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Sele) &. o) & Oneler en bd Mies Laks ‘in tn, a oe THE UNIVERSITY OF ILLINOIS . LIBRARY A\QAQ4 siieolel SIATUCRATIFG MATHEMATICS ‘he person charging this material is re- sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below.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. UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN —$<$ — —$—$————————————————————————— eeeO|}?0 a>} L161— O-1096 a 4 ; ’ ie , uiie Md ive + lis d ? as a _ ’ t HIRSCH’S EXAMPLES, FORMULA, & CALCULATIONS, ON THE BLY ERA Ce GA CUES yt B bs 3 “ ot j a LIBRARY >. TRANSLATED FROM THE GERMAN, Mir ; em as. ’ Set 4 i t Rt i ‘a REV. J. A. ROSS, A.M.,, : 4 *° res" TRANSLATOR OF HIRSCH’S INTEGRAL TABLES, LONDON: | PUBLISHED BY BLACK, YOUNG, AND YOUNG, TAVISTOCK-STREET, COVENT-GARDEN. 1k ee 2 1 Sere od ¢ rer Cth, Ger te TTS X Ye Ee ' T. C, HANSARD, Paternost a” PREFACE. Vaar I have discovered the general solution of equa- tions, my readers may very probably know already, from the announcement which I have ordered to beput in the public papers, for such a discovery deserves the utmost publicity. A complete history of the unsuc- cessful endeavours of my predecessors. for the same purpose, would appear like a panegyric on myself, and therefore I shall only relate what exactly belongs to the matter. In the tenth part of the ‘‘ Memorte di Matematica e di Fisica della Societa [taliana della Scientia,” p. 1 (1803), a celebrated Analyst, M. Ruffini, gave a proof of the im- possibility of such a solution : no very elaborate discussions are required to show its insufficiency. Read his proof, and compare it with my solution, and it will be found that M. Ruffini, in recounting the possible cases, has never thought of this mode of solution. MM. Ruffini tries to depress the equation for the assumed function, by taking as many equal formule as are proper for his purpose. I do just the contrary: with me the assumed functions, with the b il PREFACE. exception of a single condition, are always arbitrary: all their forms may be different, and the depression will be produced by making them dependent upon resolvable equations, the coefficients of which depend on other resolvable equations, the coefficients of which depend again on other resolvable equations, &c. As, for instance, with me the assumed function for the equation of the fifth degree originally depends on an equation of the 120th degree: this I reduce, first to a double equa- tion of the fifth degree ; its coefficient, which is still depending on an equation of the 24th degree, I make depending on an equation of the fourth degree, the coeffi- cients of which only still depend on equations of the sixth degree. I reduce these equations again to equations of the third degree, the coefficients of which, lastly, depend on equations of the second degree. Of all this process, there is, with the exception of the reduction to the double equation, not the slightest indication in the proof of M. Ruffini. Moreover, this Analyst has only shown that none of the methods he was acquainted with could succeed, and in this respect his proof is, no doubt, very masterly. His error can be no reflection on his well- deserved reputation, for he has shown to his successors the paths they must avoid, and has thus put them on the right course of investigation. M. Lagrange gave, in the third volume of the new Memoirs of the Berlin Academy of Sciences, an incom- parable analysis of the methods of Tschirnhausen, Kuler, and Bezout, which I have adopted in the sixth chapter, PREFACE. \ ill with a few alterations suitable tomy purpose. Heshowed, that when p is a prime number,* all these methods lead at last to a reduced equation, the coefficients of which are those functions of the roots 2’, 2”, a/”......0, which change when you change only the n»—2 last roots ‘among each other, but leave both the first in their places, and that, therefore, the coefficients depend on equations ON Che ls) 2 Gy. e0 uw second degree ; and consequently an equation of the fifth degree, on an equation of the sixth degree. For the explanation of his method, he takes the equation of the fifth degree as an example, and shows how to begin to form the reduced equation. He denotes the roots of the reduced equations by 2’, 2”, 2’, 2”, 2”, 2”, and finds the value of them in* 2’, 2”, x’, 2, x’. From this he calculates the first coefficients, and says that you can find the other coefficients in a similar way. He concludes by saying ‘‘ but we will not enter into the execution of this calculation, which, with all its im- mense labour, would not afford any clue to the resolution of equations of the fifth degree; for as the reduced equation for z is of the sixth degree, it is not resolvable, unless it is to be brought to a lower degree than the fifth. But this seems to me to be almost impossible, according to the form of the roots 2’, 2’’, &c.” But from these very forms, I affirm that the solution of the reduced equation is possible. or the functions 2”, 2” * is with him, what x is with me. * Pages 432 and 433 of Euler’s Introduction, translated by Michelsen. b 2 1V PREFACE, a”, 2, 2’ are derived from z, as M. Lagrange observed himself, when you change the roots 2’, 2”, a”, among each other. According to my notation, therefore, the Toots of the reduced equation admit of being presented by SF: (12345), ff: (12453), fi: (12534), fi: (12435), SJ: (12354), and f: (12543), and with these notations the functions 2’, 2”, 2”, 2’ 2/’, 2”, correspond according to the order in which they are here put. The three former formule /: (12345), f: (12453), f: (12534), form evidently a cyclical period of the three last roots, as well as the formule f: (12435), f: (12354), f: (12543). If we therefore combine the three func- tions 3’, 2”, 2'’, in one equation of the third degree, the coefficients of them can only have besides their value another one, namely, that which the change of «’” with a” gives. ‘These coefficients, therefore, have no more than two unequal values, and consequently they depend only on equations of the second degree, or, what is the same, the reduced equation of the sixth degree can be divided into two equations of the third degree, the one of which has the roots 2/, 2%, 2’’, the other the roots z/’, 2’, ®. That this simple observation escaped the keen penetration of a Lagrange, looks indeed like a miracle. I am not the inventor, it is he; but he did not know it. Whether I should have found the solution without him, may be doubted. I come now to my method of solution: it is very simple, uniform for all degrees, and as general as could be desired. It gives, not one solution, but as many as we PREFACE. v please; for the functions I mark with @ are quite arbitrary. However, the actual calculation is very trouble- some, and even in the sixth degree is scarcely practicable without resorting to particular artifices. We cannot escape the difficulties of calculation, when the degree of an equation is a prime number. When, however, it is a compound number, we have, no doubt, a method, which leads more rapidly to the result: it will be reserved for the third volume, in which I shall give also the solution of the equations of the 5th, 6th, and 7th degree. The Combinatorial Analysis is here of great service; and with its help I shall perhaps be able at once to exhibit the reduced equation with little more trouble than the mere combinatorial operations. A brief sketch of the contents of this volume will not be here improper. I begin with the Symmetrical Func- tions; they are the foundation of all others. ‘The two first chapters treat of them; the first gives the recurred solution, the second the independent one. Generality was the object I aimed at. The third chapter treats of the Non-symmetrical Functions. They are derived alto- gether from certain equations, which I call transformed equations. It is shown how to find the equal formule of these functions, when their nature is given by certain properties; and how to form a transformed equation of the unequal formule. The numberless references to it require particular observation. The utility of some of these propositions will appear in the sequel. The fourth chapter treats of Elimination. I was not obliged to be v1 PREFACE. so minute as Bezout in his Theorie générale des Equations Algebraiques, who confines himself exclusively to this subject. My work would have become too voluminous. Should my readers wish for further details, this may be done in an appendix. The fifth chapter treats of the properties of the roots of the equation 2’°—~1=0. Waring and Euler were my conductors. The labours of Lagrange, in the Memoirs of the Berlin Academy, gave me the materials for the sixth chapter. I desire my readers to bestow particular attention on the seventh chapter; its value will be shown in the third volume of this work. The eighth chapter treats of the General Solution of equations, but must be regarded only as a sketch. My reader is no longer the same whom I thought of in the Collection, the continuation of which I now give him: he has gone much further in the sciences. ‘The Combinatorial Analysis is no longer strange to him: he has also made already considerable progress in the Differential Calculus. Provided with this knowledge, he will, I trust, find my book not entirely useless. He will not remain where I have remained: he will look further. I do not lead him through an unfertile, but, for want of labour, an uncultivated field. or since the Differential and Integral Calculation employed the Analysts, Algebra has been little thought of. The next part will contain, besides the deeper researches about the general solution of equations, a great many PREFACE. V1l other subjects, and amongst them the important, almost inexhaustible, one, of the Analysis of Equations. I shall constantly, as far as my leisure hours permit, labour on the Third Part to hasten its appearance as much as possible. But if on all these subjects the same pains be bestowed, some time must elapse before its appear- ance. However, not to let my readers wait for what belongs to equations in particular too long, I am inclined to prepare for the next fair a supplement of about four or five sheets on this subject, and in it to communicate the complete solution of the general equations of the fifth, sixth, and seventh degree. Berlin, October, 1808. te Rhea es De a] shies lk wit aah Vigo buts aoa fate ¥ ped 1A “7 w ovatliog & ay} Ww gay divawadeon: p+ waedal aren cvcdiaieat qtr te wee Ct ! oh acts Gaiety er WK ie th wer pried ol arb bt Se. ae , P| we , Pi wf F di tl - - ' ny) s PT pt 2 ‘uu io é < as 3 ‘4 um Po ee I ‘ DU ie onal date ates OSTOR .wrmrOr fac. ty bs ten FHM (it aal Qo tony eM VE: <.goraz Mathers Kcr! ope ta/abbingy at tai aw » ub wgrtalad Mat 'n i‘ \ Dror odd 1b onaepang oP $e tHGt Ldora To Inkl aif) openteturaniaa: ar sh ft fh ‘ated is att coe standin a. MAD oY To auslnops iy ody Use coltalén vahfdio> Tres Neen selene Natt Mgr NA Satnobe tr ven earmeete | a | fics FES — ————— oe 4° . ae i ’ tr : > ; ; > ; ne oe ug oa Rs 0 a 2 gt ; ie? ees A TRANSLATOR’S PREFAC E. SIN CE the publication of Waring’s Meditationes Algebraice there has not appeared, either in this country or on the continent, so elaborate and able a treatise on the theory of Equations as that of Meyer Hrirscu. Its merit has long been recognized, and the work held in the highest esteem by those who were able to read it; a small number, undoubtedly in this country, where the German language is so rarely understood by those who devote themselves to these studies. As a treatise on equations of the higher orders, it is not less admirable for the clear and simple manner in which the theories are laid down, than for the numerous and apt examples by which they are elucidated. It is necessary to remind the reader, however, that the Solution of the General Equation on the imagined discovery of which Meyer Hirsch so warmly congratulates himself in his preface, turns out, on examination, like all the other attempts of the same kind, to be a failure. What he has written on the subject, I have, nevertheless, permitted to remain; because, in the first place, these speculations occupy only a small space; next, because they are highly curious and interesting; and finally, F ad x TRANSLATOR’S PREFACE. eng did not ‘think myself at liberty to mutilate a work which I undertook only to translate. , J. A. ROSS. Westwell-Vicarage, Kent, January, 1827. Extract from Meyer Hirsch’s Preface to his Integral Tables. « At the end of this preface I- must observe, that I was mistaken when I maintained, in my collection of Problems on the Theory of Equations, that the general solution of equations was not only practicable, but even thought I had found it. The eighth chapter of the above-mentioned work must, therefore, be read with mistrust. It is true that I have found the solution of a number of very remarkable equations, that do not admit of being analysed, but by no means their general solution, in the sense in which Euler, Lagrange, and other great Analysts take these words (general solution); for I am now convinced of the impossibility of effecting it. The mistake arose from haste, and is so readily discovered, that’ every person who reads so far, will easily perceive it.” I —On Fre : DH: > % * I.