Thus ortho-phenylene diamine yields the following products: N H N ./`N; **Xn** NZ In some cases oxidation of condensed benzenoid-heterocyclic nuclei results in the rupture of the heterocyclic ring with the formation of a benzene dicarboxylic acid; but if the aromatic nucleus be weakened by the introduction of an amino group, then it is the benzenoid nucleus which is destroyed and a dicarboxylic acid of the heterocyclic ring system obtained.

In general in space of n dimensions we have n substitutions similar to X l = a11x1 +a12x2 + ï¿½ ï¿½ ï¿½ + ainxn, and we have to express the n 2 coefficients in terms of Zn(n - I)i independent quantities; which must be possible, because X1+X2+..."IL **Xn** =xi+x2 +x3 +...+4.

=0, are non-unitary symmetric functions of the roots of a **xn-a** l **xn** 1 a2 x n-2 -...

The binomial theorem may, for instance, be stated for (x+a)n alone; the formula for (x-a)" being obtained by writing it as {x+(-)a} n or Ix+(- a) } n, so that (x-a) n =x"- 1)**xn-laF**...+(-)rn(r)**xn-rar**+..., where + (-) r means - or + according as r is odd or even.

This is expressed by saying that the sequence converges to (x+h)" as its limit; it may be stated concisely in any of the three ways, (x+h) n =lim(x"+n(1)**xn-lh**+....+ n(T)**xn-rhr**+ï¿½ï¿½.),(x+h)n =lim Sr, Sr. (x+h)n.

Then {(x+h)n - S T } /h r + l lies between n (r+1>x n-r-1 and n(r+1}**xn-r-**1(i +O)n; and the difference between these can be made as small as we please by taking h small enough.

Thus we arrive at the differential coefficient of f(x) as the limit of the ratio of f (x+8) - f (x) to 0 when 0 is made indefinitely small; and this gives an interpretation of nx n-1 as the derived function of **xn** (ï¿½ 45)ï¿½ This conception of a limit enables us to deal with algebraical expressions which assume such forms as -° o for particular values of the variable (ï¿½ 39 (iii.)).

At the Revolution he took up arms in behalf of the king, became commander of the "army of Conde," and fought **xn** conjunction with the Austrians till the peace of Campo Formio in 1797, being during the last year in the pay of England.

Suppose n dependent variables yl, y2,ï¿½ï¿½ï¿½yn, each of which is a function of n independent variables x1, x2 i ï¿½ï¿½ï¿½**xn**, so that y s = f s (x i, x 2, ...x n).

=0, a'**xn**+ (b'y+c'z)**xn-**1+...