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

166 HISTORY OF THE THEORY OF DETERMINANTS

where s = a+ 4+x4++xg. This, it will be observed, ensures that
Dfl Daf 3 / x 4 a Dy
IX1 ax2 Da3 a4, - i ',.
and al Y12 a. _ y xOxx;
aX. 0Y2 XY 9Y 1 2Y 3 4
and there remains the trouble-which is still ample-of solving four
sets of four simultaneous equations of the type
D^. 1 ay a DY2 JS D Y3 + a4 _ DJ
Yj aX7, Y 2 aXk Dy3 aX7k ay4 DX74 aDx,
in order to obtain the corresponding values of the elements of! fl Df f3 af4
ay1 aY2 aY3 y/4
At the end of the paper there is given another illustrative result
worth noting, namely, The Jacobian of the first n Aleph functions
of a set of n variables is equal to the difference-product of the variables.
This may be appropriately grouped with Tarleton's of 1867.
WOLSTENHOLME, J. (1878).
[MATHEMATICAL PROBLEMS.... 2nd edition. x+480 pp.
London.]
Here (p. 276) we have two additional examples of alternants
whose elements are goniometric functions, namely,
1 cos(/63+y) sin2(/3 —) sin2 (7y-0)
1 cos(y +a) sin (y-0) sin2 (a -0)
1 cos (a+ -) sin2(a-0) sin2 (3- 0)
= 2 sin - (y-/) sin (-a) sin -(-a)
{2 sin 20+ 2 sin(a+ + +y-0) —Esin (+y)},
and
1 cos(a+0) sin 2 (-0) sin2- (-0)
1 cos(B3+0) sin2(-0)s (a- 0)
1 cos ( + 0) sin2 (a -0) sin2 (a - 0)3
= 2 sin - (y-/) sin -(y —a) sin (3- a)
{2 sin 20+ sin (a +3+ y-0) —s sin (a + 0).