Cases of negative value of unknown quantity.

133. Corollary. When the solution of a problem gives a negative value to either of the unknown quantities, this value is not generally a true solution of the problem; and if the solution gives no other than negative values for this quantity, the problem is generally impossible.

But, in this case, the negative of the negative value of the unknown quantity is positive; so that the enunciation of the problem can often be corrected by changing it, so that this unknown quantity may be added instead of being subtracted, and the reverse.

134. EXAMPLES.

1. In what case would the value of the unknown quantity in example 25 of art. 126 be negative? why should it be so? and could the enunciation be corrected for this case?

a > b ;

Ans. When that is, when the second courier goes slower than the one he is pursuing, in which case he evidently cannot overtake him; and the enunciation does not, in this case, admit of a legitimate correction.

2. In what case would the values of the unknown quantities in examples 29, 31, 32 of art. 126 be negative? why should this be so? and could the enunciations be corrected for this case?

that is, when the first body moves faster than the second, in which case the second cannot overtake it.

The enunciation may be corrected for this case by supposing the bodies to travel in the opposite direction to

Cases of negative value of unknown quantity.

that which they are at present taking, that is, by supposing the first body to pursue the second.

Examples 31 and 32 are not, however, impossible in this case; for, from the very nature of their circular motion, the first body is necessarily pursuing the second even in their present direction; the second body must not, however, be considered as a feet or act feet behind the first, but as p―a or p― (a+c t) feet before it.

3. In what cases would the values of the unknown quantity in example 33 of art. 126 be negative? why should this be the case? and could the enunciation be corrected for this case?

which is subject to the same remarks as in the preceding question.

and c t > a, or >p+a, or>2p+a,&c.;

that is, when the first body does not start until the second body has passed it once, or twice, or three times, &c.; and if the bodies were moving in the same straight line, the enunciation would not admit of legitimate correction. As it is, however, the first body is still pursued by the second, and is pact, 2p+a-ct, &c., 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. 126 be negative? why should this be the case? and could the enunciation be corrected for this case?

Ans. When

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,

Cases of negative value of unknown quantity.

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 pact, or 2p+act, &c., feet; so that all the positive values of the unknown quantity remain as true solutions.

5. In what cases would the values of either of the unknown quantities in example 38 of art. 126 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,

that is, when the price of the most expensive wine is less than that of the required mixture.

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

In either case the problem is altogether impossible, 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. 126 be negative? why should this be so? and could the enunciation be corrected for this case?

Ans. Supposing, as we may, that m>n ;

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

Cases of negative value of unknown quantity.

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 numbers 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 numbers, and b the difference of the products obtained 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. 126 be negative? why should this be so and could the enunciation be corrected for this case?

Ans. First. When

m>n, and an <b m, or a: b<m: n ;

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

Secondly. When

m<n, and a:b>m: n;

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. 126 be negative? why I should this be so? and could the enunciation be corrected

that is, when the difference of the squares of the parts of a is to be greater than the square of the number itself, which can never be the case; for the greatest possible difference of squares corresponds to the case in which one of the parts is the number a itself, and the other is zero; and

One Equation with several unknown quantities.

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.

135. 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.

SECTION IV.

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

136. 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 val

may all be used to denote different quantities, though they generally are supposed to imply some similarity between the