The person charging this material is re¬ sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. To renew call Telephone Center, 333-8400 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN AUG 21 198) DEC 8 » I THE SYMMETRIC FUNCTION TABLES OF THE FIFTEENTHS INCLUDING AN HISTORICAL SUMMARY OF SYMMETRIC FUNCTIONS AS RELATING TO SYMMETRIC FUNCTION TABLES BY FLOYD FISKE DECKER ASSISTANT PROFESSOR OF MATHEMATICS AT THE SYRACUSE UNIVERSITY WASHINGTON, D. C. Published by the Carnegie Institution of Washington ' 1910 THE SYMMETRIC FUNCTION TABLES OF THE FIFTEENTHS INCLUDING AN HISTORICAL SUMMARY OF SYMMETRIC FUNCTIONS AS RELATING TO SYMMETRIC FUNCTION TABLES BY FLOYD FISKE DECKER ASSISTANT PBOFESSOR OF MATHEMATICS AT THE SYRACUSE UNIVERSITY WASHINGTON, D. C. Published by the Carnegie Institution of Washington 1910 1 IRfi&PV UNIVERSITY OF ILLINOIS AT URBANA-CHAMP4I0N CABNEGIE INSTITUTION OF WASHINGTON Publication No. 120 PRESS OF J. B. LIPPINCOTT COMPANY PHILADELPHIA ve.c t 5" I $.8A Ixffi H 353 THE SYMMETRIC FUNCTION TABLES OF THE FIFTEENTHIC. 0 n % cfc' My attention having been directed by Prof. W. H. Metzler, of the Syracuse University, to the desirability of having a complete table of the symmetric functions of the fifteenthic, I have completed the work here¬ with presented. Mistakes may have escaped the scrutiny of the computer, and should they be found he will deem it a favor to have them called to his attention. A brief historical summary of that part of the subject of symmetric functions which is connected with the computation and use of the tables is also given. Care has been taken to bestow credit where credit is due. HISTORICAL SKETCH. After the sixteenth century had produced solutions of the cubic and the quartic equations and had unsuccessfully attacked the solution of the quintic, attention gradually turned toward the relations between the roots and coefficients of equations. In fact, about the middle of that century, Yieta, an officer of the French Government, observed that in the equation x 2 + a 1 x + a 2 = 0 if a x is the negative of the sum of two positive numbers oq, a 2 , of which a 2 is the product, then a x and a 2 are roots of this equation. He failed to notice, however, the universality of the relation, as he recognized positive roots only. Knowledge of the relations between the roots and coefficients of equa¬ tions grew little by little until, at the time of Newton—when negative and imaginary roots had gained recognition—between the roots a 1 , a 2 , a 3 , ... . a n and the coefficients a x , a 2 , a 3 , . . . a n of the equation x n + a 1 x r '- 1 -\- a 2 x n ~ 2 H- +a r x n ~ r +- \-a n = 0 the following relations were understood, namely 2a 1 = a 1 + a 2 + a 3 +• • •+a n =—a x Sa x a 2 = a 1 a 2 + a 1 a 3 + • • • +a 2 a 3 + • • • = a 2 S a x a 2 . . . a T = a x a 2 . . . a r + a x a 2 . . . a r _ 1 a r+1 + • • • = ( — l) r a r a x a 2 . . . a n =(-l) n a„ ( 1 ) 191800 * 4 THE SYMMETRIC FUNCTION TABLES OF THE FIFTEENTHIC. A function of several quantities which is not changed when any two of the quantities are interchanged, such as Xa x a 2 , is called a symmetric function of the quantities. Such a function of the roots of the equation (1) has been designated by inclosing the exponents in brackets, expressing the repetition of a number by an exponent: thus 2afafa|a 4 a 5 a 6 would be written [3 2 2 l 3 ]. In particular, a symmetric function of the form [K] = a* + af -f • • • • +a K n is called an elementary symmetric function. Newton in his Cambridge Lectures, Arithmetica Universalis (published in 1707), adds the following relations between the coefficients and the ele¬ mentary symmetric functions of the roots of equation (1) [2] = 2cx 2 = a? A A • • • + a 2 — a? — 2a 2 [3] =2a? = a? + a!