—On THE ROOTS OF EQUATIONS, THE SUMS OF THEIR POWERS, AND THE PRODUCTS OF THESE POWERS, AND SYMMETRICAL FUNCTIONS IN GENERAL, SECTION I. In all good elementary books on Algebra it is shewn, that the first part of an equation of the undetermined nth degree......()...... x 4+- Ag? + Ba? + Ca"? + ...4 Pr+Q=0 may always be considered as a product of n simple factors of the form z—a, x—b, r—c, x—d, &c., and that then a, b, c, d, &c. are all the values of the unknown magnitude «, which verify the equation (~). If we actually multiply these factors, and compare the pro- duct resulting from this multiplication with the polynomial _ in the first part of the equation, then we have the follow- ing relations between the coefficients A, B, C, D, &c. of the equation, and its roots a, b, c, d, &c. : —A=a+b+c+d+4 &. B= ab + ac + ad + be + bd + cd + &e. —C = abe + abd +acd + bed + AS oe coat Pits, WHIVERSTP PT OS + Q=abed &. 2 Thus the first coefficient 4A, with the sign changed, is always the sum of all the roots ; the second coefficient B, with its own sign, is the sum of all the products of every two of these roots; the third coefficient C, with the sign changed, is the sum of all the products of every three of these roots; and generally the undeter- mined mth coefficient, with its sign changed or not, according as m is an odd or even number, is the sum of all the products which arise from the combination of all the roots, taken m and m together ; finally, the last term Q, (which may also be considered as the coefii- cient of «’,) with its sign changed or not, according as » is an odd or even number, is merely the product of all the roots. The coefficients 4, B, C, D, &c., are consequently no other than the aggregates of the combinations of the roots a, b, c, d, &c., taken singly (one by one), two and two, three and three, four and four, &c., or, to express myself more precisely according to Hindenburg, the ageregates of the combinations without repetitions of the first, second, third, &c. class. How these combinations may be easily represented, will be shown under the head of combinations, which are here supposed to be known only in their first principles. SECTION II, Iw the following pages, certain notations are frequently used, which, in fact, are already known to the greater part of my readers; the meaning of these notations, however, in order to prevent mistakes and confusion, I shall give in this place. 5) i. When determinate or indeterminate magnitudes are spoken of, all algebraical expressions, in which these two kinds of magnitudes are in any way involved, are called functions. We then use the formula: “ This or that expression is a function of these or those magnitudes” — because we only mention the indeterminate magnitudes, omitting the determinate. On account of the particular use which we shall make of functions in this Work, I wish it to be remembered, once for all, that here (when the contrary is not expressly mentioned), only such functions are meant, as may imme- diately be determined by means of the six arithmetical operations of Addition, Subtraction, Multiplication, Division, Involution and Evolution, so soon as the magnitudes contained in these functions are known, and when they do not contain a magnitude considered as indeterminate either as the exponent of powers, or the index of roots. 2. A rational function is one in which either there are no irrational magnitudes, or one in which at least those magnitudes which are considered as indeterminate are not under the radical sign; in the contrary case the function is an irrational one. 3. An integral function is one in which either there is no denominator, or in which, at least, those magnitudes which are considered as indeterminate, are not found in the denominator; in the contrary case it is called a fractional function. The coefficients 4, B, C, &c. of the equation (p) in 4 § I., are consequently integral and rational func- ‘tions of the roots a, b, c, d, &c., so long as these magni- tudes are considered as indeterminate. Here no reference is made to the particular properties of the magnitudes a, b, c, d, &c. themselves; consequently these may be rational or irrational, integral or fractional, and, generally, may have every possible form, or they may even be functions of other magnitudes. 4. Those functions which are here called symmetrical, are those in which the indeterminate magnitudes are so combined, that, independently of the particular values of these magnitudes, no change takes place in the value of the function, however these. Begnitudes are substituted for one another. The coefficients 4, B, C, D, &c. of the equation (Lb) m § I., are .*. symmetrical functions of the roots a, b, c, d, &c.; thus they remain the same when a is substituted for b, or 6 for c, or a for c, and 6 for d, and so im like manner of other substitutions. From this definition it immediately follows, that the sums, remainders, products, quotients, powers and roots of symmetrical functions, are again symmetrical fune- tions, provided the functions, which are combined together by addition, subtraction, multiplication and ‘division, contain all the same indeterminate magnitudes, and in the same number. Thus the expression (ab + ac + be)” + abe NF (bh ae oae is a symmetrical function of a, &, c, because ab+-ac-+be, 5 abc, a-+-6+-c, are functions of this kind. Generally, every function of one or more symmetrical functions is always again a symmetrical function, when these last contain the same indeterminate magnitudes and in the same number. Now it may pe shown that every rational, integral, or fractiofal,function of the roots of an equation, however constituted, may always be expressed rationally by the coefficients of this equation. This highly important relation between the coefficients and the roots, has thrown more light on the theory of equations than any other ; and should human genius ever succeed in fully discovering the secret of its solution, so far as this is possible, it will probably be by such inquiries as are exactly founded on this very property. SECTION III. For the sake of brevity and perspicuity, I shall use the following symbols : Let the sum of all the roots of an equation, their squares, their cubes, their biquadrates, and, in general, the sum of their pth powers, be represented by {i}, (2), Ey; GR... . . fu], so that only the exponents, but not the roots, are indicated, because the latter are not considered in the present case. ‘Therefore, I ij=a +b+cec+d +e + &. [2]J=_@4+P4+24+ P+ e + ke. [speieietere tte Fife fees I ion 4 ge aap a If the roots a, b, c, d, &c. taken two and two, be com- bined in all possible ways, and in each such combina- 6 tion every alternate root raised to the power a, the other to the power @, then the sum of all the products thus obtained may be represented by [af]. Consequently, when only four roots are assumed, the equation (vj, in § I., is of the fourth degree, [ag }= by sehen cr dey axb? + afb + arc + a’cx + ad? + abd* + boc? + bce + bed? + bed* + cxd? + ce’ d+. In a similar manner {aey denotes the sum of all the products which arise from the combination of ali the roots, taken three and three, and in each such combina- tion raising one root to the power a, another to the power @, and the third to the power y, and this in as many ways as possible. When again only four roots are assumed, we .*. have [ aey |= by, fection Wf axbecy + arbre? + a’becy + afbrcx + avb*c® + arb’c* + abedy +- axbvd? +- a®b«dy + a®byd* + avb«d? + avb’d* + a«c’dy + axcrd? + afced’ + afcrd* + arvc«d? + arvc’d* + becfdy + becvd? + bec~dy + becvd* + bred? + brc’d*. In general] a@y8 Ber apke Kk J when m is the number of the letters a, By Y. Os .o+e- k, denotes a sum of the pro- ducts which arise from the combination of all the roots to the mth class, and in each class one root is raised to the power a, another to the power @, a third to the power y, and so on, in as many ways as possible. In order .*. to represent actually the expression} agyo aoe x | find all the combinations of the roots a, 5, c, d, &c. taken m and m together, give the roots in each such combination the exponents a, @, y, «...+ , k, and then 7 “permute the exponents in all possible ways. By this method we get [z8yd] =, ahectd. ra be cd kes. + aid aan GD? C8 eh ab OdbF oF eh ee oe. c st a? bch et &e. [aaaBe] = ath? ctdhe® + atb*chdte® +... + ab boctdte® 4+ a°b*c*d?fP + at bch df? 4. ... + ab bo ct deft + &e. In order to render this notation more convenient, it would be better to use the repeating exponents in those terms where an exponent occurs more than once; thus, for instance, [a°6?] instead of [a@aaggl, and [a 6 y7| instead of [aaeeyy]. By § I., the relation between the coefficients and the roots of an equation may be thus represented : mye a ESTs on TS ea ee A One Di) LLEET ei bebe aapeid lilnae, [it Lec: From the construction of the function |a@yé...... KI, it follows, that it belongs to symmetrical functions, because it undergoes no change when the roots a, 6, c, d, &c. are substituted for one another. The function [agy6...«], or more generally [a*gb-y° dB ...k&], I also sometimes call a numerical expression. The exponents 2, 6, y, 6, ... «, are called radical expo- nents, in order to distinguish them from the repeating exponents a, 6, c, d, ... k. The radical exponents may also be negative, and then = | B 4 8 f—1] =a? + b' +67 4+ &. = —4+—4 —+&e. a Da [—2]) =a’ + b? +c? + && = : —+ ade : 5+ &e. | a Geos ie [—p] =a" + b+ c% + B= atpts + +8&e. and in the same manner | [—ag]=a-*b® + a®b-* + am*c® + afc" 4 &e. Pied Fiphi kN yd [—a—g | =a-*b-F 4- a PO-* 4 a*e7P + a Fe-* + &e. 1 1 1 rain aBe cw aptatoe + &e. and in a similar way it obtains with the numerical expres- sions, in which more than one negative radical exponent occur. SECTION Iv. Pros. To find the number of terms of which the nu- merical expression [aBy0...... x] consists. Solution,1. Let the number of the roots a, b, c, d, &c. = n, and the number of the radical exponents a, 8, y, 6, 2. K Sa git. 2, The terms which compose the numerical expression [aBys...,..«] may be found by combining the x roots a, b, c, d, &c. to the mth class, and by permuting in each such combination the m radical exponents a, @, y, 0,...« (§ III.) The number of these terms is consequently equai to the product of the number of combinations of x things to the mth class, and the number of permutations of m different things. 9 3. But the number of combinations of n things, taken m and m, is Me M—1 .N—D woose N—M+2.n—mM+ 1 me ae en eee ee i 1 eo and the number of permutations of m things is Ser S haem aw i a/c bite e'die'e m—1.m. 4. Hence the number of the terms of the numerical expression [aBy6é...... k] Me k's Id Ne veutas n—m-+1 eae Pee), fee Webb bk m xX 1.2.3...m =nN.N—1. N—2......N—M+2 -N—mM+1 SECTION V. Pros. To find the number of the terms of the nume- rical expression [a%gby° 0 ...,.. xk]. Solution.1. Let the number of all the roots a, b, c, d, &c.=n; let the number of the radical exponents, without reference to their equality or difference, or, which is the same in this case, let the sum of the repeating exponents ath+ce40+4 ...... +h, &.=m. 2. Since each term of [a* gby°od......«%] contains m of the roots a, 6, c, d, &c., in order to find the number of terms, we must here, as in the preceding section, multiply the number of combinations of 7 roots taken m and m by the number of permutations of their m radical exponents. 3. But the fottact ie nm—1 . N—2......N—-M+2.n—m+ 1 i 1 . 2g ° 3 eeeteteonertte ose m—1 ° me : Tp Cc 10 The second will be obtained by finding how often the letters a, By 75 Oy eceeee ,k, can be transposed in a series of elements a gby° Sd ...... ck, in which one letter occurs a times, another #, a third ¢, and so on. The number of these transpositions, however, 1s (as may be seen from the rule of combinations) Leech So eae PEE Ora euin eiear one m—1.m™ Sy 1.52. xd) Ba eee ew ae ee 4. If.*. these two numbers are multiplied together, we obtain the number of the terms of [a*gby° ov ...... Kh] DS Ne OE ore ee ee n—m+2.n—m+1 Wi ia te nee Tos Ba Die insist Xoo aes yrds be Corollary. If the m radical exponents are all different from one another, then, by the preceding section, the num- ber of terms = 2. n—1.n—2...n—m4+2.n—m+1. Hence it follows, that in the numerical expression [at ebyf ou ...... x&], the number of terms is less by 1 oiQon ch Mids sh oan dl Ke Cope eee, pay 7 se than when the radical exponents are different. Examp.e. For the expression [a*6?y?], when the equation to which it relates is of the twelfth degree, we haven ='12, a= 4, h = 3, ¢ = 2;'.-..m ='9. ° The number of terms, of which this expression consists, is con- sequently WEIZS. 