and either of these equations signifies that the bodies are together when the second body starts, the first body having just arrived at the point of departure of the second, and zero is, therefore, to be regarded as a real solution.

3. In what cases would the value of one of the unknown quantities in example 38 of art. 126 become zero? and what would this value signify?

Ans. When either

a = c, or b = c ;

and, in either case, these equations indicate that the price of one of the wines is just that of the required mixture, and, of course, needs none of the other wine added to it to make it of the required value; and zero, must, therefore, be regarded as a true solution.

4. In what cases would the value of one of the unknown quantities in example 39 of art. 126 become zero? and what would this value signify?

Ans. When

b=na, or = ma;

=

b;

and these equations indicate that a is itself such that, multiplied either by m or by n, it gives a product and zero may be regarded as a true solution, expressing that one of the parts is zero, while the other is the number a itself.

5. In what cases would the value of one of the unknown quantities in example 41 of art. 126 become zero? and what would this value signify?

Ans. First. When

a = 0, or b = 0,

and, in this case, zero is a true solution by regarding all numbers as having the same ratio to zero.

Cases in which the value of an unknown quantity is infinite.

and, in this case, the problem is impossible, for no two numbers can be in the ratio a: b, and, without having any thing added to or subtracted from them, acquire the different ratio m: n.

and, in this case, the problem is impossible, for no two numbers, whose ratio =a: b, and which are therefore unequal, can, by the addition of c to each of them, become equal to each other, as required by the ratio m: n=m: m = 1.

129. When the solution of a problem gives, for the values of one of its unknown quantities, any fractions, the denominators of which are zero, while the numerators are not zero; such values are, generally, to be regarded as indicating an absurdity in the enunciation of the problem.

130. EXAMPLES.

1. In what case does the denominator of the fractional value of the unknown quantity in example 25 of art. 126 become zero? and what is the corresponding absurdity in the enunciation of the problem?

and the absurdity is, that, while the couriers are travelling at the same rate, it is required to determine the time in which one will overtake the other.

2. In what case do the denominators of the fractional values of the unknown quantity in example 38 of art. 126 become zero? and what is the corresponding absurdity in the enunciation of the problem?

Cases in which the value of the unknown quantity is indeterminate.

and the absurdity is that, while both the wines are of the same value, they should give a mixture of a value different from their common value.

3. In what case would the denominators of the fractional values of the unknown quantities in example 41 of art. 126 become zero? and what is the corresponding absurdity of the enunciation ?

Ans. When

an= b bm,

that is, when a : b=m: n; and the absurdity is, that the ratio of two unequal numbers should not be changed by increasing them both by the same quantity.

4. In what case would the denominators of the fractional values of the unknown quantities in example 48 of art. 126 become zero? and what is the corresponding absurdity of the enunciation ?

and the absurdity is, that the squares of two equal numbers should differ.

131. Corollary. When the solution of a problem gives for the value of either of its unknown quantities a fraction whose terms are each equal to zero, this value generally indicates that the conditions of the problem are not sufficient to determine this unknown quantity, and that it may have any value whatever. In some cases, however, there are limitations to the change of value of the unknown quantity.

Cases in which the value of an unknown quantity is indeterminate.

132. EXAMPLES.

1. In what case would both the terms of the fractional value of the unknown quantity in example 25 of art. 126 become zero? and how could this value be a solution?

Ans. When b = a, and n =

0;

and these equations signify, that the couriers travel equally fast, and start at the same time; and, therefore, they remain together, and any number whatever may be taken as the value of the unknown quantity.

2. In what case would both the terms of either of the fractional values of the unknown quantity in example 31 of art. 126 become zero? and how could this value be a solution?

= c;

Ans. When α= 0, and C and these equations signify, that the bodies move equally fast, and start from the same place; they, therefore, remain together, and any number whatever may be taken as the value of the unknown quantity.

But, in this case, all the algebraic values of the unknown quantity but the first become infinite, as they should, because they are obtained on the supposition, that the second body has passed round the circle once, twice, &c., oftener than the first body; which is here impossible.

3. In what case would all the terms of the fractional values of the unknown quantities in example 38 of art. 126 become zero? and how could they, then, satisfy the conditions of the problem?

and these equations signify, that the wines and the mixture are all of the same value; in whatever proportion, therefore, the wines are mixed together, the mixture

Cases in which the value of an unknown quantity is indeterminate.

must be of the required value. But the values of the unknown quantities are still subject to the limitation that their sum is n.

4. In what case would the terms of the fractional values of the unknown quantities in example 39 of art. 126 be. come zero? and how could they, then, satisfy the conditions of the problem?

Ans. When

m = n, and b = na = ma;

and these equations signify, that the sum b of the products of the parts of a multiplied by m = n is to be equal to the product of a multiplied by n; and this is, evidently, the case into whatever parts a is divided.

5. In what cases would all the terms of the fractional values of the unknown quantities in example 41 of art. 126 become zero? and how could they, then, satisfy the conditions of the problem?

for these equations indicate that the two required numbers are only subject to the condition that their ratio = a: b. Secondly. When

m = n, and a : b = m : n=m: m=1, that is, a=b; for these equations indicate that the two numbers are to be equal; and that they are to remain equal, when they are increased by c, which would always be the case.

6. In what case would all the terms of the fractional values of the unknown quantities in example 48 of art. 126 become zero? and how could these values be solutions?

Ans. When α= 0, and b = 0;

and their equations indicate that the numbers are to be equal, and that their squares are to be equal, which is always the case with equal numbers.