1909-10.] Dr Muir on the Theory of Orthogonants. 
283 
Brioschi, Fr. (1854, March). 
[La Teorica dei Determinant:, e le sue principali applicazioni ; 
viii + 116 pp., Pavia. Translation into French by Combescure ; 
ix-f 216 pp., Paris, 1856. Translation into German by Schellbach ; 
vii + 102 pp., Berlin, 1856.] 
In Brioschi’s text-book, the paragraphs dealing with a “ sostitnzione 
ortogonale” are somewhat scattered, most of them appearing among the 
applications (pp. 24-26, 47-51, 62-69). 
The first deserving of notice (p. 49) concerns the product QPQ, where 
P and Q are determinants of the same order and Q is the conjugate of Q. 
Viewing the product as Q(PQ) Brioschi first uses a result of Cauchy’s to 
express any m-line minor of Q'(PQ) in terms of m-line minors of Q and PQ : 
then for the said m-line minors of PQ he substitutes with the same assist- 
ance expressions involving m-line minors of P and Q : there is thus obtained 
for any m-line minor of QPQ an expression involving only m-line minors 
of P and Q. This result may be put in the form 
(QPQ)!rl = + • • - + p SQ2*’}]> 
if 2 be put for n(n — 1) . . . (n — m+l)/l. 2. . . . m, and if generally we 
use A ( S to stand for an m-line minor of an mline determinant A, the rows 
of A taken to form A ( J$ being those whose numbers constitute the r th 
combination of m of the integers 1,2, . . . ,n, and the columns those whose 
numbers constitute the s th like combination. Putting v — s we obtain the 
expression for an m-line coaxial minor of QPQ, and thence for the sum of 
all such minors the expression 
2,-ZD”‘H p ?W + Pg , QS ) + • • • +P!?Q2‘ ) }]. 
which changes into 
2 r {P» + PSMi?+ • . • +P<?MS?}, 
if M be the determinant which equals Q 2 . Specialising still further by 
making Q the determinant of an orthogonal substitution so that 
M|V = 1 and - 0 , 
Brioschi finally obtains the important “ formula nota 
2 s (QpQ)r = 
which we may express in words for ourselves, thus —If Q be an orthogonant 