11.10 5 ORB Vee O ab oe a 1) 28 od oe We ee ered ce = 277200 > i] SECTION Vi. Pros. Let © denote the sum of all the combinations without repetitions of m letters a, b,c, d, &c. of the mth class ; further, let S/ be the sum of all the combinations Aen ore aud om of this class, which do not containa; %// the sum of all those which do not contain 6; 3S/” the sum of all those which do not contain c, and so on; find the relation be- tween >/ + 2+ S/"-+ &c. and &. Solution 1. If in 3’, >”, 3S”, &. all the combina- tions are completed, or if S=>/=>/= S/" = &e. then 4+ S”4+5/"/+&.=n. But since in these some of the combinations are wanting, consequently their sum must be less than » &. 2. But it is evident, that each distinct combination contained in S must be wanting in the sum 3/ + 3 + =/ + &c. exactly as many times as it contains elements. Then supposing the different combinations in & consist of four letters, then, for instance, the combination a bc d fails once'in >/, S/’,. 3/4, =’. 3. If.:., in general, m is the number of elements con- tained in each combination, then the sum of all the com- binations which fail in D/ + 3” + 3/” + &e. is m &. 4, Hence and from 1. it follows, that +S’ + SY” 4+ &. = (n—™m) &. Corollary. Therefore 2 + 3!’ + 3 + &c. for the first class = (n—1) =; for the second = (n—2) 3%; for the third = (n—3) 3B; and so on. 12 Examp te. Let © be the sum of all the combinations, taken three and three, of five elements a, b, c, d, e; then 3S = abc + abd + abe + acd + ace + ade + bed + bce + bde + cde, S/ = bed + bee + bde + cde, S// = acd+ ace + ade + cde, S/’/ = abd + abe + ade + bde, '” = abc +- abe + ace + bce, >” = abc + abd + acd + bcd, and 3/.4+ S/o4 3// + 27 +37 =22=(5— 3) 3; as was required. SECTION VII. Pros. There are two equations, x +t Ax’) 4+ Br? + Cr” + &e. = o et) 4 A/a? + Bg” 3 4. C’z"**§ + &e. = O of which the second has the same number of roots as the first, except a: find the relation between the coeflicients of these two equations. Solution. The second equation may be obtained from the first, by dividing the latter by x—a. By actually performing this division, we obtain a+ (a + A) a + (a? + a4 4+ B) 2”? + (8+ 0A +aB+4+C) 2%? + &. =0. Hence now it follows, that A’=a+4, B’/=a?+aA+ B, C/=@+eA+aB+C &c. and, in general, Aree ANA RnB in G tee +A When A and A! denote the mth coefficients in the first and second equations. SECTION VIII. Pros. From a given equation to find the sums of the 13 squares, cubes, biquadrates, and, generally, the sum of any power of its roots, without knowing these roots, assuming that the exponent of this power is a whole positive number. Solution 1. Let x + Ag 4+ Br’? + Ca? +......4+ Pr+Q=0 be the given equation, whose roots are a, b,c, d, &c. Further, let Pi Aa 4 Bar? ti Care dakesslo ge Ag? Blah 3 4 Cl" a4§ + &ei = 0 vl 4 Ayr? 4 Big 3 Cl y-4 4 &e. = 0 &e. be the 2 equations, which arise from dividing the given equation by r—a, x—b, x—c, Ke. successively. 2. Then the coefficients 4, B, C, D, &c. are the posi- tive or negative sums of the eee taken singly, two and ‘two, three and three, four and four, and so on, of n roots a, b, c, d, &e.; the coefficients A’, B’, C’, D’, &c., the sums of the leteous, taken singly, two and two, three and three, four and four, and so on, of the n—1 roots, 0, c, d, e, &c.; the coefficients A”, B’, C’”, D’, &c.; the posi- tive or negative sums of the n—1 roots, a,c, d, e, &c. taken singly, two and two, three and three, four and four. Then (§ VI.) | A’ + AU 4+ Al” + &. = (n—1) A Bl + B’ + BY + &. = (n—2) B C+ CY + CC” + &. = (n—3) C &e. 3. But from the preceding § ————————————— 14 A’ =a+A, A” =b4+4, A” =c+4+ A, &. If we use the symbols in § IIT., we consequently have A’ + Av + A” + & =I) 4+ nA Since further (foregoing §) B=e0+aA4+ B, B’=8 +64 + B, BY” = c + cA + B, &e. then we have B’ + B’ + BY + && = (3) | by +- nB In like manner we find C4 Ol + OW «& &e. =[s] ae Af?) + Bay + nC &e. 4, From 2. and 3. we obtain the following equa- tions : (1) + nA = (n—1) A (2) + A(1) + 1B = (n—2) B (3) + 4(@) + BQ) + 2C = (n—3) C wc: or (1) + A=o (2) + A4(1) + 2B=0 (3) + 4(2)+BQ)+3C=o0 and in general (m)+A (m—1)+ B(m—2) + «. La Cones m—1i where 4, A, denote the (m—1)th and the mth coeffi- cients when, m n, then the con- clusions which have been drawn no longer obtain, because, in this case, the sixth section, on which they are founded, ceases to be applicable. We can, however, for this case find a similar equation by another method. 15 5. Thus, if we multiply the given equation by x”~’, we obtain wp Anh Bom tas t Patt Qe" =o and if we substitute in this equation a, b, c, d succes- sively for x, we have a” + Aa”! + Ba"? + 0.1... + Pav! 4+ Qa" = 0 Dare Gee te Ee BOP Us + Pot! 4+. Qb"-" = 0 c” + Ac” we Be™-2 oe pao 1 Perr! + Qc" —o0 &e. If we add these equations together, we obtain | (m) + A (m—1) + B(m—2) + ...... + P(m—n+1) + Q(m—n) = 0. 6. If in this. equation we put m=, then, because | (o)=a+h? +c 4+ d+ &.=7n we have (n) + A(n—1) + B(n—2) +...... + P(1) +nQ=o0 7. By means of the equations found in 4. 5. and 6. we are now enabled to express the sum of every higher power by the sums of all the lower; and consequently, when these last are found, we are enabled to find the former. On account of the frequent use which is made of them in the following pages, I shall here arrange them together. (1) +A=0 (2) +A(1)+2B=0 (3) + 4(2) + B(1) + 3C=0 (4) + 4(3)+ BQ2)+ CU)+4D=0 16 (n—1)+A(n—2)+ B(n—38)+...4MO)4+(n—1)P=e (n) + A(n—1) + B(m—2) +...4 P(1) + nQ=0 (n+1)+A (nm) + B(m—1) +...4+ P(2) + QQ) =0 (n+2)+A (n+1)+B(n) +...4 P(3) + Q(2)=0 (m) + A(m—1) + B(m—2)+...... + P(m—n+1) + Q(m—n) = 0 8. From these equations we successively obtain (1)=-A (2) = A? — 2B (3) = — dA? + 3AB — 3C (4) = 44 — 44°B + 2B? + 44C — 4D ()= — 44 54B—5AB — 54°C 4+ 5BC + 5AD— 5E (6) = 4-644B+94°B? — 2B°4+64°C4+12ABC + 3C° — 64°D + 6BD + 64E — 6F &e. and consequently the sum of the powers of the roots are expressed directly by the coefficients of the given equa- tion. ExampLe. When the equation x‘ — 2° — 192° + 492 —30=0, d=—1, B=— 19, C=49, D=—30. By substituting these values in the equations in 8. we obtain C)=1, (2)=39, (3)=—89, (4) = 723, (5) = —2849, (6)=16419, &. Any person may easily convince him- self of the truth of these results, by substituting in the first equation 1, 2, 3, —5 for x. 17 Remark. The formule in 8. are known by the name of the Newtonian 'Theorem, because Newton is supposed to be the first who has mentioned it. Other proofs of this theorem, and also much information relating to the subject itself, may be found amongst other matter in Kastner’s Principles of Finite Analytical Magnitudes, third edition, p. 538, &c. also in Kliigel’s Mathematical Dictionary, part first, p. 465, &c. Art. Combination. SECTION IX. Pros. The sums of the powers of the roots of an equation, or the expressions (1), (2), (3), &c. are given : find the coefficients of this equation. Solution. From the equations in 7. of the foregoing section, we obtain by transposition A =-— (1) p--4+@ ou BO+4A@+() 3 jg pede lon So ame Ee Pian: ETS li) ie 4 &e. By means of these equations we are enabled to determine successively the coefficients 4, B, C, D, &c. when the numerical expressions (1), (2), (3), (4), &c. are given, as they are assumed to be in the problem. SECTION X. Prog. From a given equation 2° 4+ Ax + Bui? + 2... 4+ Ma? + Na? + Pr+Q=0 D 18 whose unknown roots are called a, 8, c, d, &c. find LP hiekeok another, whose roots are —, —, —, —, &e. ae bc “ad é : 1 ‘ Solution. Substitute —for 2, and then multiply the J whole equation by ¥”; we then get Qy’+ Py’ + Ny? + My + 20+ + Ay+1l=o or ’ ie N M Misery 0 ee ia ees +—+—= Pag ; 1 and this is the required equation. Jor since x= —, 1 ; then y=—-, and since a, b, c, d, &c. are the values of x, x &c. are the values of y. Lk S17 yh ; Corollary. The roots —, Saya &e. in reference to a c the roots of the given equation, we term reciprocal roots. Therefore, if 2” + Ag + Bat? + ...... + Ma? + Nx? + Pr + Q=o be any equation, and 2+ A/x” + DF eH PE. ste. + P’x+ Q’=0, be the equation for its reciprocal roots ; we then have A' = Ein sg C= ried Q Q Q SECTION XI, Pros. Find the sum of a power of roots, when the exponent of this power is a whole negative number. 19 Solution. Let Pn ey Paget Pal at SO + Mz? + Nx? + Pr+ Q=0 be the given equation, and vw" 4- Ala 4 Blatt? + CPB + + P’x + Q’=0 the equation for its reciprocal roots (foregoing section.) Then according to § VIII, when the numerical ex- pressions (1), (2), (3), &c. are taken in reference to the second equation, (1) + 4’=0 (2) + 4’(1) + 2B’=0 (3) + A’(?2) + BA) + 83C/ =0 2. But (1), (2), (3), im reference to the second equation, are precisely what (—1), (—2), (—3), &c. are in reference to the first equation; we have .°., when for : : Ie N M ° A’, B’, C’, &c. their values —, —, —, &c., are substituted QQ’ Q (foregoing section) (—1) 4- a= 0 (-2) + GED + AP = (eye )eh 3(-) © B(-) + =~ =0 &e. from which we can determine successively the sums of powers for negative exponents. ExamPLe. When the equation x* — x° — 191° + 49x — 30 = 0, we have Q = — 30, PP = + 49, N =—19, M=-— 1: we have .. (—1) ee, —2)= ——, 20 31159 —3)= : ( ) 27000 the accuracy of these results; for the roots of the given Any one may readily be convinced of equation are 1, 2, 3, —5, consequently (—1) = 1 + ror rate Wed) 2 ue 1261 Pid Lig a 30° ef, 9 | 25. 900° 1 aa pciens (—3) = pgs RS sya te. a) 7 me i 125 27000 SECTION XII. Pros. Express the symmetrical function (a) by the sums of powers. Solution. For the sake of greater perspicuity, I shall assume that there are only four roots, because this does not affect the general principle. Then (a) = a* + b* +. ct + de (6) =a? + 4+ ce + dé. If we multiply these equations together, we then obtain (a) (8) = at? + Bete 4p crt? 4 det? + ath? + a’ b* + atc? + a’ce + ax d? + a2 de + b«c? + boc + bx d? + b? d* + cx d? 4 cde, The first row of the second part of this equation = (a + @), and the remaining two rows = (af); consequently we have (X) voeees (a) (6) = (a +8) (a8) and «!*. (a6) = (a) (6) — @ + £). It is easily seen, that these conclusions obtain, let the number of the roots be what it may. Since then (a), (@), at (a+), are only the sums of powers, what was required is now done. The radical exponents a, 6, may besides be either positive or negative. For example, if a be negative, then we have (—a8) = (—a) (@) = (8—a) and when a and @ are both negative, (—a —8@) = (—a) (—6) —(—a—8). Corollary. Since we are always enabled to express the sums of powers, either for positive or negative exponents, by the coefficients of the given equation, in like manner we can always find the values of the expressions of the a? B form ab? + a’ b* + a*c? + a’c* + &e., a st ; ax * B +— + aé a’ 1 1 1 —— + &.,—— + —— 4+ — OY ale > ax bP ri a’ be ti ac? these coefficients, without knowing the roots. 1 + — — + &c. from a? c* SECTION XIII. Pros. Reduce the numerical expression (a@y), which contains three radical exponents, to a numerical expression containing no more than two radical exponents. Solution. For the sake of perspicuity, I shall only begin with three roots a, 6, c. If we multiply the equation (aR) = at? + ab + arc? + a’c* + bec? + dcx by (y) = ay + br 4+ 07, we then obtain bho hwo (y) (a8) = attr Bh 4 gh hety 4. attych 4. gh ett? 4. b2t7 ch 4 bP est + ahr he 4 atbhty 4+ atte 4 atcht1 4 bPt7¢% 4 Ot cF*7 + azh? cy + axbrc? + abb*cr + abbrc* + avb E 26 2. For, in the first place, it may be proved by a similar manner as that used in §§ XIII and XIV, that both the function in the first part, and that in the second part of this equation, contain terms widely different from one another, and that for each term in the second of these two functions, there must be one equal to it in the first. 3. Further, since all numerical expressions in the equation contain ‘m — 1 radical exponents, [A] and [aByd...cxA] excepted, of which the last contains m radical exponents; .*. by § IV, the number of the terms in the function of the first part of this equation —="nxn.n—1;sN'—2...... N= mM -- 2 and in the function of the second part ro LS mead? ee OE {hepa Cee | pee EOS m—m + 2 +- MY —1% W— 2... M—-M+2.n—mM+ I =n xXn.n-1.n—2......n—m + 2. | ‘These two functions consequently consist of the same number of terms. 