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NOTE ON AXISYMMETKIC ORTHOGKMANTS. 
By Sir Thomas Mum, LL.D., F.R.S. 
(1) It will be recalled that Cayley's rule for the construction of a 
positive unit orthogonant is to take a unit-axial skew determinant, A say ; 
replace it by its ad jugate ; multiply each element of the latter by 2/ a ; and 
then, lastly, from each diagonal element subtract 1. Since the original 
n-liue determinant, taken as it were for material of construction, involves 
\n(n — 1) arbitrary constants, this is the number of such constants in the 
orthogonant. 
(2) There is an appreciable advantage to be gained by introducing 
another constant, a say, the most natural place for it being that of each 
unit of the original diagonal. The preceding rule then has to have its 
multiplier 2/ A changed into 2a/ A, or, what is the same thing, to give place 
to the following theorem: If \a ln \, or A say, be a shew determinant with 
a \\ — a, 22 — - • • - z a >in — a , then 
2A n — — A 12 .... A 1?l 
a 
A-21 2A 22 — ^ .... A 2W 
A M1 A„ 0 .... 2A„„ - A 
1 a 
is an orthogonant ivhose basic constant is (A/a) 2 - A short verificatory proof 
is obtained by showing with the help of a result of Spottiswoode's ('Hist.,' 
ii, pp. 289-291, 315) that 
row r x row r = ( A/a) 2 
and Yow r x row s = 0. 
(3) Let us consider now the case of this where n is odd. A then, being 
skew, can be expressed as a sum of terms in descending odd powers of a, so 
that for A/a we have an expression beginning with a"- 1 and ending with a 
sum of squares independent of a. If in the result as thus simplified we 
put a = 0, Ars and A sr become equal, being conjugate minors of a zero- 
axial skew determinant. We consequently obtain when n is odd an axisym- 
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