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Transactions of the Royal Society of South Africa. 
record* the suggestion tliat "a very instructive way of establishing it is to 
combine the first two terms on the right into one, then in similar manner 
combine the result thus obtained with the third term, and so on. The 
series is thus seen to be one of those that close up telescopically." Various 
procedures are thus open for trial. Suffice it to say that the last method is 
found to be as easily applicable to the general result (X) as to Kronecker's 
and Glaisher's cases. 
(8) The inquiry about the said cases, however, led to the very interesting 
discovery that (X) in all its generality was published by Schweinsf so long 
ago as 1825 in his ' Theorie der Differenzen und DifEerentiale,' Sect. I, 
Chap. Y. With Schweins the theorem is purely determinantal, the deter- 
minants being written in a kind of elaborate umbral notation, bulky but not 
ambiguous. 
(9) The second equality (Y) is provable exactly like (X), but the 
application of the method brings out the unexpected fact that the two 
theorems are not really different, the second being, indeed, merely a case 
of the first. To convince oneself of this, one has only to take (X), which, 
as we have seen, is an equality concerning the array 
a^i .... a\n iij 
CL,i\ .... <^nn ^^n 
and change the column of %'s into a column of a's, namely, into the mth 
column where m>p and "^n. The left-hand member of (X) then vanishes, 
and the right-hand member becomes 
'K-pp^pm + + + .... 
and thus we have (Y). 
(10) Lastly, turning our scrutiny on to (Z), we find an outcome equally 
unexpected but of a different kind. The K's, which were introduced "for 
shortness' sake," do not here at all serve their purpose, for on replacing 
them in (Z) by their equivalents we obtain 
j mm — Q 
Am— lAm 
from which manifestly the A's can be struck out. This being done, the 
equality then to be considered is in Koch's own notation 
% ^pp % + l ■^p,p + l ' ^p+2 ■^p,p + 2 • • • • > 
* ' Proc. E,. Soc. Edinburgh,^ xxxiii, p. 61. 
t Muir, T., "An Overlooked Discoverer in the Theory of Determinants/' 'Philos. 
Mag.' (5) xviii, pp. 416-427 ; or ' Hist.' I, pp. 173-4. 