4. From 2 and 3 we may infer, in the same manner as in §§ XIII and XIV the accuracy of the assumed equation. From this equation, however, we obtain (Q) --.000 [aBys eee ikA] os [A] [aeys vee tk} — [atrAByd... ex] — [6 + Aayd... cx] —= ace oy sae aa [« + AaBysd... which answers the condition of the problem. Remark. The formula (@) obtains both for positive 27 and negative radical exponents, because the conclusions remain the same when the signs are changed. By means of this formula we are enabled to reduce any numerical expression [a#@y6é...A] to others, which contain one radical exponent less; and if we continue the operation with this diminished radical exponent, we at length arrive at sums of powers only, which, whether the exponents be positive or negative, may again be always expressed, by §§ VIII and XI, by the coefficients of the equation, to which the numerical expression refers. SECTION XVI. Hitherto it has been assumed, that the radical expo- nents in the numerical expression [#6y0é... A] are all different from one another. If this be not the case, and the expression is consequently of the form [a*gly"0d., ck], then the preceding formule, if we wish to apply them further, must undergo some modifications. It has already been shewn in § V, that if two numerical expressions [at gby° OU... Kk], [aByd...Z], the first with, and the other without repeating exponents, contain the same number of radical exponents, the number of terms in the first is less than those in the second by 1.2...ax 1.2 »MX1.2...0X5.....X1.2...%. The reason of this is only to be accounted for in this way, that in the case of equal radical exponents, there are exactly the same number of terms, which are equal to one another in the numerical expression [#@y0...%| for each combination of the roots a, b, c,d, &c., of which terms, only one is retained in [at gby© OU... cK], as we already know from the rule of com- 28 binations. Hence, however, it follows, in order to adapt the formule already found to this case, that each expres- sion of the form [a7 g3y° 80... x] must be multiplied by the number 1.2 5..4X1.23..BX1.2 scat Koscivin m® 1.2...%, &c. whose magnitude depends upon the repeating exponents a, , ¢, 0, &c. For shortness’ sake, in the course of the calculation, I shall denote this factor or coefficient by x, and when there are more, by x, x’, «’, &e. SECTION XVII. Pros. To reduce the numerical expression [@*], with a equal radical exponents, to others, which contain only a — 1 radical exponents. Solution 1. Assume that the numerical expression [aByd...cxA] in the equation (Q), § XV contains a radical exponents, and thate =@=y=d=...=A3 then this expression is changed to [a*]; further, the pro- duct [A] [eBys... cx] to [a] [e*"], and all the remain- ing a—1 numerical expressions in the second part of this equation to [2aa%-*]. For the reasons given in the fore- going §, if the coefficients x, x’, «’, are prefixed to the numerical expressions, we then obtain, x [at] = «/ [a] [2] — [a — 1)” [200] 2. But kal. 2.8 A E12. 3 041, KS 1.2.3,..a—2. By substituting these values, and then dividing by 1.2.3...a—1, we then get a [ai] = [a] [2 — [aa]. Consequently by this equation [a*] is reduced to the 29 numerical expressions [@~"], [2a a%-*], each of which contains only a—1 radical exponents. SECTION XVIII. Pros. Reduce the numerical expression [a%g6], with a-+h radical exponents, to others, which contain one radi- cal exponent less. Solution 1. From the equation (OQ), § XV, when we put a of the radical exponents = a, and the h remaining ones = @, we obtain K. ie >) = (oa Ga) oo aaa —hinr”’. [at 2 + Bail gb-1] 2, But (§ XVI) ae ee: | Be Le As ee edie eet We SOM at Shee Ws 1.8.38 Ae a2 i $270 752..53 $e it Vik Die Bins Erm Li Xe Mine Bc Bn cee If we substitute these values, and then divide by “1.2.3..a-1 x 1.2.3... 8 we obtain a. [x%gb] = = [2] [a-1gb] — [2aa%* eb] — [a + Bat gr which was required. SECTION XIX. _ Pros. Reduce the numerical expression [a gb y"], with a + 6 + ¢ radical exponents, to others, which contain one radical exponent less. 30 Solution 1. From the equation (@), § XV., when we put a of the radical exponents =a, b of them = #, and the remaining ones ¢ = y, we get ck. [at gby")| = Qf - [a] [at eb y"] — [aa]. e! fea at? aby!) a eH [a +e ai} gh-] y] —c.K" [at yak! gy] 2. But reget inks Rak ORS eee a as) 22 SB RR 1 4620138 2.0 Moo 1,8 .8...a—1 x1. 2238... ee aM 2, Bis A ee Osea IL © ee ee eae 7 = 1.2.38...a—-1 X11). 2). 3.0-1K%1.2. 3... r= 1.2, 8... Aa KO? Lote OX IMO IS. teed. If we substitute these values, and then divide by 1,2.3..a-1x1.2.3...8x1.2.3...¢ we obtain a [a gb y°] = [a] [at phy] — [2a at? gb 7°] ae [a+ B gil gb} y*] = [a+ y¥ gil gb yr as required, SECTION XX. Pros. Reduce the general numerical expression Jat gb y© ov . KEI, wihat+h+e4+04+..+h 41 radical exponents, to others, which contain one radical exponent less. Solution 1. If we compare the operation in §§ XVITT, XIX, XX., we shall, with very little trouble, obtain from it the following gencral equation : ( 3 GN 5655: ve a[al abl Su, .. cl] = [or] [att aby 8. — [Rar a2 gb y6 Sd... tT] — [a+ Gat) gb-1 6 80, ct] —[a + at 1gbyf180, NT] min oSeaneecdstvoessee —_ [ae + naa pa 7 OB... KEY] — [a + Aaa} Bd 80...K& AH] in which each of the numerical expressions in the second part contains no more thana +H+¢4+U+...... +k + {—1 radical exponents. The formula thus found obtains, as also the one in § XV, whether the radical exponents, a, 6, y, &c. be positive or negative, because in this case the conclu- sions remain the same. For the particular case, in which a = 1, this formula ceases to be applicable, because the repeating exponents a—1, a—2, which are contained in it, then become 0 and i which is impossible. In this case we must make use of the following formula : [a pb y* au ay KY = [a] [ gb-+* ou... ce] =: [a + gebiye OU... kT} — [a + yBdyo1 80... AY — le Oe 3: — [a + Ngbyf Sd... EAI] which may be derived fiom the same sources as the pre- ceding. Remark, By means of the equation (@) we are now enabled to reduce every numerical expression of the form [a* eb+* d¥...«&!], by a certain reduction of the repeating exponents, to another numerical expression of the form [a@yé...«KA]; and since this last, by means of the equation (Q) in § XV, may always be reduced merely to the sums of powers, and .*. may at length be expressed 32 by the coefficients of the given equation; consequently we are always enabled to express any numerical expression of the form [a gby" dd... cE] by the coefficients of the given equation. Moreover, the equation ( ¢ ) is always true, so long as we assign no determinate values to the radical exponents a, B, y, ©, &c. For determinate values, it may happen that radical exponents are equal to one another, which in the general expression were considered as different : thus, for instance, when in the equation (¢) 2a = 8, ora+ =v. Insuch cases as these, we shall do well, in order to avoid mistakes, to add the following equation derived from (©), § XV: ck. [at eby oo... cEAY] = x’. [a] [att pby dd... KRAT _ [a—1] ack ar [2a at2 gat ga ine ce I] — bik”; [2 + Bat gb1c gu... tT] —¢c¢.x”", [a + yak ebyh 180... RAT] &e. in which the coefficients, x, x’, «’, «’’, &c., have the values given in § XVI. Every integral or fractional rational symmetrical func- tion of the roots a, b, c, d, &c., however constituted, must necessarily be composed of numerical expressions of the form [a*gby*...dI]. Now since these last, as we have already seen, can always be expressed rationally by the coefficients of the equation to which they relate, conse- quently also the former can always be expressed rationally by these coefficients; to prove which was the aim of the present chapter. Sinee, however, the rule of symmetrical functions is of the greatest importance in the theory of equa- 33 tions, it is often requisite to express these functions by the coefficients of the given equation, I have .-. subjoined three tables, in which all numerical expressions, in which the sum of the radical exponents does not exceed the number 10, are fully calculated. Thus Table I* con- tains, in six small tables, the values of all numerical expressions for the Nos. 2, 3, 4, 5, 6, 7; Table IT, those for the Nos. 8 and 9; and Table III, those for the No. 10. The arrangement of these tables is evident at first sight. The letters A, B, C, D, &c. are the coefficients of the equation 2*— 42" + Ba**— Cx’ + Dx'— Ex’ + &. = 0, which is the basis of the numerical expressions, and these last, for the sake of facilitating the calculation, are assumed with alternate signs. In the horizontal lines, the numerical expres- sions themselves are found in a combination series; in the first upper horizontal series, are the different terms of their values, and in the vertical columns under them the numerical coefficients belonging to each term, accord- ing to the difference of the numerical expressions. Where there are terms wanting, or the numerical coefficients = 0, asterisks are placed. Thus, for instance, in 'T'able III [1°47] = BD—3 ABCD +3 C?°D+3 24D? —3 BD? —ABE +2 4CE +BCE —8 ADE 45224 A4°BF —BF —3 ACF +9 DF —6 4G +17 ABG —15 CG +6 42H —11 BH —6 AI +15 K. These tables were calculated by means of the equa- tions (©) and (D) in § XV. and § XX. For the more easy application of these tables, it is, however, necessary that the calculation be made successively, and * Note.—These tables are to be found at the end. Tvansiator. a. FF 34 that, in order to find the numerical expressions for any determinate sum of theadical exponents, we should first know all those for lower sums. Likewise the sums of powers must be previously calculated by means of the equations in 8, § VIII, which equations, on account of the change of the signs in the assumed equation, have the following values : (hh A (2) = A?— 2B (3) = 4 — 34B 4+ 30 (4) = 44-44B+44d4C + 2B? — 4D &e. Thus, for the successive calculation of the numerical expressions, Table III, we have the following equations : (10) = (10) (19) = (1) (9) — (9) (28) = (2) (8) — (10) (87) = (8) (1) — (0) (46) = (4) (6) — (10) 2 (5%) = (5) (5) — (40) 2 (1°8) = (1) (18) — (28) — (19) (127) = (1) @7) — (7) — @8) (136) = (1) (36) — (46) — (87) (145) = (1) (45) — 2 (5°) — (46) 2 (276) = (2) (26) — (46) — (28) ‘The numerical expressions in the first part of the equa- tions, depend here, as is easily seen, either on the fore- going, or on such numerical expressions as have a less sum of radical exponents, which, when these last are olreadv found, may successively be determined. Kine ( 35 ) IIl.—ComMPLETE SOLUTION OF THE SYMMETRICAL FUNCTIONS OF THE ROOTS OF AN EQUATION. SECTION XXII. TO solve a symmetrical function, here means no more than to find an expression for it, which contains only sums of powers. | A compound radical exponent implies one, which is compounded of more than one compound one, as a + 6+ yt+o+ &. in (a+8+y+06+ &c.), or (aa+b6+cy+dd +&c.) in (aa+b8+cy+dd+&c.) The terms of the first are a, @, y, 6, &c.; the terms of the latter are aa, bg, cy, dd, &c. In the opposite case, a and aa, in (a), (aa), are simple radical exponents. In order to show, that a numerical expression, viz. (a), can be raised to any power x, I shall merely write (a). We must then very carefully transform (a)* into (a“) ; for (a) = (a) (a) (a) (a).--3; on the other hand, (a“)=(aaa...). Soin like manner (3a+ 2) denotes the pth power of (3a+ 26), and (aa+b@-+ cy +dd+ &e.)¥ ‘the pth power of (aa + 66 + cy + dd + &c.) SECTION XXIII. Pros. Represent the numerical expressions (a8), (aBy), (aBys), &c. fully developed. 36 Solution 1. From § XII we immediately have (a8) = (8) (a) — (@ + a) 2. From § XIII we at first obtain (aly) = (7) (8) — (vy +48) — + Ba). But from 1 we have, when first y+a is put for a, and afterwards y+ for 8, (y+a8) = (@)(y + a) — (vy +6 +4) (y+8a)=(y + B)(a)— (y+ B+ 4) If these values, together with the value of (a@) from 1, be substituted in the foregoing equation, we then obtain (aby) = {y) (82) (2) — (y) +a) — (7+ 82) @ — (y+a) (6) + 1.2 (y+8+a) 3. From § XIV, we have (aBy 8)=(0) (aby) — (0 + aby) — (0+ Bay) — (6+ ya) In order to find the numerical expressions (8 + a@y), (8 + Bay), (6+ yaR), we need only successively sub- stitute 6+ for a, afterwards 6+ for @; and lastly 6+ y for y in the last equation in 2. If, after this, we substitute the values thus obtained, together with the value of (aBy), we obtain (abyo) = (8) Cy) (8) (a) — (8) Cy) +2) — (8) (7 +8) (2) — (8) (y +a) (8)41.2 (8) (y+8+4) — (8+) (8) (@) —(8+8) (y) (a) — (8+a) (7) (A) + (+7) +4) +1.2(8+6+a)(y)+1.2(8+y +6) (a) +(8+a)(y +8) +1.2(8+y+a)(B)+(8+ 6) y+a)—1.2,3(0-+y+6+a). t Qr Of 4. So, in like manner, from § XV, we have (aByde) = («) (ab y8) — (cay) — (e + Bayd) — (e+ya@d) — (c+ ay). We get the values of (e+a@y 8), (e+ Bay 8), (e+ yaR8), . (e+ a6y) completely developed from the last equation in 3, by substituting in it successively «+a for a, e+ for @, e+y for y, ande+6é for 6. The substitution of these values, together with that of (a@y6) in the foregoing equation, gives the solution required. 5. In this way we could proceed further, since we always go from one solution to another, and thus find the solutions of the numerical expressions, which contain six, seven, eight, &c. radical exponents. 6. Generally, if we have already found the solution of a numerical expression (agy6é...«), and wish from hence to derive the solution of another (a@y0é...