1899-1900.] Dr Muir on the Theory of Skew Determinants. 205 
BRIOSCHI (1854). 
[La Teorica dei Determinant^ e le sue principali Applicazioni. 
Del Dr. Francisco Brioschi. viii+116 pp. Pavia, 1854. 
Translation into French, by Comhescure, ix + 216 pp. 
Paris, 1856. 
Translation into German, by Schellbach, vii + 102 pp. 
Berlin, 1856.] 
In this, the second text-book, the same importance is given to 
skew determinants as in Spottiswoode, the first part of the eighth 
section (pp. 55-72) being devoted to them under the heading “ Dei 
determinant gobbi which Schellbach translates by ilberschlagen. 
The arrangement and treatment of the matter, however, are much 
more logical, zero-axial skew determinants being taken first, then 
the functions connected with these, viz., Pfafiians, then skew 
determinants which are not zero-axial, and lastly the use of skew 
determinants in the consideration of the problem of orthogonal 
transformation. 
The precedence given to determinants which are “gobbi sim- 
metrici” over those which are “puramente gobbi ” is explained at 
the outset by reference to Cayley’s theorem regarding the 
expressibility of the latter in terms of the former, the quite 
general theorem from which Cayley’s immediately follows being 
carefully enunciated thus : — 
“ Indicando con P 0 il determinante nel quale si pongano 
equali a zero gli elementi principali ; e con (“P^ un deter- 
minante minore principale delle’ m-esimo ordine del deter- 
minante P nel quale siensi annullati gli elementi principali 
si ha : — 
P = P 0 + X+brOP^o + AA( 2 i V)o i 
+ . . . + Aq-|$22 * • * ® nn j • 
The proof given of Jacobi’s theorem regarding the value of an 
odd-ordered skew determinant with zero, in the principal diagonal 
is essentially the same as Cayley’s proof, but fuller and clearer. 
The proof of the corresponding theorem for a determinant of even 
order resembles Spottiswoode’s, the difference lying mainly in the 
