[579 
580] 
185 
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580. 
ON THE NUMBER OF DISTINCT TERMS IN A SYMMETRICAL OR 
PARTIALLY SYMMETRICAL DETERMINANT. 
[From the Monthly Notices of the Roycd Astronomical Society, vol. xxxiv. (1873—1874), 
pp. 303—307, and p. 335.] 
The determination of a set of unknown quantities by the method of least squares 
is effected by means of formulae depending on symmetrical or partially symmetrical 
determinants; and it is interesting to have an expression for the number of distinct 
terms in such a determinant. 
The terms of a determinant are represented as duads, and the determinant itself 
as a bicolumn; viz. we write, for instance, 
/ aa 
to represent the determinants 
aa, 
ab, 
ap', 
aq' 
\ bb 
ba, 
bb, 
bp', 
bq 
/ 
pp 
- 
pa, 
pb, 
Pp'> 
pq’ 
. 
qa, 
qb, 
9P'> 
qq' 
This being so if the duads are such that in general rs = sr, then the determinant 
is wholly or partially symmetrical; viz. the determinant just written down, for which 
the bicolumn contains such symbols as pp' and qq', (each letter p, q,... being distinct 
aa 
from every letter p', q,...) is partially symmetrical, but a determinant such as J bb [ 
( cc J 
is wholly symmetrical. A determinant for which the bicolumn has m rows aa, bb, &c., 
and n rows pp, qq', &c. is called a determinant (to, n); and the number of distinct 
terms in the developed expression of the determinant is taken to be cf> (to, n); the 
problem is to find the number of distinct terms </> (to, n). 
C. IX. 
24