«A), which contains A more radical exponents, we must, in the first place, multiply merely the solution of (af y 0... «) by (A), then in this solution substitute throughout, first A+a for a, then A+ for 6, A\+y for y, &c. and change the signs of the results and the former product. SECTION XXIV. Pros. Find the law, by which the terms in the solutions of (af), (aBy), (afy9d), &c. are formed, when the coefficients and the signs are left out. Solution 1. If in the solutions of the above numerical 38 expressions in the foregoing §, we omit the brackets and the signs, and separate the radical exponents, which belong to the different numerical expressions in each term, by a comma, and all the terms by a semicolon, we then, by means of (a), find the following : Re II. B,a; Bta III. y, 8, as y, Bas y+, a3 yta, Bs y+Rt+a IV. 38, Y Bs as 8, Y Bra; o, Vo Bseas é, er ds tf 8 y+tBb+a; o+y; By a3 o+8, ¥2@35 d+a, Y B O+ys +a; O+6+a, 73 S+y+8, a; d+a,y+8 O+yta, 8; 8+8,yta; d+ y+R+a &e. 2. Hence we may perceive at first sight and from 6 in the foregoing § the law of the successive formation of the terms. ‘Thus, in order to derive the terms of a solution from the terms of the immediately preceding one, we must 43 (a) put before each of all the terms of the preceding solution, the radical exponent which is now to be added ; (b) we must connect this by the sign + with each radical exponent of every term, while, at the same time, we add the remaining radical exponents of the same term without any change. Thus, for instance, if we wish to derive IV from III, | by the rule (a) we get a 8) YB, a3 8 Y»P+a3 O5y+6, a; 6,y+a, 83 O&O y+ht+a a9 and by the rule (b) from the first term in ITT 8+ y7,04;09+8,7,4;9 +a, 783 from the second term in IIT S+y,B +a; 8+B+ay73 from the third term in ITI d+y74+2a;8 tayt+B; from the fourth term in III Stytaesd+By +a; lastly from the fifth term in ITI Oo + raise ot ee The foundation of this method is so evident from the foregoing §, that it requires no further explanation. 3. But since this mode of representation possesses this disadvantage, that in finding the following solutions, we must first add the foregoing, we can .*. with great ad- vantage make use of the Hindenburgian method of involution, which is already known to my readers from the first principles of the rule of combination. Here follows this involution, whose construction is immediately deducible from 2: 40) oO-+a, y + PB S+y+a, 8 o+8, y +a o+y+B +4 &e. It is not necessary, in the first place, to remind my readers, that this mode of representation, besides the advantage that it mmmediately gives what is sought, possesses also this one, that each solution includes all the foregoing, as the brackets denote, and this follows of course from the very nature of involution. Remarx. The involution which is here given, in- cludes, besides, as may easily be observed, all possible combinations of the radical exponents a, @, y, 6, &c. both simple and compound, consequently can be used with advantage in many other cases, in which it is required to find all the possible combinations of this kind im a given number of things. SECTION XXV. Pros. Find the law of the coefficients and the signs in the solution of the numerical expression (a@y6...A). Solution 1. From the method in § XXIII, by which the solutions are derived from one another, and from the results themselves, it may, with some reason, be presumed, that the coefficients of the terms and their signs are subject to the following laws: (a) That each numerical expression of a simple radical exponent has unity for its coefficient ; OS an 41 (6) That each numerical expression of a compound radical exponent of m terms has the coefficient 1.2. ee m—1 3 (c) That the sign — or + may be given to every numerical expression whatever, according as the number of the terms of its radical exponent is edd 2? or even. Thus if these rules-be correct, the term (a) (6 + y) (8 ++) (+3424 .«) has the coefficient 1 x 1 x 1.2.x 1.2.8, or merely 1.2 x 1.2. 3, with the sign +, because it has two radical exponents of an even, and two of an odd, number of terms. That these rules are correct for (af), (aby), (aby), one can be readily convinced of by inspection. It only remains now, by a very common method in mathematics, to prove the rule, that when they obtain for any solution, they likewise necessarily obtain for the following one. 3. For instance, let CB ig. 20 (a+ B@+yt+.-+k) (Atptov...t) be any term in the solution of (a@y...... wp). The radical exponent of the first numerical expression in (A) contains m terms, that of the second x terms. Con- sequently the coefficient of the product, according to the hypothesis = 1.2.3......mM—-1X1.2.3... mame, 4. Now let (afy....w) be another numerical expres- sion, which contains more radical exponents than the preceding by w. For its solution, the term (4) by 6, § XXIII gives the three following terms : G 42 @w)(atBt+yte te) AtMtvV+t.. typ) —CatB@tyteuntetolAatutvt i t¥) —(atBt+tyt... tk) Atptvt..t+P+wo) 5. The first of these terms is obtained from the term (A), by multiplying the latter by (w), and consequently it has the same coefficient and the same sign; which agrees with the hypothesis. 6. The second term in 4 arises from the substitu- tion of wa, wth, Wy eo. +k for a, By Ys ever Ky (6. § XXIII) and occurs m times. In like manner the third term arises from the substitution of w +A, w+), Oh as oly W for A, py vy ......W, and .*. occurs 7 times. Consequently the second term must contain m, and the third 7, more coefficients than the term (4). 7. Hence it follows, that the coefficient of the second term in 4. = 1.2.3...mx 1.2.3...n—1, and the coefficient of the third term = 1.2.3....m—1 x 1.2.3....m”. Likewise these terms have a different sign, from that of the term (A). 8. Since this agrees with the hypothesis, so it may be concluded, that, when the hypothesis is true for the term (.4), it is necessarily true for the terms derived from it in the following solution. 9. Although here the term (4) has only been assumed as a product of two numerical expressions, it is suffi- ciently evident from the manner in which the proof has 43 been managed, that it may be extended to any other number of factors, 10. If .-. the solution of [@@y ... ~] be subject to the assumed rules, so likewise is that of [@By... Ww] which follows it. But they obtain for the first solutions, they .*. also obtain for all those which follow. Corollary. In order .*. to exhibit a numerical expres- sion of the form [#@y...A], in which all the radical expo- nents are different, completely solved, we only require to perform the involution given in § X XIV, and affix to each term the coefficient determined from 1 of this § with its sign. The following example will serve as an illustration. Exampte. The complete solution of [@@yée], when - the brackets are omitted in the terms, is as follows: +1, €[8[y[ela +2,e+y+a, d, 8 —1, e/d\yi/G+e +1,2+8,6,y+a —1,edly+e, —2,<+06, y+e+a —ili,<« Oly ta, B +2, € ) yt@re —I,« o+y; Ry % —1, d+ By Yo & —1, d+, y, 8 +1, eld+y, Ba +2, d+8+a, y +2; d+y+B, a +1, d+a, y+ +2, €& o+y +a, RB +1,¢€ d+8, Y+o —6, (O+y+e+a co —6,e+y+B+a, 6 +2,e+d+y, B, & +1,¢+8, é+y; a +1,e+a,d+y, @ +2, e+0+8, y, & +l, ety, d+8, a +1, e+, +8, y +2,e+d+4, 7,8 +l,etys o+a, B +1, e+, d+a, —2,e+d+y, Bta —2, e+B+a, d+y G2 ad sett Or e+, Ys Bs & — 6, e+o+B+a,y Lene y's 8, By % —2,e+7, O+B+a —1, «+8; 85 1. @ — 6, etotytB, —1,e+, 0, ¥ B —2, +a, d+y+8 +1,e+0, y, B+a —2,e+d+a, y+ +l,ety, 6,6+2 —2,e+y+, d+e +2,e+6+a, dy 7 —6, «+d+y+, +1, e+6, y+8, & —2,6+8, d+y+e +2,e+y+@, 6, @ —2,e+0+f, y+ +1,e+a, 6, yt+B —2,e+y +a; 6+ +1,e+o, y+, B +24, e+d+y+B+e We have .*. [a@y8e] = [e] [8] [y] [6] le] — [el [8] [y] (e+¢]—L[e] [6] [y +6] «—&e. SECTION XXVI. If there are more radical exponents equal to one another in the numerica] expression [a@y ...... A], or if it takes the form [a*pby°...... x&], the solution admits of reduc- tion, because in this case more terms than one are equal to one another. By the application of the rules (a) and (b) in 2, § XXIV, we have only to take care that no term occurs more than once, and with this view it is only necessary in performing the involution, always to revert to the combinations already found, and to omit all those which occurred once before. Thus we find the involution for the terms of [a*@°] to be as follows : 45 B\B\G a a) a 2B, B+&, &, & BIB|Bia 2a 26, B+a, 2a B|a|e|3 a 20, B+2a, a BIBIB+a, a, @ 2B, B+3a BIB \B+a, 2a 3B, a, a, a BBC +2a, a 3B, a, 2a BiB B+3a 3B, 3a ) BI2B, a a, a 8B+a, a, a B28, a, Qa B+a,2B+a,a BI2 8, 3a Ba, B+a,B+a BI2B+a, a, a 3B+a, 2a RIB +a, Ba, a B+2a, 26+a B\2QB+a, 2a 2p+2a, B+a BIB+2a, B+a BB+2 a, & BI2SB+2a, a SB+3a B\2B+3a This involution contains every possible analysis of 3*+3°. In the same manner we obtain generally by the involution for [2% pby*...... x&] every possible analysis of az+he+ty+...+k«. It may .. be used with advantage in all those cases where it is required to represent analysis of this kind, with facility and without the danger of omission. In the numerical expression [a*], the opera- tion is merely reduced to a numerical analysis, respecting which information is given in the rule of combinations. But as to the coefficients of the terms, it 1s easily con- ceived, that the rules (a) and (b), in 1, § XXV, in the case where the numerical expression [a@y...A] is changed to [a%gby°...x#], must undergo many modifi- cations, and first, for this reason, because in this case 46 more terms than one of the solution vanish, secondly, for — the reason given in § XVI. But in order that we may not be obliged in the sequel, to break the thread of the inquiry by extraneous matter, we shall premise with the following auxiliary rules. SECTION XXVII. I—AUXILIARY RULE. Pros. Find in how many ways a given number of things may be placed in a determinate number of divi- sions, in such a way, that in each of these divisions there is a given number of these things. 1. Solution for two divisions, Let A be the number of all the things, a the number of these things which are to enter into the first of these divisions ; .*. 4—a the number in the second. It is evident, that this is merely to find the number of combinations of the ath class in 4 things. Now this is Ld, A—1.A— 2 oon. serve A—atl vad ch ia RA cae ena or if we multiply numerator and denominator by 1. 2. 3.,.4—a, and then in the denominator substitute a’ for A—a; in which a and a’ denote the numbers contained in the two divisions. 2. Solution for three divisions. Again, let 4 be the number of things; further, a, a’, 47 a’’, the number in the first, second and third division, ata +a/=AZ. But in the first division the number a may enter m as many different ways as there are combinations of the ath class in A things; the number of these ways is So in like manner the number of ways, in which the remaining 4—a things may be arranged in the second division in the number a’ _A—a. A=—a—1......... A—a—a'+1 Each of these last combinations may be associated with each of the first, and the number of combinations which is thus obtained is .-. the product of those two numbers. ‘The number of. ways, in which 4 things may be arranged in the first, second and third divisions, in the numbers a, a,’ a,’’ is consequently Ee Aa Aon Site terete. A—a—a’+1 Pmt AC ee ee Le aM BEPC BN nny Loe wE¥, or, when the numerator and denominator are multiplied CE y le Re A —a—da’, ohlhs pec? AA a a a | TE aan he eee vig wt ie pipe as 3. Solution for four divisions. Again, let A be the number of all the things, and a, a,’ a,’ a,’ the numbers which are separately con- tained in the first, second, third and fourth division, ~.a-+a’+a’+a"=A. The number of cases, in which 4 things may be arranged in the number a, The number of cases, in which the remaining A--a things may be arranged in the number a’ _ Aa, A—a-l ..... A—a—a' +1. ms 24, 2 ee tebase sane baveeods a’ consequently the number of cases in which 4 things may be arranged, first in the number a, and then a’ 4.A—1 ee laps Piha ay A—a-—a’'+1 Ts hee eee WoT oe a! ‘ The number of cases in which the number a’ may, in the third division, be arranged from the remaining A—a—a’ things _A—a-—a'. A—a—a'l. Sate A—a—a'—a"’ +1 A Tv AS Se ciicks cin deeee al’ ; Each of these cases may be combined with each of the preceding, .*, the number of cases, in which the 4 things are arranged in the four divisions _ A’. A=1 TEL a 8H A—a—al—a’+1 gd OP Br Sorat et OATS GE a be eer eae or, when we multiply numerator and ‘denominator by | 1S 2a a eed Ae A —a—a’—a"’ PLC IS As geeg cas sare narecpat POSS Pon of A Pal 2 OCIS a UB igh boas 2.5 gf 4. General Solution. The conclusions drawn in 1, 2, 3, may easily be extended to any number of divisions. The number of — ways in which 4 things may be arranged in » divisions, — 49 the number a’, the third the number a”, &c.; lastly, the nth division the number a of these things, is .°. pee Bah EE A—1.A Tay 2 hae lee tel. eet OX .:. Ke Examr.e. Place 16 balls in four divisions of 6, 5, 3, and 2: in how many ways can this be done >—Here A=16, a=6, a’=5, a” =3, a’”=2; consequently the required number | ee eee ey ee ES 8 es CB ae ee i Hi ~1.6.9.4.5°681.2,58.4.5xX1.28- 3x12 = 20180160. 