Cases of negative value of unknown quantity.

feet before the second, when it starts; so that
all the values given for the unknown quantity

are correct, except the negative ones.

4. In what cases would the values of the unknown quantity in example 35 of art. 94 be negative? why should this be the case? and could the enunciation be corrected for this case?

ct> a, or >p+a, or>2p+a, &c.;

that is, when the first body has passed the second
once, twice, &c. before the second begins to

move.

If the bodies were moving in the same straight line, the second body would be obliged to change its direction, and move in the same direction with the first, and even with this change of enunciation the problem is impossible, if the second body moves slower than the first.

But as it is, the bodies are still moving towards each other in the circumference of the circle; their distance apart at the instant when the second body starts being pa— ct, or 2p+a-c t &c. feet; so that all the positive values of the unknown quantity are true solutions.

5. In what cases would the values of either of the unknown quantities in example 38 of art. 94 be negative? why should this be the case? and could the enunciation be corrected for this case?

Ans. If we suppose, as we evidently may, that

a> b;

one of the values is negative,

First. When a < c;

that is, when the price of the most expensive

wine is less than that of the required mixture.

Cases of negative value of unknown quantity.

Secondly. When b> c;

that is, when the price of the least expensive
wine is more than that of the mixture.

In either case the problem is altogether impos-
sible, for two wines cannot be mixed together so
as to produce a wine more valuable than either
of them without a gain, or less valuable than
either of them, without a loss.

6. In what cases would the value of either of the unknown quantities in example 39 of art. 94 be negative? why should this be so? and could the enunciation be corrected for this case?

Ans. Supposing, as we may, that

that is, when the sum b of the products is less
than the product of a by the least of the num-
bers m and n.

Secondly. When ma<b;

that is, when the sum b of the products is greater
than the product of a by the greater of the num-
bers m and n.

In either of these cases, the problem is plainly
impossible; and, in the corrected enunciation,
a should be the difference of the required num-
bers, and b the difference of the products ob-
tained from multiplying one of the numbers by.
m and the other by n.

7. In what cases would the values of the unknown quantities in example 41 of art. 94 be negative? why should this be so? and could the enunciation be corrected for this case?

Ans. First. When m>n,

and anbm, or a b <m: n;

Cases of negative value of unknown quantity.

that is, when the first ratio is less than the sec-
ond, and the second is greater than unity.
Secondly. When m < n,

and a: b> mn;

that is, when the second ratio is less than the
first, and also less than unity.

In either case the problem is impossible, and
c is to be subtracted instead of being added in
the corrected enunciation.

8. In what case would the value of one of the unknown quantities in example 46 of art. 94 be negative? why should this be so? and could the enunciation be corrected for this case?

Ans. When b> a2;

that is, when the difference of the squares of the
parts of a is to be less than the square of the
number itself, which can never be the case; for
the greatest possible difference of squares cor-
responds to the case in which one of the parts
is the number a itself, and the other is zero;
and the difference of the squares is then just
equal to the square
of a.

The enunciation is corrected for this case by
stating it as in example 48.

99. Corollary. It follows from example 7 of the preceding section that a fraction or ratio, which is greater than unity, is increased by diminishing both its terms by the same quantity; and a fraction or ratio, which is less than unity, is diminished by diminishing both its terms by the same quantity; but the reverse is the case, when the terms are increased instead of being diminished.

One Equation with several unknown quantities.

SECTION IV.

Equations of the First Degree containing two or more unknown quantities.

100. In the solution of complicated problems involving several equations, it is often found convenient to use the same letter to denote similar quantities, accents or numbers being placed to its right or left, above or below, so as to distinguish its different values.

may all be used to denote different quantities, though they generally are supposed to imply some similarity between the quantities which they represent. Care must be taken not to confound the accents and the numbers in parentheses at the right with exponents.

101. Problem. To solve an equation with several unknown quantities.

Solution. Solve the given equation precisely as if all its unknown quantities were known, except any one of them which may be chosen at pleasure; and in the value of this unknown quantity, which is thus obtained in terms of the other unknown quantities, any values whatever may be substituted for the other

Indeterminate Equations referred to the theory of Numbers.

unknown quantities, and the corresponding value of the chosen unknown quantity is thus obtained.

102. Corollary. An equation which contains several unknown quantities is not, therefore, sufficient to determine their values, and is called indeterminate.

103. Scholium. The roots of an indeterminate equation are sometimes. subject to conditions which cannot be expressed by equations, and which limit their values; such, for instance, as that they are to be whole numbers. But their investigation depends, in such cases, upon the particular properties of different numbers, and belongs, therefore, to the Theory of Numbers.

104. Theorem. Every equation of the first degree can be reduced to the form

Ax+ By + C z + &c. + M = 0 ;

in which A, B, C, &c. and M are known quantities, either positive or negative, and x, y, z, &c. are the unknown quantities.

Demonstration. When an equation of the first degree is reduced, as in art. 88, the aggregate of all its known terms may be denoted by M. Each of the other terms must have one of the unknown quantities as a factor, and, by art. 80, only one of them, and that one taken but once as a factor. Taking out, then, each unknown quantity as a factor from the terms in which it occurs, and representing its multiplier by some letter, as A, B, C, &c., the corresponding unknown