Dr T. Muir on the Theory of Determinants. 543 
of orthogonal substitution, constant use is, of course, made of the 
functions; but there is no special notation employed, nor indeed 
anything to indicate that the expressions used were members of a 
class having properties peculiar to themselves. 
In the second memoir, which likewise is taken up with a trans- 
formation, but in which the sets of equations involve four unknowns, 
any special notation is still avoided. Expressions, readily seen to 
be determinants of the third order, are even not set down, because, 
as the author expressly states, they would be too lengthy. The 
last clause of the passage in which this statement occurs is note- 
worthy. The words are (p. 236) — ■ 
“ Dato systemate sequationum 
a u ^ X -k- y 2/ + ^ z — m, 
a u + X y y h' z = m\ 
a u + /3"a? -t- y" y + h" z = m", 
a"u + -I- y'"y -f rz = m'", 
“ponamus earum resolutione erui: 
A??z •+• Am' + A”m" -p A'"m'" = w, 
Bm + B'm' -i- B"m' ' + B'"m'" = x, 
Cm -f C'm' -f C"m" + = y, 
Dm -F D'm' -p D"m" -p D"'m'" = 
“ Valores sedecim quantitatum A, B, etc., supprimimus eorum 
prolixitatis causa ; in libris algebraicis passim traduntur, et 
algorithmus, cuius ope formantur, hodie abunde notus est.” 
On the next page, in eliminating D, D', D", D'" from the set of 
equations 
0 = D(a-a;)-j-D'5' -pD"5" +D'"5'", 
0 = D5' +D'{a +x) + T>"c'" -l-D'"c", 
0 = D5" q- D'c'" + D" (a" + x) + D'"c', 
0 = D5'" -pD'c" q-D"c' +D'"(a'" + x}, 
he arranges the resultant as one would now do who had expanded 
it from the determinant form according to products of the elements 
of the principal diagonal, viz., he says (p. 238) — 
