If we return to the equations themselves, they become

This last equation has both its members alike, and is sometimes called an identical equation. The values of the unknown quantities cannot be determined from it. In fact, since m is

zero, both couriers set out from the same point. And since they both travel at the same rate, they are always together. Therefore there is no point where they can be said to come together. The expression is here an expression of an indeter

minate quantity.

There are some cases where an expression of this kind is not a sign of an indeterminate quantity, but in these cases it arises from a factor being common to the numerator and denominator, which by some suppositions becomes zero, and renders the fraction of the form of ; but being freed from that factor, it

rator and denominator contain the factor a b, which becomes zero when a and b are equal.

Dividing by ab, the expression becomes

a (a + b)

which is equal to 2 a when ab.

Ο

It is necessary then, when we find an expression of the form before pronouncing it an indeterminate quantity, to see if there is not a factor, common to the numerator and denominator, which, becoming zero, renders the expression of this form.

The example of the couriers furnishes some other curious cases, for which we must refer the learner to Lacroix's or Bourdon's Algebra.

Let the learner examine the following examples in a similar

manner.

In Art. IX. examples 15 and 16, the following formulas, relating to interest, were obtained. How are r and t to be interpreted, when p is greater than a; and how when a and p are equal?

In Art. XXII. examples 12th and 13th, the following formulas were obtained. In what cases will the results become negative, and how are the negative results to be interpreted?

It is required to divide a given number a into two such parts, that if r times one part be added to s times the other part, the sum will be a given number b.

Ans. The part to be multiplied by r is

b

as

7-8

b

In what cases will one or both of these results be negative ? Can both be negative at the same time? How are the negative results to be interpreted? In what cases will either of them become zero? Can both become zero at the same time? What is to be understood when one or both become zero? In what cases, will one or both become infinite or impossible? Can either of them ever be of the form?

XXVII. Equations of the Second Degree.

1. A boy being asked how many chickens he had, answered, that if the number were multiplied by four times itself, the product would be 256. How many had he?

This equation is essentially different from any which we have hitherto seen.

It is called an equation of the second degree, because it contains a2, or the second power of the unknown quantity. In order to find the value of x, it is necessary to find what number, multiplied by itself, will produce 64. We know immediately by the table of Pythagoras that 8 x 8 64. Therefore

Note. The results of these equations may be proved like those of the first degree

2. A boy being asked his age, answered, that if it were multiplied by itself, and from the product 37 were subtracted, and the remainder multiplied by his age, the product would be 12 times his age. What was his age?

x x x = x2 (x2 — 37) x — x3 — 37 x.

3. There are two numbers in the proportion of 5 to 4, and the difference of whose second powers is 9. What are the numbers?

4. There are two numbers whose sum is to the less in the proportion of 15 to 4, and whose sum multiplied by the less produces 135. What are the numbers?

Let the less, and y = the greater.

Putting this value of y into the first, it becomes

Hence it appears, that when an example involves the second power of the unknown quantity, the value of the second power must first be found in the same manner as the unknown quantity is found in simple equations; and from the value of the second power, the value of the first power is derived.

It is easy to find the second power of any quantity, when the first power is known, because it is done by multiplication; but it is not so easy to find the first power from the second. It cannot be done by division, because there is no divisor given. When the number is the second power of a small number, the first power is easily found by trial, as in the above examples. When the number is large, it is still found by trial; but a rule may be very easily found, by which the number of trials will be reduced to very few. The first power is called the root of the second power, and when it is required to find the first power from the second, the process is called extracting the root.

It has been shown, Art. XXIV. that the second power of every quantity, whether positive or negative, is necessarily positive; thus 3 x 3 = +9, and also -3x-3=9. So

ax a = a2, and also αχ a = a2. Hence every second power, properly speaking, has two roots, the one positive and the other negative. The conditions of the question will generally show which is the true answer.

XXVIII. Extraction of the Second Root.

In order to find a rule for extracting the root, or finding the first power from the second, it will be necessary, first, to observe how the second power is formed from the first.