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

LESS COMMON SPECIAL FORMS (ROSANES, 1872) 481

ROSANES, J. (1872).
[Ueber Functionen, welche ein den Functionaldeterminanten
analoges Verhalten zeigen. Crelle's Journ., lxxv. pp. 166-171.]
What is here considered is a property of determinants of the form
x2 Dax Dy Dy2
D2v D2v D2zv
Dx2 Dxay a" ~ or, say, I (in, v, w),
D1w D2w D2w
Dx2 DxDy Dy2
where u, v, w are binary quantics of one and the same degree. Such
a determinant IRosanes writes in the form
f'~ f1'2 f122.f3'1 f312 f322
the functions undergoing differentiation being f1, f2, f3, and the
independent variables x,, x2. His theorem may be formulated thus:
If fl, f2, ~3, ~4 be binary m-thics; /1, '/2, 'p3, 04 be the founr
R's formable from the f's, namely, - R(f2, ~3, ~4), R(f~, f4, ~4),
-R (f,, ~2, ~4), R(f1, ~2, ~3); "fry V'12 "J3, 4/4 be formed fromt the O's
as the O's fromn the ~'s: then the 1's are proportional to the ~'s,
the comminon ratio being
- 3m? (m - 1)2 (m - 2)-5 (3 m1 - 7) 2 f R (fl, fi, f3.; Q )
The mode of proof, though effective, is neither as short nor as
direct as one could wish. Two new magnitudes y,, y2, independent
of the two x's, are introduced; and the consideration of the array
f12 fl12 f 12 J2fl2
f22 f 22 f322 f422
whose principal minors are the O's, is replaced by the consideration
of the array f f f f
8f, 6f2 6f3 8f4
82fA 822 A 82fA 82f4
where Df + f and 82f 8(8j).
f.Y1 O2+ Y
M. D. III. 2 H