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 volves. For instance, the expression a^b^c, which is of eight dimensions, may be said to be of three dimensions in a, of four dimensions in b, and of one dimension in c.

30. A compound expression is said to be homogeneous when all its terms are of the same degree. Thus 8 a^6 — a^4b^2 4- 9ab^ is a homogeneous expression of six dimensions, or of the sixth degree.

31. Different powers of the same letter are unlike terms ; thus the result of adding together 2 x"^ and 3 x^ cannot be ex- pressed by a single term, but must be left in the form Similarly, the algebraic sum of 5 a^b^ — 3 ab^ and — b^ is 5 a^b^ — 3 ab^ — b^. This expression is in its simplest form and cannot be abridged.

32. In adding together several algebraic expressions containing terms with different powers of the same letter, it will be found convenient to arrange all the expressions in descending or ascending powers of that letter. This will be made clear by the following examples.

Ex. 1. Add together 3 x^ + 7 + 6 x - 5 ic2 ; 2 x2 - 8 - 9 a:- ; 4x - 2x3 + 3x2 ; 3x3 - 9 X - x2 ; a- _ x2 - x3 + 4.

In writing the first expression we put in the first term the highest power of x, in the second term the next highest power, and so on till the last term, in which x does not appear. The other expressions are arranged in the same way, so that in each column we have like powers of the same letter. The result is in descending powers of x.

3 x3 - 5 x2 + 6 X + 7 2 x2 — 9 X — 8 — 2x3 4-3x2 + 4x 3 x3 — x2 — 9 X — x^ — x2 + X +4 3 x3 — 2 x2 — 7 X + 3

Ex. 2. Add together 3 a&2 _ 2 &3 4. ^3 . 5 a'2b -ab'^-3a^; Sa^ + 6b^; 9a'^b-2 a^ + ab^.

-2b^2 + 3ab^2 + a^3 -ab^2+ 5 a^2b — 3a^3 5b^3 +8 a^3 ab^2 + 9a^2b — 2a^3 3b^3 + 3ab^3- + 14 a^2b + 4 a^3 Here each expression is arranged according to descending powers of b, and ascending powers of a,