+ • • • +a 3 = — a? + - 3a 3 _ ^ [4] = 2ai = a 4 -|- 0-2 A ■ ■ ■ A ot-n = — 4ct 2 &2 A 2n2 A 4cqci3 — 4a 4 etc. These results were utilized for the calculation of resultants of pairs of equations as early as the middle of the eighteenth century by Euler and Cramer, who were interested in the problem of the number of intersections of two curves. Toward the close of the eighteenth century Waring published a for- - mula 1 giving the value of any coefficient in the expression for [K] in terms of the coefficients: for example, the coefficient of a 4 a|a 2 , written (4 1 2 2 1 2 ), in [10] is , 1v+a+ » -10-(l + 2 + 2-l)! ^ ; 1! 2! 2! or —60 (4) It is to be noticed that this formula is a generalization of Newton’s for¬ mulas (3). A more important contribution made by Waring to the subject was a reduction formula 2 by which any symmetric function [x Y x 2 . . . x n ] could be expressed in terms of elementary symmetric functions. This formula, 1 In general, the coefficient of a, A ia 2 A 2 . . . a,/n in [X], as given by Waring, is where J = b + • ' +*» See Waring: Miscellanea Analytica de Aequationibus Algebraicis et Curvarum Proprietatibus, Cambridge, 1762. 2 The following is Waring’s statement of the formula (see Waring: Meditationes Algebricae: also M. O. Terquem: Nouvelles Annales, 1849). Let *Sa=*i a + a: 2 a + • • ‘ + x n a Sb=X 1 b +Xj> + ■ ■ ■ +X n b S t =X/ +X 2 ‘ + ■ ■ ■ +xj HISTORICAL SKETCH. 5 in conjunction with his formula (4), provided a method for expressing any symmetric function of the roots in terms of the coefficients; but the method is a rather cumbersome one, as may be judged from the following comparatively simple application, the calculation of S a\a.\al : A = $ 4 and $*•$, B = A $4 + 3 $4 + 2 $ ry rr i rr n T n rt .O4 *^3 O 4 * ^2 ^3*^2- 3 + 2 Satalag l = A -B + 2C — $ 4 • $3 " $ 2 $2 ' $ 4+3 " = Saf • • 2a? — 2a? p _ a r $ 4 + 3+2 1 ^ o’ cr rr LO4 * |J3 * OoJ $3 ’ $ 4+2 $4 ' $3 + 2 + 2*S, -’ 4+2 2a?-2a? '4+3+2 V„6_y„4.y a 5 + 2 V a 9 -’4 ta? 3+2 at (5) During the very year (1771) that Waring announced his formula (5), Vandermonde followed it by actually calculating the values of various sym¬ metric functions of the roots of an equation in terms of the coefficients, publishing the results in the form of tables. A very pretentious work, 3 with the harmless title “A Collection of Examples, Formulae, and Exercises on the Literal Calculus and Algebra,” but announcing that the author, Meyer Hirsch, had “discovered the gen¬ eral solution of equations,” appeared in Berlin in 1808. In this work symmetric functions, their calculation, and their applications—including a method for their use in calculating the resultants of pairs of equations—- occupy an important position; and in an appendix are to be found the tables themselves up to and including the tenthic (that is, the table giving the values of the symmetric functions of an equation of the tenth degree). In his tables he arranges the coefficients according to the dictionary method, or rather according to the reverse of the dictionary method. For example, for