13. If an equation is homogeneous with respect to certain quantities, we may for these quantities substitute in the equation any others proportional to them. For instance, the equation

lx3y + mxy3z+ny3z2 = 0

is homogeneous in x, y, z. Let a, ẞ, y be three quantities proportional to x, y, z respectively.

an equation of the same form as the original one, but with a, B, y in the places of x, y, z respectively.

nominators are all of the same sign, then the fraction

lies in magnitude between the greatest and least of them.

a

Suppose that all the denominators are positive. Let be t

least fraction, and denote it by k; then

where is the greatest of the given fractions.

In like manner the theorem may be proved when all t denominators are negative.

15. The ready application of the general principle involv in Art. 12 is of such great value in all branches of mathematic that the student should be able to use it with some freedom any particular case that may arise, without necessarily introduci an auxiliary symbol.

Each of the given fractions=

sum of numerators sum of denominators

=

x + y + z
a+b+c

.(1).

Again, if we multiply both numerator and denominator of the three given fractions by y + z, z+x, x+y respectively,

Multiply the first of these fractions above and below by x, the second by y, and the third by z; then

16. If we have two equations containing three unknown quantities in the first degree, such as

we cannot solve these completely; but by writing them in the form

It thus appears that when we have two equations of the type represented by (1) and (2) we may always by the above formula write down the ratios xyz in terms of the coefficients of the equations by the following rule:

Write down the coefficients of x, y, z in order, beginning with those of y; and repeat these as in the diagram.

Multiply the coefficients across in the way indicated by the arrows, remembering that in forming the products any one obtained by descending is positive, and any one obtained by ascending is negative. The three results

are proportional to x, y, z respectively.

This is called the Rule of Cross Multiplication.

Example 1. Find the ratios of x y z from the equations 7x=4y+8z, 3z=12x+11y.

By transposition we have 7x-4y - 8z=0,

or

(-4)x(-3)-11x (-8), (-8) x 12-(-3) x 7, 7x11-12x (-4),

that is,

denoting each of these ratios by k, by multiplying up, substituting in (1), and dividing out by k, we obtain

а1 (b2с3 − b¿C2) + b1 (¤‚a3 − с ̧a ̧) +с1 (a2b3 — açb2)=0.

This relation is called the eliminant of the given equations.

whence

k {be (b-c)+ca (ca) +ab (a - b)}=(b−c) (c− a) (a - b),

k{(b-c) (ca) (a - b)} = (b−c) (c − a) (a - b);

.. k=−1;

x=c-b, y=a−c, z=b− a.