198. The following Examples will serve to illustrate Examples. the application of this rule. (1) .333.... The repeating period is 3, commencing from the decimal point. The equivalent fraction is, therefore, The repeating period is 125, commencing from the decimal point; and the corresponding fraction is The repeating period is .02439, and the fraction cor (4) .0232558139534883720930232558.... The repeating period consists of 21 places, commencing from the decimal point; and the fraction corresponding is 23255813953488372093 999999999999999999999 when reduced to its lowest terms. (5) .63 64 64 64 .......... The repeating period is 64, and the non-repeating part 63: their difference is 1: and the fraction corres6300 + 1 6301 ponding to it is, therefore, (6) 1.142857142857.... The non-repeating part is 1, and the repeating period is 142857: the fraction corresponding is, therefore, 1142857-1 1142856 8 = 999999 CHAP. VIII. ON INVERSE OPERATIONS IN ALGEBRA, AND ON THE Ex- of detecting of a pro 199. If the product of two or more algebraical quan- Difficulty tities be required, the process for finding it is general the compoand certain; but if the product alone is given, and it is nent factors required to find the factors, the question becomes more duct. difficult, and in many cases it admits not of solution, and as far as the signs of the factors are concerned, it is always to a certain extent ambiguous. It is not our intention at present to enter upon the general question of the resolution of an algebraical product into its component factors, as we shall have occasion to discuss it when we come to the general theory of equations: we shall merely mention in the first instance, a few cases, where such a resolution may be effected, and afterwards proceed to the specific consideration of the extraction of the square, cube and other roots. It must be kept in mind that our attention is confined to the determination of such factors as present themselves under a rational and possible form. 200. The factors, if any exist, of any homogeneous Factors of expression may be found, where the symbols are equally any expres involved. sion may be found which is homogeneous and For the factors must be symmetrical expressions of the symbols, and the dimensions of one of them at least must symmetrinot exceed one half of the dimensions of the original product. cal. Examples. It is merely requisite, therefore, to form a series of symmetrical functions of the symbols, and so find by trial those which will succeed. 201. To find the factors, if any, of the expression a2b+ a°c + ab2 + b°c + ac2 + bc2+3abc. Since the symbols a, b, c are equally involved, it follows that a+b+c must be a factor in this case, if any exist, and it is found by trial to succeed. The two factors are a+b+c and ab+ ac + bc. (2) To find the factors, if any, of a1+b*+ c1 + a2b2 + a2c2 + b2 c2-2abc-2b2ac-2c2 ab. a2 + b2 + c2 ab + ac + bc a2 + b2+c2 + ab + ac+bc a2+b2+c2-ab-ac-bc. The two last of which succeed. It is not necessary to make trial of every symmetrical function which may present itself, whose dimensions are within the required limits, as there may be some which clearly cannot succeed: of this kind is ab+ac + be; for in case it was a factor, there could be no such terms as a1, b1, c1. (3) To find the factors, if any, of 2a2b2+2a c2+2b2 c2 — a1 — b1 — c1. The first factor, a+b+c, of which trial is made, is found to succeed, and the result of the division by it, gives us a2b+a2c + ab2 + ac2 + b°c + bc2 — a3 — b3 — c3-2abc. - If this quantity be resolvible into other factors, and if one of them be b+c-a, there must be other two, which are a+b-c and a+c-b; otherwise the three symbols a, b, c, would not be equally involved, and the expression would not be symmetrical with respect to them: they will be found by trial to succeed. (4) To find the factors, if any exist, of a3 — b3 + c3-2ab+2ab2 + 2a°c + 2 ac2 + 26°c -2bc-3abc. A very slight examination of this expression would shew that it would be symmetrical with respect to a, b and c, if the sign of b was changed: in other words, it is symmetrical, with respect to a, b and c, and the factors, if any, must be symmetrical functions or combinations of a, -b and c. The only such combination of one dimension is a-b+c, which is found to succeed: the other factor is a2+be+c2 - ab+ac-bc. sions where same sym 202. We have already considered the resolution of Expres expressions into their component factors, in which one one power power only of each symbol appears (Art. 173. Ex. 10): only of the in this case, the same symbol can present itself in one bol is factor only, and therefore the product of all the others must be a factor of its coefficient: the following are examples: (1) To find the factors of 60xyz +72xy +75xx+80yx+90x + 96y+ 100≈ + 120. The coefficient of x is 60y≈ + 72y + 75≈ + 90, and if the whole expression be divided by it, we get a finite (since there is no fraction in the product) is one of the found. Examples. factors required. Z |