1887.] Dr T. Muir on the Theory of Determinants. 471 
oritur alia forma 
/A A' A"\ 
\B B' B'7 
quam forme 
. . . . F 
fa a 
a \ 
U b' 
v) 
• . f 
adjundam dicemus. Hinc rursus inuenitur, denotando breui- 
tatis caussa numerum 
abb + AAA + a"b"b" - ad a" - 2 bb'b" per D, 
BB - A' A" = aT), B'B' - AA" - AD, B"B" - AA' = A'D, 
AB - B'B'' = bD, A'B' - BB" = V D, A"B" - BB' - b" D, 
unde patet, forme F adjunctam esse formam 
/aD, AD, A'D\ 
\6D, AD, A'D/ . 
Numerum D, a cuius indole proprietates forme ternarie / im- 
primis pendent, determinantem hums forme uocabimus (xv. 2) ; 
hoc modo determinans formae F sit = DD, sive aequalis quadrate 
determinantis formae/, cui adjuncta est.” 
In this there is no advance so far as the theory of modern deter 
minants is concerned, the identities given being those numbered 
(xx) and (xxi) under Lagrange. On the same page, however, an 
extension is given of Lagrange’s theorem (xxii), regarding the 
determinant of the new form obtained by effecting a linear substi- 
tution on a given form. Gauss’ words in regard to this are — 
“ Si forma aliqua ternaria / determinantis D, cuius indeter- 
minate sunt tJC 1 y *)(j y tD (puta prima = «, &c.) in formam ternariam 
g determinantis E, cuius indeterminate sunt y , y\ y ", trans- 
mutatur per substitutionem talem 
a =a -y +y y’\ 
A =dy +(3y' + fy" , 
d' — d'y +/3Y + /V', 
ubi nouern coefficientes a, /?, &c. omnes supponuntur esse 
numeri integri, breuitatis caussa neglectis indeterminatis 
simpliciter dicemus, / transire in g per substitutionem (S) 