the tenthic he uses the equation in the form x n — Ax n ~ l A Bx n ~ 2 — • • • —Ix n ~ 9 A Kx n ~ 10 — • • • =0 where the positive integers a, b, . . . t, are all different and A=S a - Sb- . . .St B S(i\b ^ Sfi 1 c ^ A So- lSa-S b ' S a -S, 1 r *s \ r S a +b * Sc ■ :i BB=A [s a .s b -s c ss d + ---\ ■pi _ 4 j _ *S>a+fr+c+ a 3 a 2 a i + c 7 a% I Next express [321] in terms of the s’s by means of (5) thus [321] — S 3 S 2 S 1 SgS^ s 4 s 2 S 3 T 2sg II then apply ( 6 ), making K = 6 —|c 1 = 2 , or c 4 = —12 4 M v Brioschi: Annales de Tortolini, Rome, 1854. HISTORICAL SKETCH. 7 then making K = 5 — i(c 2 «i + c 1 a 1 ) = —>S X and since = — a 4 c 2 = 7, making K = 4 — t (c 3 a 2 + c 4 af + c 2 a\ + c^o) = - S 2 = — (a? — 2 a 2 ) and by equating the coefficients of the a’s, c 3 + c x = — 8 and c 2 + c 4 = 4, giving c 3 = 4 c 4 = — 1 making /v = 3 —| [ 2 c 5 a 3 + c 6 a 4 a 2 + ch (c 3 a 2 + c 4 af) + a 2 c 2 a 4 + cla] = s 4 s 2 — 2s 3 from which are obtained c 5 3 c 6 — 1 Finally c 7 is found to be 0, and [321] = — 12a 6 + 7 a b a x + 4a 4 a 2 — 3a 4 al — 3 a§ + a z a 2 a 4 By the use of a determinant and symbolic multiplication, Brioschi also expressed Waring’s reduction formula (5) in a much simpler form than did its discoverer; thus where and in general ^11^12 • • • U ln • ..*»] = R-21^22 • ■ • U 2n (7) V'nlV'ni • ■ • U nn 'M' rs U 8r = [* S + *r] U rs U st . . . U vr = [x r + x 8 + • • • +*,] About the same time (1857) Cayley 5 republished the Hirsch tables, using the equation ( 1 ) and reversing the order of the coefficients in the tables, so that Hirsch’s principal diagonal became his sinister diagonal. He proved that each of its elements must be ( —1)”’, where w, called the weight, is defined for [x 4 x 2 . . . x„] by the equation W = x 1 + x 2 + ■ • • +x n (8) for example the weight of 1,a\a 2 a z is 7. Since if a\'a 2 - . . . a„ n , written (/fik 2 . . . k n ) is to have a coefficient other than 0 in the expression for [x 4 x 2 . . . x n ] in terms of the coeffi¬ cients, the relation ^1 + ^2+ ‘ ' ‘ + X n — ^1 + 2k 2 + • • • must obtain, we may also calculate the weight from the equation w = / l 1 + 2k 2 + • • • + uk n 5 A. Cayley: Philosophical Transactions of the Royal Society of London, vol. 147, 1857. 8 THE SYMMETRIC FUNCTION TABLES OF THE FIFTEENTHIC. Cayley used the term conjugate partition, due to Ferrers, the meaning of which may be shown by the following example exhibiting its calcula¬ tion. To get the partition conjugate to 7, 3, 2 2 a row of seven dots is written; under these, beginning at the left, a row of three, under these two rows of two dots each; then the dots in the columns are counted and . . found to be 4, 4, 2, 1, 1, 1, 1 respectively, and accord- 4 ; 4 ) 2, 1, 1, 1, 1 ingly 4 2 , 2, l 4 is called the conjugate partition of 7, 3, 2 2 . In general the partition conjugate to x u x 2 , . . . x n is l* 1 -* 2 , 2 K2 ~ K3 , ■ • • (ft— ft*n Cayley’s theorem in regard to the sinister diagonal elements may now be given the form, coefficient of (P) in [Q] is ( — 1)“ (9) if Q is the partition conjugate to P. Cayley noticed a symmetry in the constituents of the tables, observing that coefficient of (P) in [P] is same as the coefficient of (P) in [P] (10) In the following year the Italian Betti 6 independently observed this symmetry and showed the necessity for it, since which time it has been known as the Cayley-Betti law of symmetry. Cayley