The theory of determinants in the historical order of development, by Sir Thomas Muir.

AXISYMMETRIC DETERMINANTS (GLAISHEIR, 1878).115
Another is identical with Schultze's of 1871 and a third is a slight
variant of Janni's of 1874.
This last is also Picquet's main algebraical result.
GLAISHEIR, J. W. L. (1874, 1878).
[On the solution of the equations in the method of least squares.
Monthly Notices R. Astron. Soc., xxxiv. (1874), pp. 311-334.]
[Questions 4418, 5530. Educ. Times, xxvii. p. 69; xxxi. p. 21.
Solution by E. J. Nanson in Math. from Educ. Times, (2) ii.
pp. 95-96.]
What is given in the Educ. Times is an expression in the form of a
series for d'Arrest's ratio of an axisymmetric determinant to one
of its coaxial primary minors. A perusal of the paper on the Method
of Least Squares will show how the expression originates * (see,
~~ 5, 6). When freed of awkward notations it is, in the case of the
fourth order,
_________ _ ja~b4f12 a~b2C4J12
Ia,.b2Cd4 4 a. aI ab2lj a.b2j H a.b2C31
We may note for ourselves that the corresponding identity when
axisymmetry is not insisted onl is (Hist., ii. p. 40; iii. p. 51)
ctlb2c3d4 d14 a4d1 a~b4 I a~d~l I a~b2c41 I alb2d.1
a~b2C.1 a, a, Ia~b21 I1alb21 a~b cf1
and that a very instructive way of establishing it is to combine
the first two terms on the right into one, then in similar manner
combine the result thus obtained with the th ird term, and so on,
the successive 'approximations,' so to speak, being
(1~ a~d4 a~b2d4 a~b2c d 4
a. I a'lb a~b2C3j
The series is thus seen to be one of those that close up telescopically.
*Other papers on the same subject areGEER, P. VAN, "1Over het gebruik van determinanteln bij de methode der
klienste kwadraten," Niesuw Archief veer Wi~slcunde, i. (1875), pp. 179-188.
CATALAN,1 E., "Remarques sur la th6orie des moindres earr6s," Mgm. de
t'Acad.... ctde Belgique,'xliii. (1878), pp. 24-33.