698 
Proceedings of the Royal Society of Edinburgh. [Sess. 
its original meaning, so that the principal term of an 7i-line determinant 
has \n{n — 1) derangements, seeks to show that the sign of any other term 
having /x derangements is ( — 1)^-1) 
Mainardi, employing Cauchy’s “ clefs algebriques,” finds himself also 
face to face with derangements, and seriously advises that in counting them 
we should say, not 1, 2, 3, 4, 5, . . . , but 1, 2, 1, 2, 1, . . . , the sign being 
— or + according as we end with 1 or 2. 
Gallenkamp, W. (1858). 
[Die einfachsten Eigenschaften und Anwendungen der Determinanten. 
12 pp. Sch. Progr. Duisburg.] 
A workmanlike twelve-page exposition. 
Sperling, I. (1858). 
[Teorija opredelitelej i eja vaznejsija prilozenija. C. 1. St Petersburg.] 
This dissertation I have failed to see. In English the title is, The Theory 
of Determinants and its most important applications. The letters used 
here in transliterating the Russian title have German values. 
Casorati, F. (1858, September). 
[Intorno ad alcuni punti della teoria dei minimi quadrati. Annali di 
Mat, i. pp. 329-343.] 
The title here refers only to the latter half of the paper, the other half 
being concerned with an auxiliary series of theorems on the product-deter- 
minant. The first of these theorems is avowedly old, being that which 
concerns the so-called product C of two non-quadrate arrays 
a \\ rt 12 ®13 • • ' a l n ^12 ^13 * ‘ ’ ^1 n 
^22 ^23 " ' ' ^ 2 n ^21 ^22 ^23 • • • 
^ml ^m2 ^mS • • ■ 5 ^ml ^m2 j 
where n>m. The second, though not so spoken of, is only new in form, 
and concerns any primary minor of C. Unfortunately, Casorati does not 
observe that any primary minor of C is a determinant formed exactly like 