reduced the number of coefficients to be calculated in a table by observing that a k would not occur in any symmetric function involving less than k numbers in its partition: for example, in the fifthic a 4 occurs in [1 3 2] and [l 5 ] only. He observed that it is possible to keep the sinister diagonal elements ( — 1)™ by arranging the partitions representing the sym¬ metric functions in the same order as those representing the coefficients, giving the table a triangular form. He saw also that the table could be made symmetrical by arranging the self-conjugate partitions (such as 4 3 3) at the middle, in one order for the functions and in the reverse for the coefficients; but that the table could not at once possess both properties. He expressed his preference for the latter method. Cayley called attention to the ease with which resultants could be calculated if the appropriate symmetric function tables were at hand. He corrected the score of mistakes in the Hirsch tables, making them reliable for computation. They may be found in Salmon’s “ Modern Higher Algebra.” Faa Di Bruno, who a little later took up the work, had a formula giving multipliers for the constituents in a row or column such that the sum of the products (except in the case of the row or column with the 9 M. Betti: Annales de Tortolini, Rome, 1858. HISTORICAL SKETCH. 9 partition l fc ) would be 0. For example, the coefficient of a 3 a 2 in a function of the fifthic would require the multiplier gf*! or 10 (11) and the complete check for [41] would be 5-1-1 -5-5- 10 + 1 -20 + 3-30-1-60 + 0-120 = 0 In general the coefficient of . . . X n ) in the ic-thic would require the multiplier w\ ( 12 ) He published in 1875 the first ten symmetric function tables, adding to the list the eleventhic. They 7 may be found in his “ Theorie der Binaren Formen.” In them certain misprints may be noted: the values of the coefficients where the misprints occur are here given, ing the coefficient of (P) in [Q]. Eighthic: Q signify- Ninthic: 7 They may also be found in the Gottingen Naehrieht (1875). 10 THE SYMMETRIC FUNCTION TABLES OF THE FIFTEENTHIC. Tenthic—continued: In 1881 a formula giving the number of symmetric functions of a given weight was announced by Forsyth. 8 An application of it, to get an idea of the lengths of tables not then computed, gives the number of symmetric functions of weight 15 to be 176, of weight 22 to be 1001, and of weight 30 to be 5595. Hammond, in the Proceedings of the London Mathematical Society for 1881-82 (vol. 13), gives a convenient method for calculating those terms in [x x z 2 . . . * n ] which contain no a with a subscript greater than a given number by identifying the coefficients with coefficients in tables of lower weight. Thus [54321] =a 5 X (terms in [4321] containing no a with a subscript greater than 5) + terms containing a’s with subscripts greater than 5: that is [54321] = a 5 X [ (4321) — 3 (43 2 ) — 3 (4 2 1 2 ) -f 4 (4 2 2) — 3 (52 2 1) + 4 (531 2 ) + 5 (532) — 5 (5 2 ) ] + terms containing a’s with subscripts greater than 5. He also gives a formula for the computation of symmetric functions by the use of auxiliary functions. A. R. Forsyth: Messenger of Mathematics, vol. 10, 1880-81. HISTORICAL SKETCH. 11 In the same year Rehorovsky 9 published the eleventhic and the twelfthic, with the aid of an additional formula giving the sum of the constituents in a row or column; for example, it gives the sum of the constituents in 3 2 2 2 1 as ( 1N2+2+l ( 2 + 2+l) ! ^ ’ 2! 