275 
1909-10.] Dr Muir on the Theory of Orthogonants. 
A - 2(a 2 + b 2 + c 2 + 6 2 ) 2(f0 + a + bh - eg ) 2 (gO + b + cf - ah) 2(h0 + e + ag - bf) 
2( -fO -a + bh -eg) A - 2(g 2 + h 2 + a 2 + 0 2 ) 2( - cO - h+fg - ab) 2(b6 + g + hf - ca) 
2(-g0-b + cf-ah) 2(c6 + h +fg - ab) A - 2{h 2 +f 2 + b 2 + 0 2 ) 2 (- aO -f+ gh-bc) 
2(-h6 - c + ag - bf) 2( - b6 - g + hf- ca) 2(a0 +/+ gh - be) A - 2(/ 2 + g 2 + c 2 + 6 2 ) 
where 0 = af+bg + ch and A = 1 + a 2 -\-b 2 c 2 +f 2 g 2 + h 2 + 6 2 . 
Before leaving Cayley’s very interesting paper it should he noted that 
the essential part of it is contained in the first few lines, where in effect 
he says that if we put 
0 j + X0 2 + fxd 3 + . . 
. = Xj 
— + 0 2 + v0 3 + ■ • 
. == Xc 
+ 
CO 
+ 
CN 
1 
<3T 
a. 
i 
. = X, 
0 l -X0 2 - . . . = 
X0 1 4- 0 2 - v6 3 — . . = £ 2 
| f x0 1 +v0 2 + 0 3 - • . • = £ 3 
then x lf x 2 , x 3 , . . . and £ 2 , (p 3 , . . . are orthogonally related, the co- 
efficients of the linear substitutions connecting them being rational 
functions of A, /a, v, . . . The rest of the paper is taken up with the finding 
of these coefficients, that is to say, with the elimination of 6 1 , 0 2 , 0 3 , . . . and 
the expression of each of the remaining variables as a linear function of all 
the variables of the set to which this variable does not belong. 
Hermite, Ch. (1849, January). 
[Sur une question relative a la theorie des nombres. Journ. ( de 
Liouville) de Math., xiv. pp. 21-30.] 
The' problem here solved has only a distant connection with our subject. 
What is given is a set of mutually prime integers forming the first column 
of a determinant, and the requirement is to find all the other elements so 
that the square of the determinant may be 1. 
Spottiswoode, W. (1851). 
[Elementary Theorems relating to Determinants, viii + 63 pp., 
London.] 
Following Cayley, Spottiswoode places the construction of an orthogonal 
substitution at the opening of his section (§ 9) on skew determinants. The 
mode of treatment differs from Cayley’s in being verificatory rather than 
investigative. Starting with the two sets of equations 
^11^1 + ^12^2 “t ^13^3 ^14^4 
