OBS. In algebraical analysis it is frequently useful to observe whether the algebraical expressions under consideration are homogeneous or not, that is, whether the 'dimensions' of every term be the same or not; for, if this homogeneity be found at first, no legitimate operation can destroy it; or, if it be not found at first, it cannot be introduced; and thus an easy test is afforded, to a certain extent, of the accuracy of each succeeding step in the analysis.

For example, if the equation

ax2 + b2 x + c3 = 0,

be proposed for solution, in which every term is of three dimensions, that is, which is homogeneous, every step of the solution will present an homogeneous equation, if it be correct.

As a simple case it may be well to observe that, if the proposed equation be homogeneous, the final result must be so. A proper attention to this observation will frequently detect an error in the process of solution.

PROBLEMS.

IN reducing a Problem to an Equation, the course to be pursued is stated in Art. 194; but much depends here, as in the solution of equations, upon a practical acquaintance with particular artifices, by which the most convenient unknown quantities are assumed, and the problem most easily translated into algebraical language.

The general question always is, having certain known quantities, represented by given symbols, and one or more other unknown quantities, represented by one or more of the letters x, y, z, &c., to connect the known and unknown symbols together by the conditions of the problem, so as to produce as many independent equations as there are unknown quantities.

There is also one general property of a large class of such problems, viz., that the increase or decrease, the selling or buying, &c., is after a uniform rate. Thus, if A is said to perform a piece of work in a days, he is supposed to work equally every day. If A is said to travel p miles in q days, he is supposed to travel one uniform distance each day. And so on, unless the contrary be expressed. So that the following Rule is of constant application, seeing that uniform increase or decrease of every sort may be represented by uniform

motion:

If v represent the space described, by a body moving uniformly, in 1 unit of time, (whether it be 1 second, 1 hour, or any other known unit), and s the space described by the same body in t such units of time, then s = tv.

PROB. 1. In the year 1830 A's age was 50 and B's 35. Required the year in which A is twice as old as B.

PROB. 2. In what proportions must substances of "specific gravities" a and b be mixed, so that the "specific gravity” of the mixture may be c?

DEF. By the "specific gravity" of a body is meant the number of times which its weight is of the weight of an equal bulk of water.

To 1 cubic foot of the first substance let a cubic feet of the second be added.

Then,

since 1 cubic foot of the first weighs a cubic feet of water, and Ꮳ .... feet second...... bx...

.....

.......

.. the whole 1 + x cubic feet of mixture weighs a + bx cubic

feet of water.

But since c is the specific gravity of the mixture, the weight of 1 + x cubic feet is c (1 + x) cubic feet of water,

that is, for every cubic foot of the substance whose specific

gravity is a there must be

whose specific gravity is b.

a-c

с b

cubic feet of the substance

PROB. 3. From a vessel of wine containing a gallons b gallons are drawn off, and the vessel is filled up with water. Find the quantity of wine remaining in the vessel, when this has been repeated n times.

Let x1, x2, 3,..., be the number of gallons of wine remaining in the vessel after 1, 2, 3,...n drawings off respectively.

(2) x2 = a-b- quantity of wine in b gallons of first anixture.

Now, a gallons contain a -b of wine;

And so on, for each succeeding mixture; so that, generally,

X

(a - b)"

PROB. 4. If A and B together can perform a piece of work in a days, A and C together the same in b days, and B and C together in c days; find the time in which each can perform the work separately.

Let w represent the work,

and x, y, z, the times in which A, B, C, can separately do it :

A's daily work + B's daily work.............. (1),

To shew that the denominators of these fractions are necessarily positive:

By the Prob. B alone could not perform w in a days,

But B and C together can perform w in c days;