2! 1! -30 (14) In the same year Durfee showed that the scheme of arrangement in a triangle is always possible in two ways, and in the American Journal of Mathematics he published the twelfthic arranged in one of them, checking his results by the Cayley-Betti law of symmetry (10) as well as by the law for the sum of the constituents in a row or column (14). Major MacMahon, in his article in The Proceedings of the London Mathematical Society for 1883-84 (vol. 15), gives several symmetric func¬ tion formulas, four of which will be here noted. The first formula enables one to write down all those terms of the highest degree in the n’s of a sym¬ metric function from a function in a table of lower weight. For example: [3 2 2 2 ] = a\a 2 — 2 a b a 3 a 2 — a 5 a 4 a 1 + 5a| + 2 a 6 a| -(- 3a 6 a 3 a L — 9cc 6 a; 4 —7a 7 a 2 a l + 6a 7 a 3 + 7a 8 a\ + a 8 a 2 — lSagftj + 15a 10 Therefore [3 3 2 2 ] = ( — 1) 13-10 (a5« 3 — 2a 6 a 4 a 3 — a 6 a 5 a 2 + + 2a 7 a| + 3a 7 a 4 a 2 — §a 7 a b ax — 7a 8 a 3 a 2 + §a 8 a i a l + 7cc 9 a 2 + — \ba w a 2 a l + 15a n a 2 ) + terms of lower degrees. (15) Another of his formulas enables us to compute any function from a single function of lower weight by means of a differential operator V _ r . In particular F _ 1 = cr 1 d da 0 + 2 a 2 d da 7 + 3c 3 d dan + and by means of his equation V- r ty] = ( — l) K x[_x l r] (16) we may calculate, for example, [41] from [4]. We first write in the homogeneous form [4] = — 4a 4 «o + 4a 3 a; 1 ao + 2a|«o — 4a 2 a 2 a 0 + cc\ Then [41] = —\V _ l[4 ] = — 7 [ — 12a 4 a 1 ao + 8a 3 ala 0 + 4 ala 7 a 0 — 4 a 2 a\ + 8a 3 a 2 al — 1 + 8a 2 a\ 4- I2a 3 a 2 al — 12 a 3 a\a 0 + 16a 4 a,rto — 20a, 5 a ( ] ] or [41] = 5 a 5 — a 4 a! — 5a 3 a 2 + a 3 a\ + 3ala 7 — a 2 a^ •Rehorovsky: Wien Denkschriften der Kaiserliche Akademie der Wissenschaften in Wien Mathe- matisch-Naturwissenschaft-liche Classe, vol. 46, Vienna, 1882. 12 THE SYMMETRIC FUNCTION TABLES OF THE FIFTEENTHIC. A third formula, based on a system of operators developed by Ham¬ mond in The Proceedings of the London Mathematical Society for 1882-83 (vol. 14), provides a method for calculating any symmetric function of the n-thic from the elementary symmetric functions of weight n and lower, l—l checks being provided for a coefficient of a term containing l a’ s. The operators are unusual in that, while calculation by means of them is very simple, they themselves are functions of elementary symmetric functions. One application of them to the calculation of symmetric is given by Hammond in his paper, his illustration being the calculation of [3 4 1] from [3 4 ]. A fourth formula in MacMahon’s article provides a check by giving the sum of the coefficients of the terms of a given degree in a symmetric function, or the sum of the coefficients of a given term in all the symmetric functions with a given number of parts in their partitions. He first introduces a generalized definition of weight 1T V —ri/l y _i+ r 2 k v + r 3 A, y+ i+ • • • where r x =K + }K(K-l)(r-2) that is, where r K is the kth of the r-gonal numbers; a definition which, it is to be observed, makes w 2 = W. The formula for the sum of the coefficients of the terms of the mth degree in the a ’s in the value of [xf’xf 2 . . . x* n ~] (or the sum of the coefficients of (xf'xf 2 . . . x„ n ) in all of the symmetric functions with m-part partitions) is where _ )K+m - AK-l)\W m+l 1 K x \K 2 \...K n \ K=K 1 +K 2 +- --+K n (17) For example, the sum of the coefficients of terms of the fourth degree in the a’s in the value of [S^ 1 ! 