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

LESS COMMON SPECIAL FORMS (HIRST, 1859)

469

s cos si -cos s~ s sin sp
(s+l)cos(s+I)q -cos(ss+l)0 (s+l)sin(s+l)0 -sin(s+l1).................. ~.....
and so finding for its square a determinant whose every element
is independent of s, the element in the place i,j being in fact
(j-i)cos(j L)~0.
He does not note, however, that such a determinant is zero-axial
and skew, and that its value is thus readily seen, by a theorem of
Cayley's, to be
(cos2 -4 cos22~ + 3 cos 0 cos 30)2,
i.e.
(- 4 sin40)2.
CAYLEY, A. (1859).
[Note on the value of certain determinants, the terms of which
are the squared distances of points in a plane or in space.
Quart. Journ. of Math., iii. pp. 275-277; or Collected Math.
Papers, iv. pp. 460-462.]
The five results given in the paper are more important than
the title would imply, being true -when instead of Cayley's
elements 122, 13,... we write any elements whatever, namely,
12, 13,... This change being made, the fourth and fifth are
12 13 14
21. 23 24: 1221.34 43 - 1223 24 41,
31 32. 33
41 42 43.12 13 14 15
21. 23 24 25 1223344551
31 32. 34 35 = 2
41 42 43. 45 1 2331.455
51 52 53 54.
where the I's cover 3, 6, 24, 20 terms respectively. No
commentary is added, nor any indication of a law including