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Inhaltsangabe
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1856 Excerpt: ...interest--play an important part in the Theory of Equations. For instance, it is the occurrence of what I have proposed to term critical forms of those functions which prevents us from reducing the general solution of equations of the sixth, to that of equations of the fifth degreef. On the other hand, there are similar forms which facilitate the discussion of certain problems. In what follows it will be seen that I avail myself of the labours J of Mr. Jerrard on Symmetric Functions, without adhering implicitly to his notation. Let F»(#) = 0 (1) denote the general equation of the nth degree in x; also let y = F + Qxt +etc (2), Philosophical Magazine, ser. iii, vol. xxviii., p. 191. f Ibid, p. 395. J Mathematical Researches, by George B. Jerrard, A.B. then if we eliminate x between (1) and (2), we shall have a result of the form y+Ptf"-1 +p&n-2 +... + pry-r+...+pn= 0....(3). Now Mr. Jerrard has shown that (in such a case as the present) Pr = t/ntfo+Qq +etc.y (4), the development of which last expression, or of similar ones, will be seen on referring to Mr. Jerrard's Researches f. I shall apply the relation (4) to the investigation of some interesting properties of certain symmetric functions; and, first, let us consider the expression 2»j»2--(re--I) pi1, which we may denote by o. Then we have a=«/n(Po + Q? + etc.f--(n--1) /„(Po + Q? + etc.) or, as we may write it, a = AP + 2BQP+etc., and P2 and QP disappear from a. So if, in the expression for y, Q'a:«' be any term subsequent to Qxq we might show, in the same manner, that Q'P disappears from a, and we may hence infer that a is altogether free from P, whatever be the number of terms in the expression for y, and a is, consequently, the critical function of the second deg...
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Reseña del editor
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1856 Excerpt: ...interest--play an important part in the Theory of Equations. For instance, it is the occurrence of what I have proposed to term critical forms of those functions which prevents us from reducing the general solution of equations of the sixth, to that of equations of the fifth degreef. On the other hand, there are similar forms which facilitate the discussion of certain problems. In what follows it will be seen that I avail myself of the labours J of Mr. Jerrard on Symmetric Functions, without adhering implicitly to his notation. Let F»(#) = 0 (1) denote the general equation of the nth degree in x; also let y = F + Qxt +etc (2), Philosophical Magazine, ser. iii, vol. xxviii., p. 191. f Ibid, p. 395. J Mathematical Researches, by George B. Jerrard, A.B. then if we eliminate x between (1) and (2), we shall have a result of the form y+Ptf"-1 +p&n-2 +... + pry-r+...+pn= 0....(3). Now Mr. Jerrard has shown that (in such a case as the present) Pr = t/ntfo+Qq +etc.y (4), the development of which last expression, or of similar ones, will be seen on referring to Mr. Jerrard's Researches f. I shall apply the relation (4) to the investigation of some interesting properties of certain symmetric functions; and, first, let us consider the expression 2»j»2--(re--I) pi1, which we may denote by o. Then we have a=«/n(Po + Q? + etc.f--(n--1) /„(Po + Q? + etc.) or, as we may write it, a = AP + 2BQP+etc., and P2 and QP disappear from a. So if, in the expression for y, Q'a:«' be any term subsequent to Qxq we might show, in the same manner, that Q'P disappears from a, and we may hence infer that a is altogether free from P, whatever be the number of terms in the expression for y, and a is, consequently, the critical function of the second deg...
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