98 
Proceedings of Royal Society of Edinburgh. [sess. 
SCHWEINS (1825). 
[Theorie der Differenzen und Differentials; u. s. w. Yon 
Ferd. Schweins. vi. + 666 pp. Heidelberg, 1825, Pp. 317— 
431 : Tlieorie der Produde mit Versetzungen .] 
It may be remembered that Schweins’ large volume contains 
seven separate treatises, that the third treatise deals with deter- 
minants ( Producte mit Versetzungen ), and is divided into four sec- 
tions (Abtheilungen). The first of the four almost entirely concerns 
general determinants, and consequently an account of it has already 
been given. The second section (pp. 369-398) now falls to be 
undertaken, its heading being “ Determinants in which the upper 
index denotes a power” ( Producte mit Versetzungen , wenn die 
oberen Elemente das Potentiiren angeben). 
His first theorem is 
. h . h . It » ~ht 
Wr- A ,' 
A “l A “2 A a 3 
t\j J 1 1 1 2 3 
which is seen to be an extension of one of Cauchy’s ; but, besides 
this, in the first chapter there is practically nothing worth noting. 
The remaining four chapters, however, are full of interest, and 
deserve every attention, as until the present day they have been 
utterly lost sight of and contain a theorem or two which are still 
quite new. 
The second chapter concerns the multiplication of an alternant 
of the n th order by the sum of the p-ary combinations of the 
variables in their h th power. In Schweins’ notation this product 
is represented by 
ai , a 2 
1 A 2 
in later notation, the case where n — 3, p = 2, li = 5 would be 
written 
( a°b b + a 5 c 5 + 5 5 c 5 ) . 
a r a s a t 
b r b s V 
c r c s d 
or . |a r 5 s c^| . 
The case where p = 1 is first dealt with, and the proof is written 
