Hence b c is a factor of the given expression; and we can prove in a similar manner that c a and ab are also factors. Now the given expression is of the fourth degree; hence, besides the three factors we have found, there must be one other factor of the first degree, and as this factor must be symmetrical in a, b, c, it must be a + b + c.

Hence the given expression must be equal to

L (b − c) (c − a) (a − b) (a + b + c),

where L is a number.

We can find L by giving particular values to a, b, and c; or, by comparing the coefficients of a3, we have at once

=

Thus b3 (c- a) = − (b − c) (c − a) (a − b) (a + b + c).

154. The following is an important identity: a+b+c3-3 abc = (a+b+c) (a2 + b2 + c2 — bc — ca- ab). It should be noticed that

a2+b2 + c2 — bc-ca-ab-1} (b-c)2+(c−a)2+(a−b)2}. ca—ab=1{(b−c)2 Since a+b+c is a factor of a3 + b3 + c3 – 3abc, it follows that a3 + b23 + c3 — 3abc = 0 if a+b+c= 0. Hence a+b3 + c3 : 3 abc for all values of a, b, and c,

=

provided only that a+b+c= 0.

That is, the sum of the cubes of any three quantities is equal to three times their product, provided that the sum of the three quantities is zero.

For example, the letters involved being a, b, c,

(bc)33 (bc) (ca) (a - b),

Σ (b + c − 2 a) 3 = 3 (b + c − 2 a) (c + a − 2 b) (a + b − 2 c) ; also, since (x a) (b − c) + (x − b ) ( c − a) + (x — c) (a − b) = 0, we have

(x − a)3 (b −c)8 = 3 (x − a) (x − b)(x − c) (b−c) (c − a) (a - b).

EXAMPLES XLII.

Let the student employ the Σ notation for the purpose of abbreviating such of the following expressions as admit of its use.

3. Show that

+ c2 + bc + ca + ab).

(b − c)5 + (c − a)5 + (a − b)5

= 5 (b − c) (c − a) (a − b) (a2 + b2 + c2 − bc − ca — ab).

4. Show that

(a+b+c)+ − (b + c)+ − (c + a)± − (a + b)1 + a1 + b2 + c1 · 12 abc (a + b + c).

=

-

a1 (b2 — c2) + b2 (c2 − a2) + c1 (a2 — b2)

- (b + c) (c + a) (a + b) (b − c) (c − a) (a − b).

9. Find the factors of be (b2 - c2) + ca (c2 — a2) + ab (a2 — b2).

10. Find the factors of a (b + c − a)2 + b(c+a−b)2+c(a+b−c)2

11. Find the factors of a2 (b + c − a) + b2 (c+a−b)+c2 (a+b−c) - (b + c − a) (c + a · b) (a + b − c).

12. Find the factors of a (b+c) (b2 + c2 − a2)

+ b (c + a) (c2 + a2 − b2) + c (a + b) (a2 + b2 — c2).

13. Find the factors of

(yz)3 + (z - x)3 + (x − y)3 — k(y − z) (z − x) (x − y).

CHAPTER XIV.

FRACTIONS.

155. To obtain the arithmetical fraction, we must divide the unit into 7 equal parts and take 5 of those parts. So also to obtain the fraction, where a and b

are positive integers, we must divide the unit into b equal parts and take a of those parts.

156. The numerator and the denominator of a fraction, defined as in Art. 155, must both be positive integers: we cannot, for example, with that definition, have

a

We must therefore suppose the letters in to be re

b

stricted to positive integral values, or we must alter the definition of and as we cannot restrict the values of

the letters, we must entirely dispense with the fractional form, or make some modification in its meaning.

Now, with the definition of Art. 155, to multiply the

a

fraction by b, we must take each of the a parts b

times; we thus get ab parts, the parts being such that each b of them make up a unit, and therefore the whole ab parts will make up a units.

Thus

a

Now we may define the fraction as that quantity which

b

when multiplied by b becomes a; for, as we have just seen, this new definition agrees with that of Art. 155, whenever the definition of Art. 155 has meaning; and by taking this new definition we do away with the necessity of ascribing only positive integral values to the letters.

a

b

We may similarly define the fraction as the quotient obtained by dividing a by b.

Hence, instead of the definition of Art. 155, which is inapplicable to an algebraical fraction, we have either of the following equivalent definitions.

Def. I. The algebraical fraction, where a and b are

b

supposed to have any values whatever, is that quantity which, when multiplied by b, becomes equal to a.

Def. II. The algebraical fraction

obtained by dividing a by b.

a

is the quotient

b

The fractional form has already been used with the b

meaning a÷b, and henceforth the notation a/b will also be frequently employed to denote a fraction.

157. We now proceed to consider the properties of algebraical fractions; and we shall find that algebraical fractions are added, subtracted, multiplied, divided, and simplified, precisely in the same way as arithmetical fractions. It will be assumed throughout the discussion that the quantities involved are all finite and different from zero.

158. The value of a fraction is not altered by multiplying its numerator and denominator by the same quantity. We have to prove that

Divide by bm, and we have x = am ÷ bm ;

Thus the value of a fraction is not altered by multiplying its numerator and denominator by the same quantity.

159. Since, by the last article, the value of a fraction is not altered by multiplying both the numerator and the denominator by the same quantity, it follows conversely