152. THEOREM I.-If several fractions are equal, a fraction whose numerator is the sum of the numerators and whose denominator is the sum of the denominators of these fractions, is equal to each of these fractions.

and, in adding these equations member to member,

a + a' + a" = (b + b'+b")q;

since a+a+a" is the product of b+b'+b" by q, q is the quotient of a+a+a" divided by b + b' + b′′.

Thus it follows:

a + a' + a"

b + b' + b "

and let m, m', m", be any positive or negative numbers whatever; it follows that

since each term of a fraction can be multiplied by the same number without altering its value (131), one has

and, on applying the preceding theorem to the equal fractions

THEOREM II.—If several fractions are equal, each of them is equal to a fraction which has for its numerator the square root of the sum of the squares of the numerators, and for its denominator the square root of the sum of the squares of the denominators of these fractions. Thus, if one is given

Since these last fractions are equal, their square roots are equal, and we have

It may be demonstrated, as in the preceding theorem, that whatever the numbers m, m', m' are, one has

EXAMPLE 2. Show that (+12 + rr') (h — h') and (r2h — r'2h')

H

H H2

X

h h h2

Hence, on omitting the common factor in the denominator, it follows