Moreover, if the values of all the unknown quantities take

the form

0

0

we cannot affirm certainly that the given equations

are consistent, but not independent. For example, suppose the equations to be

ax+by+cz=d,

ax+by+cz = d', ax+by+cz=

d";

here it will be found that the values of all the unknown quan

0
0'

tities take the form but the equations themselves are obviously

inconsistent, unless d, d', and d" are all equal.

216. We may shew that if the numerators in the values of x, y, and z, all vanish, the denominator will also vanish, assuming that d, d' and d" are not all zero.

For supposing these numerators to vanish we have

d (b′′c′ – b'c') + d′ (bc" − b′′c) + d′′ (b'c — bc′) = 0,

d (c'a' — c'a') + d′ (ca′′ – c′′a) + d′′ (c'a — ca′) = 0,

d (a′′b′ — a′b′′) + d′ (ab” — a′′b) + d′′ (a′b — ab′) = 0.

Let us denote these relations for shortness thus,

Ad+ Bd' + Cd" =0, A'd + Bd'+Cd" =0, A"d+ B"d′+C"d" = 0.

By Art. 213, since d, d' and d" are not all zero the following relation must also hold,

A (B′C" – B′′C'′) + A′ (B′′C − BC′′) + A′′ (BC′ — B′C) = 0.

It will be found that

BC" – B′′C′ = a {a (b′c′′ – b′′c) + a′ (b′′c – bc") + a′′ (bc′ – b'c)} ; and B"C-BC" and BC' - B'C may be similarly expressed, so that finally the relation becomes

— {a (b ́c′′ — b′′c) + a′ (b′′c − bc′′) + a′′ (bc′ — b’c)}2 = 0.

This establishes the required result.

217. If we adopt the method of indeterminate multipliers given in Art. 207, it may happen that the equations for finding I and m are inconsistent; we will examine this case. Suppose then b'c-b'c" = 0, so that these equations are inconsistent (Art. 205). In this case the value of x may be obtained from the

second and third of the given equations, without using the first. For multiply the second of the given equations by c", and the third by c', and subtract; thus the coefficients of y and z vanish, and we have an equation for determining x. For example, suppose the equations to be

4x+2y+32= 19,

x + y + 4z

=

9,

x+2y+8x=15.

Here the value of x may be found from the second and third equations; we shall obtain x = 3; substitute this value of x in the three given equations; from the first we have 2y + 32 = 7, and from the second or third y + 4x=6; hence y = 2 and z = = 1.

Again, the values of 7 and m may take the form

0

so that the equations for finding them are not independent; we will examine this case. Here we have b′′c′ – b'c′′ = 0, bc" – b′′c = 0, and b'c-bc' = 0; these suppositions are equivalent to the two relations

c"

с

==

cpc, and that b" qb, and therefore c" qc. Thus the given

=

equations are

ax+by+cz= =d,

=

a'x+pby + pcz = d′,

and they may be written thus,

=

a"x+qby+qcz=d",

Here x may be found from any two of the equations; if we do not obtain the same value from each pair, the given equations are of course inconsistent; if we do obtain the same value for x, then the given equations are not independent; and in fact we shall in the latter case have only one equation for finding by + cz, so that the values of y and z are indeterminate. For example, suppose the given equations to be

x + 2y + 3z = 10,

3x+4y+6z=23,

x+6y+9z=24.

From any two of these equations we can find x=3; then substituting this value of x in any one of the three equations we obtain 2y+32=7, and thus y and z are indeterminate. If, however, the right-hand member of one of the given equations be

altered, we shall not obtain the same value of x from each pair of the equations, and thus the given equations will be inconsistent.

218. In the preceding articles we have supposed the given equations to be solved, and from the peculiar forms of the solutions have drawn inferences as to the nature of the given equations. We will now take one example of investigating a relation between the equations without solving them. Suppose, as before, that the equations are

ax+by+cz= d,

a'x+by+cz=d',

a"x+by+c"z=d" ;

and let us find the relations which must exist among the known quantities, in order that the third equation may be deducible from the other two by multiplication by suitable quantities and addition. Suppose then that by multiplying the first equation by λ, and the second by μ, and adding, we obtain a result which is coincident with the third equation. Thus,

(λa + μa) x + (λb + μb') y + (λc + μc') ≈ = λd + μď′

Hence in order that the third equation may be deducible from the other two in the manner proposed, we must have the following relations among the known quantities,

a"d'-a'd" b"d′-b'd" c'd - c'd"

=

=

ad" - a'd bd"-b"d cd"-c"d'

It is easy to shew that if these relations hold, the values of

0

x, y, and z take the form For by multiplying up we obtain

0°

results which shew that the numerators in the values of x, y, and z vanish; and then by Art. 216 the denominator will also vanish.

(a+b+c) (a3 + b3 + c3 + abc) − (ab + bc + ca) (a2 + b2 + c2) = a* + ba + c*.

6. A person leaves £12670 to be divided among his five children and three brothers, so that after the legacy duty has been paid, each child's share shall be twice as great as each brother's. The legacy duty on a child's share being one per cent. and on a brother's share three per cent., find what amounts they respectively receive.

shew that the sum of the products of every two of the quantities

x-α, x-b, x − c,.

......

will be equal to the sum of the products

of every two of the quantities a, b, c,

XVI. INVOLUTION.

219. If a quantity be continually multiplied by itself, it is said to be involved or raised, and the power to which it is raised is expressed by the number of times the quantity has been employed in the multiplication. The operation is called Involution.

Thus as we have stated (Art. 16), a xa or a2 is called the second power of α; αχαχα or a3 is called the third power of a; and so on.

220. If the quantity to be involved have a negative sign prefixed, the sign of the even powers will be positive, and the sign of the odd powers negative.

221. A simple quantity is raised to any power by multiplying the index of every factor in the quantity by the exponent of that power, and prefixing the proper sign determined by the preceding article.

mn

Thus a raised to the nth power is am"; for if we form the product of n factors, each of which is a", the result by the rule of multiplication is am". Also (ab)" = ab × ab × ab... to n factors, that is, a xa xa... to n factors × b× b× b... to n factors, that is, a" × b". Similarly, a2bc raised to the fifth power is a1o13c3. Also a raised to the nth power is a"", where the positive or negative sign is to be prefixed according as n is an even or odd number. Or as a" - 1x a", the nth power of a" may be written thus (1)" x am" or (-1)"a"".

mn

-

222. If the quantity which is to be involved be a fraction, both its numerator and denominator must be raised to the proposed power. (Art. 142.)

T. A.

9