1912-13.] Dr Muir on Axisymmetric Determinants. 
49 
V. — The Theory of Axisymmetric Determinants from 1857 
to 1880. By Thomas Muir, LL.D. 
(MS. received June 24, 1912. Read November 18, 1912.) 
My last communication in reference to the history of axisymmetric deter- 
minants dealt with the period 1841-1859 ( Proc . Roy. Soc. Edin., xxvii. 
pp. 135-166). The present paper continues the history up to the year 1880, 
but in addition contains an account of one contribution belonging to the 
previous period, namely, by d’Arrest (1857). 
d’ Arrest, H. (1857, August). 
[Beobachtungen des Cometen in., 1857. Astron. ISTachrichten, xlvii., 
col. 17-19.] 
From consideration of three special cases d Arrest ventures on the 
generalisation that if in the application of the Method of Least Squares 
the normal equations be 
( aa) x + (ab)y -I- ( ac)z + . . . = O' 
(ab) x + (bb)y + (bc)z + . . . = 0 
(ac) x + (bc)y + (cc)z -f . . . = 0 
then the weights of x , y , z , . . . are 
AAA 
[aa\ ’ [bb] ’ [cc] ’ 
where A is the determinant of the set of equations and \aa\ is the cofactor 
of (ad) in A. In the same serial in 1866 (vol. Ixvii. pp. 174-175) a proof 
of the proposition is given but unaccompanied by any author’s name. 
Ferrers, N. M. (1861). 
[An Elementary Treatise on Trilinear Co-ordinates (chap. iii. pp. 
58-71). xii-(-154pp. London.] 
One of the results given at the close of the short chapter on deter- 
minants is 
. 1 1 1 
1 . a+b a+c .... 
1 b cl . b c .... 
1 c+a c+b 
abc . . . \ +- + 
\a b c 
VOL. XXXIII. 
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