In order to express this problem in an equation, it is sufficient to set up the condition that the two particles have required the same time in travelling till they meet, the first travelling the distance A R and the second the distance A'R. Let t be the time required; since the distance AR is x- -a, it follows that

whence

x — a = vt,

The following convention concerning the signs of the magnitudes, distance, velocity, and time is adopted:

1. The distances OA, OA', OR are reckoned positively from left to right, and negatively from right to left.

2. The velocity of a particle is regarded as positive or negative according as this particle moves from left to right or from right to left.

3. Finally, the time which has elapsed between the moment that the particles passed A and A' and the moment that they meet at R is reckoned as positive or negative according as the moment of passage of the point R took place after or before the moment the particles passed A and A'; the formulae (1) and (2) are applicable in every case. Therefore, the equation (3), which was established for a particular case, is true for all cases.

205. Discussion. Consider equation (3), 204,

(v — v') x = va' — av',

and examine the various cases which can arise:

1. If v-v0, the equation has a root

then the two particles meet at a point to the right of the point 0 if

is positive, and to the left of the point O if this fraction is negative.

2. If v — v' = 0, two cases must be considered, according as va' av' is different from zero or equal to zero.

(a) If va' — av' is different from zero, the equation does not

have a finite root; the two particles will not meet.

It is easy to explain what this result means a priori. Since one has by hypothesis

it follows that

v = v' and va' — av' # 0,

v'a' — av' # C,

and, therefore, that a is different from a'. Since the two particles are always at some distance apart, a' a, and move at the same rate, they will never meet.

av' is zero, the equation is satisfied for

(b) If, however, va' any finite value assigned to x,

In this case the two movable particles do not separate; since, if ༧ = v' and va' — av'

it follows that

a = a'

and the points A and A' coincide. Therefore, the two particles are at A at the same time, and as they travel to the right, with the same velocity, they do not separate.

The results of the preceding discussion can be arranged in the following table:

v — v' =/= 0; the particles meet.

va' — av' = 0; the particles do not meet.

\ va' — av' == 0; the particles do not separate.

NUMERICAL APPLICATION OF THE SAME PROBLEM

Certain applications are now given in order better to fix the meaning of the formula in the previous discussion:

1. The points 0, A, A' are arranged as indicated in the figure annexed; the two particles travel from left to right. The distance OA is 12 ft. and the distance O' is 14 ft. ; the velocity of the particle which passed A, when the second particle passed A', is 2.5 ft.; the velocity of the second particle is 1 ft.

The particles meet at a point which is 31.333 ft. to the right of the point 0, and after the particles passed the points A and A'.

2. The points O, A, A' are arranged as above, but the particles travel in opposite directions; the particle which passed A, when the other passed ', moves from left to right, with a velocity of 1.5 ft.; the other travels from right to left with a velocity of 1.4 ft.

The particles meet at a point which is 1.448 ft. to the right of the point 0, and after the particles have passed the points A and A'.

3. The points O, A, A' are arranged as above, and the particle which passed A when the other passed A', travels from right to left with a velocity of 2.5 ft; the other travels from left to right with a velocity of 4 ft.

The particles meet at a point which is 2 ft. to the left of 0, and

before the particles passed A and A'.

CHAPTER VI

SIMULTANEOUS LINEAR EQUATIONS

SYSTEMS OF EQUATIONS IN TWO UNKNOWN QUANTITIES.

A SINGLE

EQUATION IN TWO UNKNOWN QUANTITIES HAS AN

INDEFINITE NUMBER OF SOLUTIONS

206. Indeterminate Equations.-Often two unknown quantities satisfy an equation of the first degree. Assign arbitrarily any finite value to one of the unknown quantities; then the other will take a finite and determinate value. Consider the equation

Give any value whatever to y; then for determining x, the above is an equation of the first degree, whose root is

On the other hand, a might be given any arbitrary value and the corresponding value of y would be determined by the equation,

y =

7x 18 2

x=

18+ 10
7
18+ 12

=4

2

x =

=

7

7

Any such set of corresponding values of x and y satisfies the given equation, and therefore gives a solution.

An equation which, like the above, has an infinite number of solutions is called an indeterminate equation.

Observe that equations (1) 206 and (2) have one common solution, namely, x 4 and y = 5. Later it will be shown that these equations have only this solution in common. One might suspect that this is the case by comparing the two systems of values of x and y for equations (1) and (2).

With the single exception, x = 4 and y = 5, equations (1) and (2) have no solution in common, and are for this reason called independent equations. If every set of values of x and y which satisfies (1) were also a solution of (2) the equations would no longer be independent.

are not satisfied by any common set of values of x and y. For, by 1206 and 1207, equations (1) and (2) are satisfied by the values x= 4, y = 5. But equation (3) is evidently not satisfied by these values because 2 4 – 5 – 8 — 5 is not equal to 5. These equations are said to express three independent linear relations between x and y.