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

COMNPOUND DETERMINANTS (TANNER, 1877) 15

TANNER, H. W. LLOYD (1877).
[Question 5247. Educ. Times, xxx. p. 20; Math, from Educ. Times,
xxviii. pp. 41-43.]
The theorem here proved is not essentially different from Bazin's
of 1851, but the proofs given are both new, and the second, by
G. S. Carr, is noteworthy. When applied to the first example
given by us under Bazin (Lust., ii.- pp. 206-208) and somewhat
improved, it is as follows, IA'1B 0D41 beiug, the adjugate of 1234:
5234 6234 _ a5A + b5 B + c5C, -1 d,5D1 a6A1 + b6B1 + C(AC + d6D1
1534 1634- a. A 2+ b5B2 + c5C2 + d5D 2 a6A ~ bGB2 ~ c6C2 + d6D9
A2B2C2D2 a,,bc~d6
- IAB2fI1-1a5b.1 +.... + I C1D2fIc d6I CA
- alb2c.d4 I c.3Id4l -Ia b,1 +. I- a.,b d.cd I
- a~b cd4 a5bCA I
MERTENS, F. (1877).
(Sdtze tiber Determinanten und Anweud-ung derselben zum lBeweise
der Siitze von Pascal und Brianchon. Crelle's Journ., lxxxiv'.
pp. 3055-359.]
Mertens might quite appropriately have adopted the title of
Hunyady's paper (1875-6), their common subject being the eliminant-forms connected with six points on a conic. The one form
dealt with by both writers is the doubly compound determinant
I I Y24 I Z23X56 IIX34Y61 IIor A
The other forms which Mertens deals with are
xi2 yii2 Z12 yJz1 zJX1 X~y1
X22 Y2 2 Z2 2Y2Z2 Z2X.2 X2Y2 orPsy
xj2 y62 Z(;2 Yf6Z6 Z6X-6 X6Jt6
and
XlX2 YlY2 Z1Z2 yIz2 +Y2Z1 zIX9.+Z2X1 Xly2 -X2Y1
X23 Y2Y3 Z'2Z08 Y2Z3 +Y3Z2 Z2X 5AZ3X2 X9-YS3 +X3Y2 or, Q -say;
X6XT1 YA~ Z6Z1 lYoZi+YIZ6 Z6X1 +Z1XG X6YJ1 -X1Y6