318 Transactions of the Boyal Society of South Africa. 
He then contrived to satisfy himself that for the first element of each 
column he was entitled to substitute an expression which turned out 
to be the sum of the remaining elements of the column. The vanishing 
of the determinant was thus concluded to be inevitable. 
3. Now it is readily seen that in effect this was equivalent to saying 
that the determinant could be expressed as the sum oin — 1 determinants 
of the nth order, that is to say, as an aggregate of {n - 1) (n !) terms of the 
nth. degree in the first differential-coefficients oi tt, v, . . . . ; whereas 
being an aggregate of {n - 1) ! terms of the {n - l)th degreee in the said 
differential -coefficients, ^-^ must be an aggregate of {n - 1)\ {n — 1) I 
0 Jb 1 
terms in each of which a second differential- coefficient occurs, and 
therefore must be an aggregate of [n - l)\n\ terms of this latter 
kind. Such being the case, doubt is at once thrown on the so-called 
demonstration. A little examination of the reasoning employed fully 
justifies us in setting it aside as misleading and fallacious. 
4. Fortunately our scepticism leads us at the same time to produce as 
a substitute a valid proof not hitherto made known. 
Recalling the fact that the differential-coefficient of an n-line determi- 
nant can be expressed as the sum of n determinants of the same order, 
c)A 
we see that — ^ can be expressed as an array of n rows with n — 1 
determinants in each row. If then the elements of each column in this 
array be added, it will be found that the result is zero, in accordance with 
a well-known theorem of Kronecker's regarding vanishing aggregates of 
determinants. 
Thus, when n = 4, we have on writing u^.^ for 'd^uftx^T^Xg 
^Xj^ 
'i)X^ 
^A. 
o 
ILK 
31 
U 
1L\ 
v. 
o 
UK 
u^. 
U2I 
u 
32 
10. 
u^ 
24 
u 
10, 
34 
10, 
+ 
+ 
U2 
10^ 
^'21 
u, 
10^ 
U^ 
V. 
23 
to J. 10^ 
■U2 
^32 
Wo 
'^14 
10^ 
V 
24 
to. 
u^ 
'^'34 
10, 
+ 
+ 
^2 
'^2 
u^ 
u^ 
^3 
10,^ 
'^3 
10 
23 
10 
32 
u^ 
^4 
^4 
U4 
^4 
W3J, 
c)A 
4 _ 
^x. 
U 
42 
to. 
and therefore by addition 
u 
43 
to. 
Uj 
V 
41 
to. 
tL 
V 
42 
to. 
u, 
'^43 
to. 
7\ A 
2^ ^04-0 + 0. 
ox. 
Uj 
10 
41 
tl^ 
to 
42 
V, 
to 
43 
