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

258 HISTORY OF THE THEORY OF DETERMINANTS

functions of n +h variables shall be connected by an equation independent of the said variables is that the Jacobian of the functions with
respect to any n of the variables shall vanish. Two of them bear on
the theorem concerning the condensation of the Jacobian into one
product (Hist., i. p. 391). The first is familiar, namely, he alters
x=pcosO i -IP-x-
y psin 0 sin lr into x = pcosO
z = p sin 0 cos 4 = psinosin V/,
and so obtains
D(x,y,z) = DZ Dax~ Dy P( p sinO)psinc ' = -p2sinO,
'a (X''Z) 'z 'ax'ay P(-p sin 0) p sin 0 cos sin 0
N(pO,4) Dp aO av
In the second the data are
XI = cos 01
X2 = sinmqcosq52
X3 = sin 01 sin 7O2 cos 43
n= Sin 1 sin k2... sin 95, Cos n
and the result
a(x1, X2,..., xi) = sinn /, sinn-1-2* si952..-2. 3 sin'y4,.
D()(P, (I2, ' ', n)
What may be viewed as an illustration of another theorem of'
Jacobi's (Hist., i. p. 389) is the result
a (X~1 X', x2x'.. X., 11x' 1"
D(x1, X2,,J xi)
where x12-x22+. 2= 1.
WOLSTENIOLME, J. (1863).
[Question 4892. Educ. Times, xxviii. p. 252; or Math. from EducTimes, xxvi. pp. 104-105.]
The theorem here dealt with is that of ~ 15 of Jacobi's memoir
of 1841 (Hist., i. pp. 378-380). Any fresh interest is due to the
example given in illustration of the simplest case of the theorem,.
namely, the Jacobian of
a2+b2+c2, ax+by+cz, X2+y2+ Z2, bz-cy, cx -az, ay-bX