3 ] is ( n« +4 -i (5-l) 1(5-1) 1 ’ 3!1! 1! 20 since K x = 3 K 2 = 1 K 3 = 0 K, = 0 K 5 = l K 6 = 0 K 7 = 0 . . . Z n = 0 K = 5 M = 4 TF S = 5g • 1 = 5 In the following year the same writer published an article in the American Journal of Mathematics in which he repeated formula (17) and included the additional check that any function except [1 A ] would vanish if for each coefficient the reciprocal of the factorial of its suffix were sub¬ stituted, (18). Accompanying the article is the thirteenths, arranged like Durfee’s twelfthic and checked by the CaylejMBetti law of symmetry, (10), as well as by his own checks, (17), (18). HISTORICAL SKETCH. 13 A misprint may be noted in It should read = 1 . In 1887 Durfee published in the American Journal of Mathematics the fourteenthic arranged according to his second (dictionary) plan and checked not only as was his twelfthic, but also by MacMahon’s formula (17). In 1898 a formula 10 was published by Metzler which, when the value in terms of the a’s of say ^a\ala\a° 4 is given, makes it possible directly to write all those terms in say 2a? _0 at _1 a i~ 2 a\~ z , involving only the coefficients di, d 2 , d 3 , d 4 . s From the equation for ^,alalala 4 in the homogeneous form i a\a\a\u . 4 = d 3 d 2 d x d 0 — 3d%d% — 3d 4 d\d 0 + 4a 4 a. 2 a 0 + 7d 5 d x dl — 12a 6 tto is derived the equation V 4-0 4-1 4 CL o rt - Z a ^9 ~ 2 a\ 3 = d 4 _ 3 d 4 _ 2 d 4 . 1&4-0 3d 4 - 3 d 4 . _ 4 cq _ icq _q -l - 4n 4 _ 4 n 4 _ 2 n 4 _o + terms involving d 5 , d 6 , etc. giving in the nonhomogeneous form ^a\alala 4 = d 4 d 3 d 2 d 4 — 3d\d\ — 3cc 4 a| + 4a 4 cc2 + terms involving d 5 , d G , etc. (19) During the year 1899 there appeared in the American Mathematical Monthly a series of articles on symmetric functions by Roe. Not content with Cayley’s remark as to the ease with which resultants could be calcu¬ lated if the appropriate symmetric functions were at hand, he carried the matter farther, identifying the problems of calculating all resultants and calculating all symmetric functions. From a consideration of the former problem he derives a complete set of formulas for the latter. He arranges them in three classes (fundamental, reduction, and normal) and expresses them by means of a new symbol (*) which represents the coefficient of (P) in [P]. His fundamental relations consist of the Cayley-Betti law of symmetry (10) and Metzler’s formula, (19). His reduction formulas consist of four which he designates as reduction (or derivation) formulas and one which he terms the formula for the completely reducible form. Of the four reduction formulas, one is Hammond’s formula, another provides for the calculation of [l x ], the third for the reduction of d% [x 4 x 2 . . . x n ] 10 W. H. Metzler: Proceedings of the London Mathematical Society, vol. 28, 1897-8. 14 THE SYMMETRIC FUNCTION TABLES OF THE FIFTEENTHIC. where m is greater than x x , while the fourth states that a function of the roots of an equation of the nth degree has the same coefficient of the term involving (X^ . . . X r-1 ) as the same function of the roots of an equation of the (r— l)th degr'ee. His completely reducible form is © t where P is the partition conjugate to Q, the formula giving Cayley’s result (9). Roe’s normal forms are those for which his fundamental and reduction formulas do not provide a reduction. The formula is +1 (T"“lh . . . n /n yn©(ra—l)^ . . . r=n . V r= 1 (1 + M r ) X 0 + + 1 Ri . . . (r — 1 )X- 1 +,?