692 
Proceedings of the Royal Society of Edinburgh. [Sess. 
the theorem Rubini is the more successful. Taking the 7i-line determinant 
whose element in the place r,s is a rs + b rs , and denoting by A the determin- 
ant of the a’s, and by A,, a determinant obtainable from A on substituting 
for r columns of a’s the corresponding r columns of 6’s, he writes the 
expansion in the form 
A 4 - + 2A 2 + .... + + A n . 
Bellavitis, G. (1857, June). 
[Sposizione elementare della teorica dei determinante. Memorie . . . 
Istituto Veneto, . . . vii. pp. 67-144.] 
Notwithstanding its place of publication, this writing of Bellavitis’ is 
exactly what its title implies ; and as a text-book it could scarcely have 
failed to be useful, so simple and clear is it in style. It consists of two 
chapters, one on determinants in general (pp. 3-30), and one on special forms 
(pp 30-72) : a note of six pages on permutations appears as an appendix. 
To Bellavitis we owe the modification of Laplace’s notation which is 
now in common use. The passage introducing it is : “ Quando gli elementi 
sieno indicati in modo die chiaramente apparisca la loro formazione, noi 
porremo tra le due | | i soli elementi della diagonale ( intend endo sempre 
per diagonale quella da sinistra verso destra discendendo). Cosi 
«lV 3 
equivalera a 
| equivalera a 
a m a M 
ec. 
Throughout the exposition this notation is employed. “ Riga ” he uses 
either for a “ fila orizzontale ” or a “ fila verticale,” and “ colonna ” for a, 
“ fila perpendicolare a quella che s’intese per riga.” 
Two well-known developments he specifies thus : — 
17 I / 0 , , 0 0 
IVV-3--- 1 = K^ + S Ti + %-+ • 
J I a i^2 C 3 ■ ' * I 
(a I A +aA _A_+ .... +a j h 
0 2 
, 0 2 
+ a 3 h i=r~w + • 
0a 3 00 1 
. . -aji^ 
0 a 2 0/? 1 
_ 02 _ 
0a 3 0/q 
+ 
0 2 
0aj0 1 + 
| Vs • * • ]l n | = 
