Two couriers, A and B, in the course of a journey between the towns C and D, are at the same moment of time at A and B. A goes m miles, and B, n miles an hour. At what point between C and D are they together? It is evident that the answer depends upon whether they are going in the same or opposite directions, whether A goes faster or slower than B, and so on. But all these, as we shall see, are included in the same general problem, the difference between them corresponding to the different forms of the letters which we shall have occasion to use. After solving the different cases which present themselves, each upon its own principle, we shall compare the results in order to establish their connexion. Let the distance AB be called a. Case first.-Suppose that they are going in the same direction from C to D, and that m is greater than n. They will then meet at some point between B and Let that point be H, and let AH be called x. Then A travels through AH, or x, in the time during which B travels through BH or x-a. But, since A

D.

the point H together. The answers in this case are therefore the same as in the last. Case fourth. Similarly, if they are moving from D to C, and A moves faster than B, the answers are the same as in AH the first case, since this is a reverse of the first case, as the third is of the second. We reserve for the present the case in BH which they move equally fast, as another species of difficulty is involved which has no connexion with the present subject. We shall return to it hereafter. Case fifth. · Suppose them now moving in contrary directions, viz.: A towards D and B towards C, Whether A moves faster or slower than B, they must now meet somewhere between A and B, as before let them meet in H, and let AH = x.

B

Case sixth.-Let them be moving in contrary directions, but let A be moving towards C, and B towards D. They will then have met somewhere between A and B, and as this is only the reverse of the last case, just as the fourth is of the first, or the third of the second, the answers are the same. We now exhibit the results of these different cases in a table, stating the circumstances of each case, and also whether the time of meeting is before or after the instant which finds them at A and B.

The time elapsed is

=

na m + n

which

wise changed, becomes is the value of B H in the fifth, but in a different form. But we observe, that BH falls to the left of B in the fifth, whereas it fell to the right in the first. Again, in the first and sixth examples, which differ in this, that A moves towards D in the second and towards C in the sixth, the value of AH in the sixth may be deduced from that of A H in the first, by changing the form of m, which change

α

m+n

-m

α

or

which is of a different form

from that in the sixth; but it must also be observed, that the first is after and the other before the moment when they are at A and B. In the fifth and sixth examples which differ in this, that the direction in which both are going is changed, since in the fifth they move towards one another, and in the sixth away from one another, the values of AH and BH in the one may be deduced from those in the other by a change of form, both in m and n, which gives the same values as before. But if

m and n change their forms in the expression for the time, the value in the

sixth case is

Also the time in the fifth case is after the moment at which they are at A and B, and in the sixth case it is before. From these comparisons we deduce the following general conclusions:

1. If we take the first case as a standard, we may, from the values which it gives, deduce those which hold good in all the other cases. If a second case be taken, and it is required to deduce answers to the second case from those of the first, this is done by changing the sign of all those quantities whose directions are opposite in the second case to what they are in the first, and if any answer should appear in a negative form,

such as

ma

m'

n-m

which may be written thus it is a sign that the quantity which it represents is different in direction in the first and second cases. If it be a right the cases, such as A H, it is a sign that line measured from a given point in all A H falls on the left in the second case, if it fell on the right in the first case, and between the moment in which the couthe converse. If it be the time elapsed riers are at A and B and their meeting, it is a sign that the moment of meeting is before the other, in the second case, if it were after it in the first, and the converse. We see, then, that these six cases can be all contained in one if we apply this rule, and it is indifferent which of the cases is taken as the standard, provided the corresponding alterations are made to determine answers to the rest.

This detail has been entered into, in order that the student may establish, from his own experience, the general principle, which will conclude this part of the subject. Further illustration is contained in the following problem:

A workman receives a shillings a day for his labour, or a proportion of a shillings for any part of a day which he works. His expenses are b shillings every day, whether he works or no, and after m days he finds that he has gained c shillings. How many days did he work? Let x be that number of days, x being either whole or fractional, then for his work he receives ax shillings, and during the m days his expenditure is

x=

bm-c

α

which only differs from the

former in having -c instead of +c. It appears then that we may alter the solution of a problem which proceeds upon the supposition of a gain into the solution of one which supposes an equal loss, by changing the form of the expression which represents that gain; and also that if the answer to a problem which we have solved upon the supposition of a gain should happen to be negative, suppose it -c, we should have proceeded upon the supposition that there is a loss and should in that case have found a loss, c. When such principles as these have been established we have no occasion to correct an erroneous solution by recommencing the whole process, but we may, by means of the form of the answer, set the matter right at the end. The principle is, that a negative solution indicates that the nature of the answer is the very reverse of that which it was supposed to be in the solution; for example, if the solution supposes a line measured in feet in one direction. a negative answer, such as - c, indicates that c feet must be measured in the opposite direction; if the answer was thought to be a number of days after a certain epoch, the solution shows that it is c days before that epoch; if we supposed that A was to receive a certain number of pounds, it denotes that he is to pay c pounds, and so on. In deducing this principle we have not made any sup position as to what -c is; we have not

asserted that it indicates the subtraction

of c from 0; we have derived the result from observation only, which taught us first to deduce rules for making that alteration in the result which arises from altering +c into c at the commencement; and secondly, how to make the solution of one case of a problem serve

first magnitude to 0, the fraction passes from its first value through every possible greater number. Now, suppose that the solution of a problem in its most