526 Proceedings of Royal Society of Edinburgh. 
( 2 ) 
1 
a 
1 
1 2 
a a 
2 2 
12 3 
a a a 
3 3 3 
12 3 n \ 
= aaa a 
12 3 n 
12 3 n 
a a a a 
n n n n 
The first of these is proved from first principles, and not by the 
immediate use of theorems xlvii., xlviii. above. The second is 
proved by noting that any other term is got from the first 
12 3 n 
aaa a 
12 3 n 
by permutation of the under-indices, that any such permutation 
will introduce one or more elements whose upper-index exceeds the 
lower, and that such are all zero. (vi. 5.) 
SCHWEmS (1825). 
[Theorie der Differenzen uud Differentiale, u.s.w. Von Ferd. 
Schweins. vi.-|- 666 pp. Heidelberg, 1825. Pp. 317-431 ; 
Theorie der Producte mit Versetzungen.] 
With much of the preceding literature, Schweins, our next author, 
was thoroughly familiar. Cramer, Bezout, Hindenburg, Rothe, 
Laplace, Desnanot, and Wronski he refers to by name. The one 
notable investigator left out of his list is Cauchy, whose important 
memoir bearing date 1812 might have been known, one would 
think, to a writer who knew Desnanot’s book of 1819 and Wronski’s 
memoirs of 1810, 1811, &c. Still more curious is the omission of 
Vandermonde’s name, whose memoir, as we have seen, is to be 
found in the very same volume as that of Laplace. 
Schweins’ portly volume consists of seven separate treatises. It 
is the third, headed Theorie der Producte mit Versetzungen^ 
which deals expressly and exclusively with the subject of deter- 
