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

BORDERED DETERMINANTS (VERSLUYS, 1876)

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VERSLUYS, J. (1876): MANSION, P. (1877).
[Resolution d'un systeme d'equations, dont une est du second degre
tandis que les autres sont lineaires. Archiv d. Math. it. Phys.,
lx. pp. 128-137.]
[Resolution d'un systeme de n equations 'a n inconnues dont
Nouv. Corresp. Math., iii. pp. 376-381.]
Versluys' solution of the set of equations
ax2+by2+cZ2+2fyz+2gzx+2hxy = 0)
px+qy+rz = Of
rests on the fact that by using with the first equation the multiplier
ahg~p
hbfq
fq or ac say,
g f c r oray
p q r
he obtains as a substitute the equation
A H: G: P x
H B F Qy
- G F C R z0=0
P QR
x y z
in accordance with an identity first foreshadowed by Sylvester in
1850 (Hist., ii. p. 118). Performing on the five-line determinant
here the operation
r row, + q row2 + prow1,
followed by the operation
r col3 +-F-q col, + p col,,
and utilizing the second of the given equations, he thus readily
eliminates z, the result being
iA H P x
AH.P
12
P Q
x Yy