— l(f-j- 1)^+1 . t _ \ 1 (m— l)!“i . . . (m —r — l)^-i(m — — r + !)/“>•+1 . . . O ot /_ Thus he reduces his normal coefficients to the sums of coefficients of tables of lower weight. For example R) 2 134° , 40 3 , / 0 2 13\ /0 2 3 N R 4 40V (1 + 0) \30h (1+0)1 ' 0 T HO 2 . (—3+1=4 that is, coefficient of a 3 a l in 2a? = — (1 +0) X coefficient of a 3 in 2a? — (1+0) X coefficient of cq in 2a x = 3+1=4. Roe gives also a formula for calculating the constituents in each of the first four lines (or columns). Those for the first two follow: First line w(?l 1 + ^-2+ • • • + X n —1) ! ) xfx 2 !. aj ( 20 ) where ra = X 1 + X 2 + • • • + X n . Second line (po^i n (w-l)10. . .0, 1 J/ (w — 1) (/C+X 2 + • • • 2)! 1-1 ^ -l)ra,!...JL! + +n — 1 ) w(x i +X 2 + Xi! X 2 V ( 21 ) where h = 2 when m = 2 and h = 1 when m > 2 . Roe’s formulas were used by the writer in the calculation of the fifteenthic here appended. The work has been verified by the use of the law for the sum of the coefficients in a row or column, (14), and by Roe’s formulas, ( 20 ) and ( 21 ). ILLUSTRATIONS OF USES OF SYMMETRIC FUNCTION TABLES. 15 ILLUSTRATIONS OF USES OF SYMMETRIC FUNCTION TABLES. To illustrate a use of symmetric functions we may solve the quartic equation x 4 —x 3 — 8x 2 + 2x + 12 = 0 ( 1 ) with roots say a 1} a 2 , a 3 , a 4 . We may begin by forming the cubic resolvent in Z, where Z has the three values of the function The equation is a k a h + a i 3 a u 11 Z 3 -Z 2 Xa x a 2 + Z2 a\a 2 a 3 — (Sa?a 2 a 3 a 4 + 2ta?a|a|) = 0 (2) By the use of tables of symmetric functions this may be written Z 3 + a 2 Z 2 + (a x a 3 — 4 a 4 ) Z + 4a 2 cc 4 — aj — ct?a 4 — 0 (3) for the general quartic X 4 + a x x 3 + a 2 x 2 + a 3 x + a 4 = 0 (4) or for (1) Z 3 + 8Z 2 — 50Z — 400 = 0 (5) Solving the cubic (5) 4 = a 4 a 2 + a 3 a 4 = —8 (6) @2 ~ a l a 3 + a 2 a 4 “ 5^/ 2 (7) (3 3 - oqoq + a 2 a 3 = — 5yj 2 (8) and since a x + a 2 -f a 3 + a 4 = 1 (9) we may solve the set (6), (7), ( 8 ), (9), for the a’s and get oq — 3 a 2 = — 2 a 3 = yj 2 a 4 = — -yj 2 To illustrate, in particular, the use of the fifteenthic, let us find the resultant of a cubic equation and a quintic equation, say f 1 (x)=x 3 -Sx 2 -2x+l = 0 ( 10 ) and 3, we may write = <4 12 — 2 (- 4a|a 2 — 2 a|a? + 4aia|a! — a 3 a$) + 3 (— 5a|a! + 5aj$a| 4- 5a|a 2 a? — ba 3 a 2 a x + a 2 ) + 4 (— 4a§a! — 2 a 3 a 2 2 + 4:a 3 a 2 a\ — a 3 a\) — 6 (— 3a| + 7a 3 a 2 a x — a 3 a\ + 2a 2 — 4 a\a\ 4- a 2 af) 4- 9 (5 a 3 a 2 — 5 a 3 a\ — 5a 2 a x 4- 5 a 2 a\ — af) 4" 8cl 3 4- 12a 2 + 18cii -1- 27 and since = —3, a 2 = —2, and cr 3 =l, R Ml = 2619 12 To read the value of Saja|a| from the fifteenthic, we use the numbers in the row designated 5 3 for coefficients. The first coefficient is in column headed 15 and therefore signifies—5a 16 . Thus we have 2aja|a|= —5a ]5 + 5a u a x + 5o, 3 a 2 + 5a 12 a 3 + etc. ill fiiiM I| ■1 m ill 111 ■*■ lllliill l m ■ ,■ ! m . .. :: .. y S: iiiiiii ■ isiiiiaiiiiM '' ftwpnMpn^ MM i i ! I —BI BH|| ||inBii it|ii i v i$f|iiiiiiii|§ 11 1 s |iii||is |i MB i ii i lB SKill &A&: i ■ i i I . 1 ■:jw : ife: 18 ,, SiAttSL. 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