2 SECTION XXVIII. Il,—Auxiliary Rule. Pros. The numbers A, B, C, &c. of different kinds of things are given: find in how many different ways these, viet tees numbers may be arranged in a given number, When in each determinate division there shall be a given number of things of each kind. Solution 1. For the sake of perspicuity, I shall assume x the particular case, that there are only three numbers of Ages a different kind given, which are to be arranged in four divisions, so that in the first division there are a things of the first, 6 things of the second, and c things of the third kind; and that a’, 8, ¢/; a”, BY’, c/3 al’, b”, c”, denote the same for the second, third, and fourth divisions, as a, b,c, do for the-first, .-. a+a/+a/+a/"= 4, b+ 664-0" +b = B, c+ 4+e/4+e"=C. Further, let H LiKe See axel Rie! eR iver Aseria ede BB oe serene seeneatsneneneseneeneens Bi aiiee Sal ¢ 2.3.0; Rad eeMe a sod 62 sO) Ke ee Wik 2 2 2 8 verter Be eee ase? Guat C Ty Oc lera.. 2. ecm ce |. Clete, Oi Gaede 2. Then, from the preceding §, A is the number of ways in which 4 can be arranged in four divisions of a, a’, a’’, a’’’ things, \/ the number of ways in which B things can be arranged in four divisions of b, b/, b’’, b/’ things, and \” the number of ways in which C things can be arranged in four divisions of c, c’, c/’, c/’. 3. But it is evident, that each of these divisions may be combined with each of the other two divisions in all possible ways. Now, since the number of these combina- tions is equal to the product A A’ A”, in like manner the number which arises as often as the numbers 4, B, C, agreeably to the proposed conditions, are combined together = AGN 4. What has been here proved for three numbers and four divisions, can, in a similar way, be proved for every other number. If ..-. /, A”; N,N, AY, &e. are similar expressions for the numbers B, C, D, E, F, &e. as is for the number 4, then the required number is always =X NA”! ANY NY, &e. ExamrPie. There are 40 balls of four colours, viz; 10 red, 14 blue, 9 green, and 7 white: show 5] in how many different ways these 40 balls may be arranged in three divisions, so that in the first division there may be 7 red, 5 blue, 3 green, and 2 white ; in the second division 2 red, 6 blue, 4 green, and 1 white; and in the third 1 red, 3 blue, 2 green, and 4 white. Here A=10, B=14, C=9, D=7; a=7, b=5, oe 2: =a o b/=6, (/=4, d=1; a“ =1, . bY! —3, cha 2, di’: 3°. aie REA EA ae C295 oT 4 Se oT Se 8 561 1.2.3.4.5.6.7.8.9.10.11 .12.13.14 Al hen : = 168168 1920.4; 6.0.1. 2.5.4.5) P2838 16816 Lit-Qer 3 SARS OPCs AG le err 1960 Presser Ane 16.8 mee T 2 : | Palak: docks a Sia a Rg AN ele ete el eS Ciel Oh RA er ah a ee Lee Boe et The required number of possible divisions is .°. = AN AYN” = 8009505504000. SECTION XXIX. L11,-— Auxiliary Rule. | Pros. Let there be more numbers 4, B, C, &c. of things of different kinds. It is required to arrange these things in pty! +p +p’ + &e. rows, so that in each of the js rows there may be a things of the number J, b things of the number B, c things of the number C, and so on; in each of the py’ rows, a’ things of the number 4, b/ things of the number B, c’ things of the number C, and so on; in each of the px’ rows, a” things of the num- ber 4, 0” things of the number B, c’” things of the O2 number C, and soon. Find the number of all the pos- sible divisions. Solution 1. Vf all the rows were different from one another, as, for instance, when they are represented by different numbers, then we could have applied the formule of the foregoing §to this case. Then y we 2B tte es ANT GL 2 REV KC ES a Pox OL. 2 ee Xe Be. Leo UBORE Kors. De. Sicha ees REOTVR Me Bn 2 8 cerette teeters se tetenteeeeterees Cel. CO. CCK re CERI Ne. ee x Cll ae ac pe wc. &e. and the number of all the possible divisions, as in the preceding §, = AX A” VN” &e. N 2. But since the problem only requires in general, that the « rows should be arranged with the combina- tions of a, b, c, &e. things, the py’ others with the combi- nations of a’, b’, c/, &c. we must .*., as we know from the rule of combinations, dittdsty the number just found by bi Zi.ip & be daipowd 28 ww KE BU x &e. 3. The required number of all the possible divisions, according to the conditions of the problem, is .-. Wh RM” Se. Li 2m KER kK EQ i! & Be Remark. Any one of the numbers a, b,c, &c. a’, UV’, o; &c., &c. may be = 0. This happens, for instance, when 53 in a Certain row, or in more rows at the same time, one or other of the different kinds of things is altogether wanting. In this case, it is only necessary, for self- evident reasons, to omit from the denominators of A, A’, A”, &c. those of the products 1.2...a, 1.2...6, &e. which refer to the deficient numbers. SECTION XXX. Pros. Find the coefficients and the signs of the terms in the solution of [a gay ... «&]. Solution 1. If in the numerical expression [a@y0d...w] more radical exponents than one are equal to one another, consequently, when this expression assumes the form [a*eby® ... xk], then the terms of the solution have the following general form : (ae +b@ +cey +... + fd) x (Wat WB+cdyt+... + UA) x (a/at B+ e/yt+ ... +70) r &e. epee 2. Since in each term of the solution of [a@y6...w], all the letters a, 8, y, ...w are divided into simple and compound radical exponents (Remark, § 24) ; then, like- wise, in the case when this numerical expression assumes the form [a4 gby° .., c&], in each term (yp) atd+a’+if&=a,64+0U4 0 + & =8, e+tce4+e+ &&=¢, &e. or, in other words, the radical exponents of the numerical expressions, which occur in each term as factors,. are no other than the divisions of aa + b@+ cy +... + kk, + HS 54 as was already observed in § XXVI. We must now, in the first place, find how many terms of the solution of fay cnet w] must be combined to form a term of this 3. If we consider those radical exponents in [a@y...w] which = a, as things of one kind, whose number = 4; those which are = @, as things of a second kind, whose number = {; those which are = y, as things of a third kind, whose number =, and so on; the question merely becomes, to find how many terms of the solution of [a@y...w] are to be combined with the term ({); in how many ways the numbers a, b, ¢, .... & of things of different kinds may be arranged in divisions or rows of the respective numbers a, b,c, ..... SYR Ete, TOL AS RUWD SiCe ae Te 4. Since this is exactly the problem in § XXVIII, when we put d=a, B=h, C=r, &c. we obtain, when NW denotes the number of the terms of [a@y... w], which are to be combined with (WL) Neu etn oe Raat are PR PRYS Calsie' stents ne eee a—-l.a 1 aa Heel CER vai LOG a a WO dhs aa 1 AS Nags thane a gs ea teat 5 Ot et 1. 2...0x1.2...0'x1. 2...” x &. * eye RaW lites ce eonice Hearts aad sie c—l.ee 1.2..8%1.2..0°xX1.2..07 x &e. * &e. 5. Each of the terms which is combined with (W), by § XXV. 1. (4), has the coefficient si) Cr I 2...m=+IX1 6 2.0.0! 1K 1 2m = 1K ce: , when we puta+b+c+8&e.=m, ad + Wf +e 4+ &e. = m’, a” + 0” 4+ ce’ + &e. = m”, and so on. 6. By § XVI., when [2@y...w] is transformed into [a*gb+°...«k], the latter expression must have a coeffi- cient x, by which .*. each term of the development, consequently also the term (W), must be divided. But by the same § ee es HK ie ell OC) Bee es oh i mee 7. From 4, 5, 6, it follows, when the coefficient of the term (w) is represented by A, that jane 1.2.-9n— 1X 1.2,..00 — 1 xX 12...m¢—1 x &e. or if for N its value in 4 be substituted 1. 2...m—1 x1. 2...m/—1xX1. 2...m//—1 x &e. eee penne Wea 1 ae Disa 22, Sco" x SOL LD Sc SC ben SO Se 1 De. Chee, G1, Sar Gl 2. KL 2 cl! x &e, &e. K= 8. Hitherto it has been assumed, that all the nume- rical expressions which occur im the term (yw) as factors, are different from one another. If this be not the case, or if the term have the form (aa +08 + ey + ww. + fo) + (va +B +ey +... + LAMY + (aa + WB + cy + see Bro” + &e. 56 then the number of divisions (§ X XTX.) which the numbers a, b, cr, &c. admit of, is r 1 1 2 RAM x Le pec A 3 a ee Pap) netlen a ee eg (1.2...a)"x (1.2...0)" x (1.2...) x &e. * (Oe BEB oo. 2 ie ae en bags CU Pb) 4s (ae SOO oa) 22 hab) eee, 19 Qh Breer aa tatecnce teenies time are hH—1.¢ (1.2.00) (1 (2.00 hx (1. 2...” + &e. 7 &e. 9. But from § XXV. 1. (6), the coefficient of every term which is combined with the term () = (1.2...m—1)"x(1 .2...m/— 1)" x CIM... La oe Further, as in § VI. rn A ee een XL OX rat 10. We .°. have for this case i » 2...m—1)" x (1. 2...m/—1)"” x 4 (1 .2...m/—1)"” x &e. | 1G Fo OTE 2 SER CAT NUE? SECT SOx LOO or, when for XN its value in 8 is substituted, Bi -2...m—1)*+(1.2...m’ —1)" x (12.0. mn! — T= &e: | K= «(Li ae (Dee oa Ux (1) 2p aencnre X-(1 62 OKC T. beix (1 ee ax Ore: x (1 22,56)" x. 2a 0 pax ees Oe Miten dean eh &e, 175 eee LEX G22. Mints San We Fa eee 57 iC A a Ltex (1.2 2 Gitar 1) x (1.2...m/—1)” x &e. | TE. oi TR | WC Te Base 01.2. DUK RB 20.60% one x &en MAC Vie 220-0 X18... ble Seec a Ser MOM Lae sad KL sd 2 on a eee cue Ne” Be beste ts &e. - 12. This expression for K includes that in 7, because we obtain the former when we substitute w=p/ =p’ = &c. =1 in the latter; it is consequently quite general, and obtains for all imaginable cases. In the case where one of the numbers m, m/, m’’, &c. say m,=1, we must, instead of the product 1 . 2...m—1, merely put 1. 13. The sign of the term (pb) is always the-same as that of the expression (Fm)! x (Fm’/yY! x Gr)”, when every time we merely give the numbers m, m/, m’’, &c., the sizn—when even, and the sign + when odd. The reason of this follows from § XXV., 1. (c.) Examere. Determine the coefficient and the sign. of the term (3a+76 x 2y +40) (5a+ 47 +0)* (5a)* of the solution of (a°*@7- 1487). Here a3, b=, cm2, d=4; a/=5, W=0, f= 4, w/—1; a’=5; further [i fit ¥ P pl! = 4 Suse ee iba a+b+c+d=16, m/ =a +b/+e¢+d/=10, m’/=a’/=5. We have consequently from the formula in 11. 58 ogee »1LS)PXCl 2. 3,..0) RC eS. Se c See jee x (1.2.38 %1.2..7Kh2 x12. 3, 4). | | x ped thy CN Dre Bs Sp a EM a Bie fa ae be LX Peg .8.4.5)4 4 500594094. Pee sae : The sign of the term is the same as that of the ex- pression (—16)' x (—10)* x (+5)‘, consequently +. SECTION XXxXT. In order to render what has been already advanced more intelligible, I shall here give the complete solution of the numerical expression (a°°), which has already been made use of in § XXVI. as an example for the representation of involution. ‘The terms are arranged as they are there found. (a6?) = as (a)* (8)? qo (a) (2a) (6)? + ys (Sa) (8) : — 24 (a)°(2)"(a + @) + 2(2a)(4)"(a + 8) +9(a)(6)° (2a +2) — 2 (8) (3* + 8) — x2 (a)? (6) 28) + 2 (@) 2a) ) (28) —s (3a) (8) (28) + 3(a)’(8) (a + 28) + (a) (8) (a + BP — 3(2a)(6)(a+ 28)—(8)(a + 6)(2a+ 8B) —3(a)(6)(2a+ 28) +2 (@) (3a+28) + 2(a)?(28) (a+ 6) — 5 (2a) (28) (at) —4 (a) (28) (2a + 8) + 3 (28) (3a + 8) + 3 (a)? (38) —s (a) (2a) (38) + 3 (3a) (38) — 3 (@)? (a + 38) — () (a+ 6) (a + 28) — 5 (a+ 8) + 3 Za) (a+ 38) + (a+ 28) (2a-+ 6) +3(a+@) (2a+ 28) + 2 (a) (2a+ 38) — 19 (3a + 38). From this example we shall clearly perceive, how we are to proceed in every other case, and it seems to me unnecessary to add any thing more. Vil.—On THE VALUES OF THE, FUNCTIONS OF THE ROOTS OF AN EQUATION WHICH ARE NOT SYM- METRICAL, AND ON THE METHOD BY WHICH TO MAKE THESE VALUES DEPENDANT ON EQUATIONS. SECTION XXXII. SYMMETRICAL functions differ from the others in this, that in the first place they contain all the roots of the given equation; and secondly, these roots are so combined with one another, that the functions mwhen tes rep enaitian, are changed. For this kind of functions, it is sufficient only to mention in general the form of the combination, without referring to the roots themselves, because we are always sure before-hand to obtain the same results in the composition. Thus, for instance, the expression (12) is fully determined, although by the notation nothing more is indicated, than that we are to take the sum of all the products which arise from thescombination of every root with the square of another. A function of this kind .*. can only have a single value, and this value is no other than that, the way to find which, was shown in the foregoing §. It is, as we have already seen, in reference to the coefficients of the given equation, always rational, and it must necessarily be so, because otherwise the function would have more values. But this is not the case with the other functions. If 60 we merely wished to give the form of their combination, we should not by this means alone be able to determine the function. If, for instance, a, b, c, be the three roots of an equation of the third degree, then indeed the sum of all the three roots a+ 6-+-c, can only be expressed in one way; on the other hand, the sum of two roots, in three different ways, viz. by a+6, a+c, b+c; and the difference of two roots may be expressed in as many as six different ways, viz. by a—b, b—a, a—c, c—a, b—c, c—b. + &. =0. 4. Let n=2m+1. In this case, when 4/t is put for x, we have (¢?