Lehrbuch der Chemie: Lehrbuch der Chemie

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Multivolume work

Persistent identifier:: 836202945

Author:: Berzelius, Jöns Jacob

Title:: Lehrbuch der Chemie

Edition title:: 5., umgearb. Orig.-Aufl.

Year of publication:: 1843

Place of publication:: Dresden

Identifier (digital):: 836202945

Language:: German

Publisher of the original:: Arnoldi

Document type:: Multivolume work

Volume

Persistent identifier:: 836326741

Author:: Berzelius, Jöns Jacob

Title:: Lehrbuch der Chemie

Sub title:: Mit einem besonderen Abdruck der Atomgewichts-Tabellen

Scope:: VIII, 1266 Seiten

Edition title:: 5., umgearb. Orig.-Aufl.

Year of publication:: 1845

Place of publication:: Leipzig

Identifier (digital):: 836326741

Signature of the source:: Chem. 88(3),5

Language:: German

Publisher of the original:: Arnoldi

Usage licence:: Public Domain Mark 1.0

Publisher of the digital copy:: Technische Informationsbibliothek Hannover

Place of publication of the digital copy:: Hannover

Year of publication of the original:: 2015

Document type:: Volume

Collection:: Chemistry

Chapter

Title:: [Specielle Beschreibung der Salze.]

Document type:: Multivolume work

Structure type:: Chapter

Chapter

Title:: 5. Salze von Barium.

Document type:: Multivolume work

Structure type:: Chapter

[631 631] SYNOPSIS OF THE THEORY OF EQUATIONS. 7 e y n — 1=0, which roots; it is shown : a + /3i such that form cos 9 + г sin 9; or of the equivalent equations p x = Xa, &c., then a n — p x a n ~ 1 + ... =0, b n — pj) n ~ l + ... =0, Ve have thus the &c., (viz. it is for this reason that a, b, ... are said to be roots of x n — p x x n ~ l +... = 0); ilso obtain a like We are thus led and, moreover, that conversely from the last-mentioned equations, assuming that a, 6,... are all different, we deduce alued function, viz. is is p x = Xa, p 2 = Xab, &c., and x n —p x x lv ~ l + ... = (x — a) (x — b) .... denote the ?i-valued Observe that, if for instance a = b, then the two equations a n — p x a n ~ l +... = 0, b n — p x l') n ~ 1 + ... = 0 would reduce themselves to a single equation, which would not of itself express that a was a double root, that is, that (x — a) 2 was a factor of x n — p x x n ~ x + &c.; but by considering b as the limit of a + h, h indefinitely small, we refer here to the obtain a second equation na n-1 — (n — 1) p x a n ~ 2 + ... = 0, all which follows in ifications) applicable subject, it will be which, with the first, expresses that a is a double root; and then the whole system of equations leads, as before, to the equations p x = Xa, &c. But this in passing: the general case is when the roots are all unequal. We have then the theorem that every rational symmetrical function of the roots is a rational function of the coefficients; this is an easy consequence from the less general theorem, every rational and integral symmetrical function of the roots is a rational and integral function of the coefficients. In particular, the sums of powers Xa?, 2a 3 , &c., are rational and integral functions of the coefficients. зге are n quantities з they may become An ordinary process, as regards the expression of other functions Xa a b p , &c., in terms of the coefficients, is to make them depend on the functions Xa a , «Sic., but this is very objectionable; the true theory consists in showing that we have systems of equations p x = Xa, ing with the roots a, x — b,... in the laving the n roots f P'2 = 2a&, \ p x = Xa? + 2Xab, l Pz = Sa6c, \p x pz = Xa 2 b + 3 Xabc, le the existence of iven, then we write \ p x 3 = Xa 3 + 3Xa 3 b + 6Xabc, &c., &c. where, in each system, there are precisely as many equations as there are root-functions on the right-hand side, e.g. 3 equations and 3 functions Xabc, Xa 3 b, Xa 3 . Hence, in each system, the root-functions can be determined linearly in terms of the powers and products of the coefficients.

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