be apparently, if not absolutely, indeterminate, in consequence of our being unable to assign the position in the period to which it corresponds: thus, the values of cos 0 and sine are periodical, their limits being 1 and 1, through the intervals of which they pass, whilst 0 passes through intervals of value equal to 2π. What then is the value of cos∞ and sin ∞? What is the value of tan∞ or cot∞? The answers to these questions are founded upon indirect considerations, to which we should never recur, when other resources are within our reach: they form points of transition between the demonstrated and acknowledged truths of deductive science and the less definite results of metaphysical speculation. • See Mr De Morgan's Memoir above referred to, Sect. : it is usual to assign 0 as their values, inasmuch as 0 is the mean of their successive periodic values for it is equally probable that the value in question is a or - a: and if the value is assumed to be unique, O is the only value which the doctrine of chances, or similar considerations, would assign to it. :: CHAPTER XXXIX. Rational algebraical ON THE DECOMPOSITION OF RATIONAL FRACTIONS WITH COM- 948. It is usual to distribute arithmetical fractions into two fractions classes as proper and improper (Art. 95), according as the numerator distinguish is less or greater than the denominator: and the same denominations are extended to rational algebraical fractions according as the dimensions of the symbols in their numerators are lower or higher ed into two classes as proper and improper. Improper fractions 949. In the same manner, that improper numerical fractions are reducible, by the division of the numerator by the denominator, to an integer and a proper fraction, so likewise improper expression algebraical fractions are reducible by a similar process of division to an integral expression and a proper algebraical fraction: in are reducible to a rational and a proper Example lution of an algebraical expression and and similarly in all similar cases. a proper algebraical fraction: 950. But the proper fraction, which is thus obtained, is of the reso-farther resolvible into other and more simple fractions, when its algebraical denominator is resolvible into factors: thus suppose it was refraction. quired to resolve the fraction x + 1 x2-7x+12 into partial and more simple fractions, whose denominators are the factors - 3 and x - 4 of its denominator where A and B are unknown and are required to be determined. If we add the partial fractions together, we get Inasmuch as the value of x is indeterminate, this equation can only exist for all values of x, by supposing the corresponding terms of both its members to be severally identical with each other; we thus get A+B=1, 4A+3B=-1, and therefore A=-4, and B=5: we consequently find 951. The principles which are involved in this process, Principles though simple and obvious, are of considerable importance and of very extensive application. In the first place, the relation between x + 1 and (A + B) x − (4 A + 3 B) is not merely one of equality, but also x2 - 7x+12 of identity, where x may have any value whatsoever, provided it is simultaneous in both: and we thus get as many conditions, or equations, as there are corresponding terms on each side, and therefore as many equations as there are unknown quantities A and B to be determined. which are involved in this pro cess. Identity and equality. General process for resolving rational fractions with de composable denomina tors. When the factors of the deno minator are More generally, if any two series, proceeding according to the same law, such as a + bx + cx2 + dx3 +......(1)* a+ẞx+yx2 + dx3 + ......(2) be identical as well as equal, the corresponding terms of each series must be identical also, which will furnish as many equations as there are terms in each: for if not, the value of x can be no longer indeterminate, admitting of all values between zero and infinity inclusive: we have assumed the truth of this proposition in the investigation of the series for a* in Art. 898. If the members of the equation were merely equal to each other and not identical, the symbols A and B would be indeterminate, and the value of x (as would appear by the ordinary solution of the equation) would be expressible in terms of them, and therefore be dependent upon them: it is the additional hypothesis of the identity of form as well as of the equality of value of the two members of this equation, which furnishes the equations for the determination of A and B, and leaves x indeterminate, as is the case with the symbols involved in any other identical equation. 952. The following method of resolving rational fractions with decomposable denominators, into an equivalent series of partial fractions, is general, and will serve to illustrate the preceding observations. M Let be a proper and rational fraction, and let a+bæ be N a factor of N: if we make N = (a + bx) Q, we get For if not, by subtracting the equal expressions (1) and (2) from each other, we get 0 = a−a + (b −ẞ) x + ( c − y ) x2 + (d − d) x3 + &c., where can be neither zero nor infinity, unless a - a and the other coefficients are also zero. If we now suppose M to become m and Q to become q, 953. If (a+bx)' be a factor of N, where r is greater than 1, When some it will be found, if we make N = (a + bx) Q, that a + bx will still be a factor of Q, and consequently P(a+bx) of the factors are will become O possible and equal. and not necessarily 0, when a + bx=0: under such circumstances we make N=(a+bx)' Q, and assume wise P would be infinite: it follows therefore that a+br is also a factor of M-AQ, and if this factor be obliterated by division, the value of P will be found. |