+ Be? + Dt”? + Fe"? + &e.) Jt = Ai” + Ct"" + Et” + &e. and when both sides of the equation are squared, and the terms properly arranged, we obtain the same equation as in 3, except that we get 2m+1, instead of 2m. 5. For both cases of the equation we consequently have + (2B—A*)t + (2D—2AC + B)t + (2F-2AE + 2BD—-C’) t'?+&c. = 0. Remark. If we had not already known how to find the expressions (2), (2°), (2°), &c. by another method, we could have found them immediately by these means, by placing the equations in 1 and 5, opposite one another, and 79 putting the coefficients of the same powers of ¢ equal to one another. ‘Thus for instance we obtain — [2] = 2B — A’ [2°] = 2D — 2AC + B — [23] = aF — 2AE + 2BD— CO [24] = 2H — 2AG + 2BF — 2CF + D’ &e, from which the law is easily seen. If we denote the coefficients of the given equation, in order to denote the places which they occupy in it, by PT fai ft Sediiinsicads dh eAwB CO, Dihostha’ ley will be still more easily perceived. ‘Then we have 2 ony — [2] =2A — AA 4 LS per [2°] = 24 — 244 + AA 6 Ley ' 4 .2 om — [29] = 24 — 2444 224A — AA 8 mek | (ap eh? SOMES ee [24] = 24 — 2444 2AA+ AA AA and in general 2n 2n—1 1 2n—2 2 2n—3 3 + [2"]=24-244+2AA—-2AA+... n+1n—1 nn Oit +4AA+ AA | the upper sign obtains when n is even, and the lower \ when 7 is odd. Euler uses these formule for finding the impossible roots of an equation [Complete Introduction to the Differential Calculus, translated by Michelsen, Part III. p. 135], but he gives no proof of them, but merely says, that they may be found by the theory of combinations. A proof of these formule different from the above, may be seen in Kliigel’s Mathematical Dictionary, Art. Combination, p. 469, &c. - LO 80 SECTION XLIV. Pros. From the given equation a? — Ax’ + Br’ * — Cr" + &. = 0,7 find the equation for the mth powers of its roots. Solution. The roots of the required equation are 2’”, a//™, x///™, &e. Hence we obtain in the same way as in the foregoing §, the equation t*—[m] t+ [m?] &?—[m9] P+ [mt] ne sesnee Ee [ae” | t+ [m"] = 0 The numerical expressions here are all of the form [a"], — and .*. may easily be found. SECTION XLV Pros. From the given equation Pi Ar +. Bae — Or + oc, =.0; find the equation for the differences of its roots. Solution 1. The number of the different values which the function 2’ — 2’ contains by the substitution and— transposition of the roots, is equal to the number of variations (in the sense in which Hindenburg uses this _ word)* of n different things of the second class, .*. = m. (xn—1). The required equation is consequently of the n.(n—1)th degree. Further it is evident from 5 XXXVI, that this equation contains only even powers of t; and that .-., when for the sake of brevity we put 1 « ai 1) = 2m, it has the following form : t?’ se Alpe—2 x Bie ot C/t2"-6 a3 &e. = 05 * See vol. 1. p. 83, note. 81 and the roots of these equations are («/ —«/’)?, (a/ —ax/’)?, (x! —a/")?, &e., (0 —a!)?, (a —a/")?, &e. &e. The coefficients 4’, B’, C’, &c. may be determined in the same way asin § XXXVI. yet the calculation by this method is attended with many difficulties, and besides» the law of the terms cannot be easily discovered. ‘The following method, which will be frequently used in the sequel, is more simple and general. } 2. For this purpose, put SL = (alae!) 2b (00 mae!)2 4 (00 meel”)2 et (ae mal)2 Fs, SQ = (a/—a!/)4 + (00 mal!) 44 (00/man!?) 4 pase ee (00m!) 4, S13 = (alma!) F 4 (ane) (ae ma "Fee (00a!) 4, &e. then the expressions S1, $2, 83, &c. are no other than the sum, the sum of the squares, the sum of the cubes &c., of the roots of the equation for ¢. 3. Since the expressions S11, $2, $3, &c. are the same for the transformed equation, as the expressions (1), (2), (3), &c. are for the given equation, consequently the formule found in § IX. are equally applicable to the coefficients 4’, B’, C’, &c., when throughout $1, S2, S3, &c, are put for (1), (2), (3), &c. and also— 4’, +.B’, — C’, for 4, B, C, &e. Thus we have A’ = S§1 A’S§1 — §2 1 aS cles ae TL ats Ls 2 = B’S1 — A’S2 + S3 i OM aan aa 1 p — CSL = Bis2 + A'S3 — S4 ae 4 &e. M 82 If we had calculated the expressions $1, $2, S'3, &c., we should have found, by means of these equations, the coefficients 4’, B’, C’, D’, &c. 4. If the expressions $1, S2, 83, &c. are solved, we obtain Si = (n-1) (22 + x? 4 2/2 4 &e.) — 2 (a/x! + ax! + a4! + &e.) = (n—1) (2) — 2(1?) S2=(n-1) (2/4 +04 44/4 4 &e.)— Aa! oe! 4320! + aoe!!! fe y/8y/ aa! 3 4 ap! /3.g/// ak &e.) 4. 6 (2/22? + gl Pall? 4 gl/2y//!2) = (m — 1) (4) — 4(13) + 6 (2’) S3= (n-1) (rte aie q// ale 1/6 “ke &c.) EaAG (a/a? 4 ar peal al 54 gly! 4 ll gll5.4. al l5ep!ll 4. Be.) 4 15(a2a0!"4 py ll2 4 gpl yll4 4. alba! ll2 4. pl Beg lIA A gl 4yQlI2 4 Be.) — 20 (a!33//3 4/8/34. 8/3 + &e.) = (n— 1) (6) — 6 (15) + 15 (24) — 20 (3°) &e. These values of S1, S2, 83, &c. need only be substituted in the equations in 8, in order to find the coefficients A', Bl, Cl, &e. 5. But these values may at any time be reduced, by means of the two equations (a8) = (a) @) —(a + 8) 2 (a®) = (@)? — Ga) _to sums of powers only, and then we have sano fOS9) 83 S2 = (n-1) (4) — 4[() (3) — ()] + 6[ OO) (n-1) (6) —6[(1)(5) — (6)] + 15[(2)(4) — (6)] 20] =O] &e. or after the proper reduction, 6. S2 = n(4)—4(3) (1)4+ pee S3 = n(6)— 6 (5) (1) + 15(4)@) - ao? and in general Spw=n(2u) — —2y(24—1(1) + "4B PR (au—2)(2)—.. ne ape: ae RE ut 1(u)? oe Bate Bry Peery ig Ae Bi dveves ait 2 These formule will be of great use in the sequel in the theory of imaginary roots of equations. Remark. From Elementary Books on Algebra it is : A known, that when in the equation «”— Ax” + &c., # + 7 is substituted for x, its second term vanishes. But since by this substitution, all the roots of this equation are diminished in an algebraical sense by the magnitude A "ak their differences remain the same, consequently also the equation for their differences undergoes no change. ee 84 We obtain, however, this advantage by losing the second term, that the values of the expressions (1), (2), (3), &c. S1, $2, 83, &c. are much more simple. ‘Thus we have (1)iS310 (2) = — 2B (4) = 2B°— 4D (5)= —5BC + 5E (6) = — 2B° + 3C? + 6BD &e. Further, from 6, because (1) = 0, we get Sl =n (2) S2 = n(4) + 3(2) S3 = n (6) + 15 (4) (2) — 10(3)* &e. These formule may be used with advantage in the ease, when in the values of S1, S2, S3, &c., which have been found in 4, there are such numerical expressions as exceed the limits of the annexed tables. If this be not the case, it will be better to retain the former formule, SECTION XLVI. Pros. From the given equation a — Ax" + Br? — Cr’? + &. = find the equation for the sums of every two of its roots. Solution 1. The required equation must have the following Yoots : al < , ‘ x +e", Mal”, a! 4 al", eee Egle Ee hi De awl + al? eoeese and the number of these roots is equal to the number of | | | | | } 85 combinations of x things taken two and two, consequently n.n—I1 ERE: m. ‘The required equation will consequently be of the mth degree ; it may be represented by (tm A) + Bi — C'S +e. = 0. The coefficients 4’, B’, C’, &c. may be most easily de- termined from the method used im the foregoing §. , for which, for the sake of brevity, I shall put 2. For this purpose, then, put Si= pay +2”) + (a/ +2’) 2 (a’ +2’”) =.) + (a +24”) a mn S2= (a! ta!” 4 (opal! 4- Fal" Ee ba). yo (av +2”)? + (0! + 2e!)8 + (a’ + 20/”) + . + (e+ a//)3 4. a &e. so that the expressions S1, $2, §3, &c. denote the sums of the first, second, third, &c. powers of the roots. Ifwe have determined the values of these equa- tions in any way, then the coefficients in 3 of the fore- going § give the coefficients 4’, B’, C’, &e. 3. If we solve these expressions, we get Sh (n—1) (1) S2 = (n—1) (2) + 2(1’) S3 = (n—1) (3)-+ 3(12) S4 = (n—1) (4) + 4(13) + 6(2?) S'5 = (n—1) (5) + 5(14) + 10(23) PF cc. whence the law may very easily be discovered.\ I 86 4. If we wish to represent the values of the expressions S1, S2, S83, &c. immediately in sums of powers, we only need proceed as in 5 of the foregomg §. We then obtain S1 = (n—1) (1) :, (ye—(2) S2 = (n—1) (2)+2 | : | S3 = (n—1) (3)+3[(1) 2)—G)] 2 4 Se ne (4) +4 £0) (8) ay] +6 SS nat | S5 = (n—1) (8) +8101)(4) —(8)] + 10002.) —(2)] &e. or after the’ proper reduction * §1 = (n—1) (1) $2 = (n—2) (2) + ao S3 = (n—2’) (3) + 3(2) (1) $4 = (n—2°) (4) + 4(3) (1) +62) S5 = (n—2*) (5) + 5(4) (1) + 10 (3) (2) &e. and in general Su=(n—2) (uw) +m (u—1) 0) +4 en (2) pl 2 ni cS u-3) (3) +. and the last term of this series is either iD fe e —j. —2 e@eteeveae tee —— 1 = a be Sees Wee Bie setae: ie | | : ; : | | | | or ts pL pl 12 e 9 itl, pl may a) ee 1 2 Didsnderses 2 according as m is an even or an odd number. The abbreviations in the note to the preceding §, may, moreover, be also applied here. SECTION XLVII. Pros. From the given equation a” — Ax) + Br’? — Ca’? + &. = O find the equation for the function (axv’ + bx’’)*; when p is a whole positive number. Solution 1. Since in the function (ax’ +62)’, we can put every other root of the given equation for the roots x’, x’’, consequently the number of the values, which the function can contain, = n . » — 1, for which I shall sub- stitute m. ‘The required equation is consequently only of _ the mth degree, and may .:. be represented by the equation ” — At 4 Bier? — Ct" +. &e. = 0. 2. The method which has been made use of in the two preceding sections, for determining the coefficients 4’, B’, C’, &c. may likewise be applied here. Thus, if we denote by S'1, 2, S83, &c. the sums of the first, second, third, &c. powers of the roots of the transformed equa- tion, we have \ 88 = (ar! + bx!) ? + (ar + bx’) *+ (ax! + ba!) ? +. S2 = (ar/+ba!’)\” + (ar + bx’)? + (ar! + bv’)? +... S3 = (ar + br’)? + (av! + ba’)” + (a2x’ + ba’)? +... &c. Having determined these equations, the equations in 3, § 45, give the values of the coefficients 4’, B’, C’. It 1s now only necessary to determine the expression §1; having found this, we then obtain the remaining ones, S2, 83, &c. when we successively substitute 2p, 3p, &c. for p. To find .S'1, we shall arrive most readily at the object in the following way. 3. Make the new expression ...... (db) cvenee = (ax! + bz) + (av + bz)? + (ax’” + bz)? + (ax’” +- bz) + &c. ; in which z denotes any unknown magnitude hitherto unde- termined. If we solve this expression by means of the binomial theorem according to the powers of z, we obtain — S =a (ve? + ar? + ol? + ov’? + &.) + pad (a?) 4 allel 4 glPV 4. W/V &e.) & + &ec. | 2=a"(p)+par" b(p—1) . z+ -b— saa po Pat B(p— 2). 3% + &e. This equation must always be true, whatever we substitute for z. 4. Now, if we successively put a’, a”, a’, a/”, &c. for 89 z, and denote that which & becomes by these means, by >, =”, D”, =”, &., we have in the first place from the equation (@) WY =(a+b)r’? + (ax” + ba’)? + (aa +b’? + &e. 4 = (ar! + br” Pf + (at by”? + (Car + br! + &e. yy’ = (axr’ ay bx!!)? ah (ax// a bait ys ae (a + b)Pa ll? + &e. &e. Hence it follows that B+ + B+ Be. = (at by [p]+ St Further, from the equation () we obtain ea a’[p| + pa? bf p-1] . a” a r ——— a?~*}?[ pm2|.u? + &e. x _— at|p]+pa’0| pat] .a!” 44 os Freer CABAL al ea > _ a'Lp]-+parB[p-1) a!” 42 Par 8[p-2]. a!” + Be. &e. and hence + 4 3" + &e. = natl p] +pa?*0] p21] [1] +P w*"{ p-2] [2] +8 5. If we put the two values found for &/-+ 2// + 2/’&c. equal to one another, we obtain a+b|’ [p|] + S1 opt oe pa) e2e Sar} [p=2] [2] + &e. + N 90 and hence S1 = [na?-(a+b)| [p] + pa? b [p—1] [1+ BP—Te-382 [p—2} [2] oe See [3] 6. If in this expression for $1, we combine the first and the last terms, the second and the last but one, and generally every two terms, of which one is as distant from the first, as the other is from the last, and at the same time keep in mind that [0] = a°+a/°+4a/+ &e. =m, we then obtain S1= (n(@ +5) — (@ + bY) [p] + p (ab + abt) [p—1] [1] + BeP—Nar-2 4 ab) [p—2] [2] + PePX PF 8 4 a6) [p—3] [5] + &e. The last term of the expression, when p is an even num- ber, 1s S s y | fe hore. Lascl But if p be an odd number, then the last term is Pa 9 Bo. 7 Tear, ia): | (F] i lg ame Pp-p—tl.. 7, If in the expression for §1, we substitute for p, | OF 2p, 3p, 4p, &e. successively, we obtain the values of the expressions $2, $3, S'4, &c. and the substitution of these values in the formule in 3, § 45, gives the values of the assumed coefficients 4,’ B’, C’, &c. SECTION XLVIII. Pros. From the given equation a” — Ax" 1+ Br" ?— Cr" 3+ &e. = 0 find the equation for the function (ax’ + bv’’)~*, when p is a whole positive number. Solution. From the equation found in the preceding § for t, which has the roots (av’+b2”’)’, (ax”+br’), (ax’+ bx’), &c. another may be derived, which has (§ 10) the reciprocal roots (ax’+bx”)*, (ax” + bx’), (ar/ + be’), &e. and this will be the equation which is here sought. SECTION XLIXx, From the foregoing problems it is sufficiently seen, what must be done, in order to find the equation on which a given function of the roots of an equation depends. By these, then, we arrive merely at the two following points : 1. To find all the possible values of which the given function 1s capable. 