(iii) w = 4; the quadric 2ars^r^«=0 (r, 5=1, 2, 3, 4), degenerates
into a cone if its determinant is zero; into a pair of planes if all the
first minors are zero ; into a pair of coincident planes if all the second
minors are zero.
It is convenient to have a name for the least number of indepen-
dent variables in terms of which a quadratic form can be expressed ; we
shall adopt the ter m rank (G erman Rang) for this number.
6. We shall give next a method of transforming a single quad-
ratic form A. Consider the determinant



K =



^11 > ^12 >

^21 > ^22 >






^k\i ^k2i •"} f^Kki ^k
^\i -^2> •••> -^fc> A

where X^^ Xa, ..., X^ have the same values as in Art. 4; but now (as
in Art. 5) «„ = (^n-y and of course k is less than n.
If we multiply the first k columns of iTby x^^ x^, --, x^ in order,
and subtract from the last column; and then treat the rows in the
same way, it will be seen that JTno longer contains any of the variables
Xi, X2, ..., Xk» Now these operations do not alter the value of the

* First stated explicitly by Sylvester, Phil. Mag. series 4, vol. i, 1851, p. 121 ;
Coll. Papers J vol. i, p. 221. But the substance of the theorem is contained in an
earlier paper of 1850 (Coll. Papers, vol. i, p. 145); this contains the further
theorem that all the A;th minors are zero it ^(k + 1) (k + 2)ot them (properly chosen)
are zero.