2. From these values to form the required equation. I shall begin with the first. In order to find all the possible values of a function, 92 we must transpose its roots in as many different ways as possible with the other roots of the given equation, and with each other; and of all the results or values thus obtained, we only retain those, which are actually different from one another. If all the roots of a given equation are in a function, it is only necessary to permute these roots in all possible ways. Consequently, if the given equation be of the nth degree, then a function of this kind gencrally contains 1.2.3... values, because x things can be permuted this number of times. But if the form of the function is such, that more permutations than one generate equal results, then the number of the values is often less; and if all the values be equal, then the function is symme- trical. If in the function there is only a number pu of the roots of the given equation, then these m roots enter into the function in as many different ways as there are com- binations in z things taken «x andj; and the number of — these combinations Erte g! Jou EL mane ona cone SE he PLott fe we 1 ° 2 . 3 eeeseeCeCeeetgeoeeettenaeeee ft Every such combination, however, allows of 1.2.3... — permutations of the roots it contains; consequently the — number of the values which such a function generally contains is =N.N— 1. NK eZ .ee N— Nt i .*. itis equal tothe number of the variations of m things — taken. andy. But if amongst these there are equal values, then this number will often be much less; although in ie Eo a Sag 93 the assumed case they never can be less than , because the number of variations never can be less than the number of elements. Consequently the transformed equation in this case can never be of a lower degree than the given one itself. In the general inquiry respecting functions, it is always allowable to assume, that all the roots of the given equa- tion are contained in them, because in the contrary case, only each of the roots which are wanting, considered with a coefficient=o, can be added to the function. SECTION Le EXPLANATION 1. Functions are said to be homo- geneous, when they contain the same roots, and when in all the transpositions of these roots, they either at the same time change, or remain the same. Let, for instance, the functions a! a al al”, al al all” and likewise the functions al’ al! a’ =, +. — +—, glbyliyl"" ue allyl ty!" + al? ylty!"" x x x be homogeneous. ‘Then the first two have no more than the following 6 different and corresponding values : a tall — al! — al", xa!’ — a!'x!” a’ + all’ at. al! wt till al yl’ Nee allalY vl teal”! — oll all, all” — alla!!! vl gl? al — al, a lal” — aol! ag’ “| af” — v A gh ll ala!” en a/yl" IO el alle iad rial «el All, ane a / 94 and the two last no more than 2, viz. / alll “i f / fi & yi + oy ‘ 4. 7) 5 elPyl yl" te 7, MP yl 1/4441 + all lP alta f i as ae TT a tlle SE AeA Ll hs ied Z 2. The letter f prefixed to the roots of an equation, or even to other magnitudes, in the sequel, always denotes a rational function of these roots or magnitudes. Thus f: (a’) denotes a rational function of. a’, f: (2) (x’’) a rational function of x and 2”, and in general f: (2) (a) (a) (#’) «2.2. (@) a rational function of xl, al, a’, ax/”,......0, and so in like manner of other . magnitudes. In order to distinguish the functions, some- times also the letters #’, @, ~ are made use of instead of f. In this notation of the functions, it is preferable with each set of brackets ( ), which follows the letter f, to attach a definite representation of the manner in which the magnitudes contained in it are combined with the others; so that when any permutation of these magni- tudes under the symbol f is intended, one of the permu- tations corresponding to it in the expression represented by it, must be denoted. Thus, if f: (4) @”) (@”%) = @’e" — 2”) (a —2), we then have sis (v’) (x) (v’) = (0’e! — 2) (a! — x’) f: (e”) (2) (2!) = (aa! x") (a — 2’) fi: (’) (2!) (2) = (ala! — x’) (c/” x) fs (a) @’) (c) “ (a4! —a/’) (e/—2'”) if: (c’”) (ve) (2) = (aa —2’) (a — a!) Sit ie ee ee ee f ‘ ‘ Oar Ce Ave snes. iuillen a ree er ee te § dd i‘? - fy ¥ iL a 95 3. In order to distinguish the values which a given function contains by the permutation of its magnitudes, from the symbols by which this permutation is denoted, I shall call the latter types. Thus, the function f: (v’) (2!) (x/) has six types, viz. : f: (2’) (e) (2), fi: @&) (we) a”), £22”) Q) 0), FO”) OD CD, F 2:4) ©); F2@” @%)@)3-and generally there are always as many types of a function as permuta- tions of the magnitudes under the functional symbol. The types are, as it were, the representatives of the values which a function contains, and in general inves- tigations of functions may be used with very great advantage. If, for instance, we wished to show that any particular function had such a form, that the values arising from these or those transpositions were equal to one another, instead of actually expressing these permutations, which would often be attended with a great deal of trouble, it is only necessary to give the types, which correspond to the equal values. SECTION LI. When a function has such a form, that any two of its values are equal to one another, then the function must always necessarily have more than two equal values. Thus, if the function be such, that Fe) @) = fi @) then also must FQ) @) =f) Fs (0) (2+) (0) = fe”) (9 = w 4 96 For the first equation shows, that the value of the type ft: (a’) (a/) (2) remains unchanged, when the roots of the two last sets of brackets are transformed, .*. the types f 2 (0) @YV QQ”), f: (2) (#*) @4 m the same transformation of the brackets, must remain unchanged, because the equality, in the sense in which it is here taken, by which is meant no more than identity, 1s not derived from the numerical nature of the roots, but merely from the nature of their combination, consequently from the form of the functions. SECTION LILI. Auxiliary Rule. If we combine a series of elements Fis DR AS ae p, with as many numbers 1, 2, 3, 4, Sake 7, arranged in any order according to a cipher, for instance, as follows: NT LO AE REN RAK abcdef ght....—p after this, permute the elements in the manner denoted by the ciphers placed over them, and from the permutation thus obtained, derive another, from this again another, and generally from every permutation last found, derive a new one, always observing in the transposition the law denoted by the ciphers: now I affirm, that by continual permutation, we must necessarily return again to the first permutation. 583214796 Thus we obtain from the permutation 4,=«edefg ht for the figures placed oyer it no more than 9 permutations | A., A;, A, As, Ag, Ay) As, 40, Ag ' ; | 583214796 A, =abedefghi Ady A, =ehc bad g OH & iss onchaty 4, =aichebg fd CD he A, =efctah gdb A, =adcefe tobh A; =ebcdafg hi 4, =ahcbedgif A, =eichab gs fd 4, =a fecieh gdb PS og tg bh If we proceed with the last permutation in the same manner as we did with the preceding ones, we again obtain the first. Proof. Let LES Ose ap le de EI py RN erated sritep ann ees denote the permutations which may successively be de- rived from the expression 4,, = abcd Cha viersy pp according to the law of any one cipher. Since the number of transpositions, which an expression can generally contain is always limited, we must .-. necessarily once come to a permutation 4,, which is equal to one of the preceding 4,. But if 4, = A,, consequently also 4, = 4,,_,; for if the permutations A,_,, A,_,, were not equal to one another, then also the permutations 4,, 4,, could not be equal, since A arises from 4,_, by the same transformation of the elements as A,, arises from 4, ,. In the same way we may further conclude from A,_, = A,_,, that also A,_. = A,_2» and hence again, that 4,_;= 4, and soon. Consequently | © 98 A, (4-1) Ayu) = — A,. We have .*. a permutation A,_¢,-1) which is equal to the first. Q. E. D. The contents of all the permutations obtained accord- ing toa given rule of transposition, are, for the sake of shortening the expression, called a period, because we always obtain the same permutations again, however long we continue the transposition. SECTION LITII. From the foregoing § we deduce the following propo- sitions. 1.—That all the permutations of a period are different. from one another. For if in the period 4,, 4,, 43, 4s +++: Ais ace oe . A, there are two equal permutations A,,, A,, then also must 4,.,= 4,1, 4). = A,» and so on; con- | sequently also 4,_q-1) = 4uwen = 413 but in this case A, could not be the last permutation of the period. | II.—Let B denote any permutation different from Aa 7 | now it may belong to the period 4,, 7 bas Mle a A, or | not; further, let B,, B;, B,, Xe. be the permutations derived from B,, according to the same rule of transpo- sition by which 4,, 4, A,, &c. was derived from 4,3 I affirm, that in this case the two periods arising fod A, and B, consist of the same number of permutations. For since the rule of transposition, which is denoted by the figures, does not refer to the elements themselves, but only to their places, ‘so it 1s quite the same, in re- . ference to the number of the permutations of which a 99 period consists, which elements are in the different places of the first permutation. Iif.—If B, is equal to any one of the permutations Vo BA a, Cop pee A,, of the first period, then the two periods consist of the same permutations. This proposition is an immediate consequence of the foregoing §. IV.—If B, be not equal to any of the permutations A), A,, A;, ...... A, of the first period, then the per- mutations of the two periods are all different from one another. For if in the period B,, B,, By ...++ Ee eed A there be any permutation B,, which is equal to a permuta- tion 4,, of the period 4,, 4,, A; ...... vert ac Aten likewise must B,,, = 4,,,,, because B,,,, is derived from B, by the same rule as 4,,, is from 4,3; and when we further conclude in this way that B,,. = Ayia Bus = A,+3, and so on, lastly Byy.: = A,iur+1. But since tee = 12 £3 2 Bi ed en Pp abGeek then must da ire bp which is contrary to the supposition that 6, 1is different from all the permutations contained in the first period. SECTION LIV. Pros. Let a function be such, that any two given | types are equal to one another : find all the equal values | of the function which arise from this supposition (§ 51). | } Solution. For the sake of perspicuity, I shall confine a ee ee ee 100 myself to a single case, because it will be sufficiently clear from it how we are to proceed in every other one. 1. Let F229) @) A”) 7) @) denote any function for which the two types As hess Ss Ca ae) Ce) ar) Acct sssgre DCG tes ire er) are equal to one another. Compare these types, and observe how the roots are transposed in the first, second, third, fourth, and fifth brackets, when J, is de- duced from 4,. If we retain this transposition im our memory, and then derive from 4, a new type by the same rule by which 4, was generated from 4,, from this last derive another, and continue this proceeding till we return again to the first, we shall obtain the following period, consisting of five types : | A Bees aa uss AY CMD CA Cadi Fe pp ice i ese CHO CAG COR CA G8 Agnes Gy RL CA) Moa Ail EO TU CREE EG eer ee of SAO EEO ELA CAO TOSS EASY: Tene gy the) AOn ONE ay hee! ) 2. These types must necessarily be all equal, because they have all been derived from one another by the same_| rule.. Now, since the function f: («’) (@) (a/”) (a’’) (v") has exactly as many types as there are transpo-— sitions in five magnitudes, consequently 120 types, it only remains that we take from the remaining 115 those which, under the hypothesis that 4, = .4,, are equal to one another. 101 3. But this can be done very easily. For we only require to take away from the remaining types any one whatever, and from this derive a second period by the same rule, then again from the 110 types which still remain take away another, and by the same rule form a third period, and continue this till there are none remaining. 4. In this way we obtain 24 periods, each of which consists of five equal types. But if these be already found, then also the equal values, as soon as they are known, of the function itself corresponding to them, can be found. ExaMPLe. Suppose we have observed, that the function al lll 3 4 gl ylV 2G V3 4. gM gl 4 gl gll0q V3 4 g/V aq VB /2 remains the same, when in the terms in which the roots a, xl, a’, «/”, x” are found, we put the roots 2”, 2/”, x”, x’, a’/ respectively, or more briefly f: (a) (a) (a) Cie arty f= Ce) a) (ae) (2), (a), fSince (this is exactly the equation which was used in the solution for the illustration of the operation, it is .*. certain, that the function has 24 times five equal values. Thus the period in 1 contains the five following values : Bae Cat) (ae) Caiasy aay (cat) = a! y//2y//3 + aly! V2y¥3 -f eV y/2y//3 + al zll124/V3 -+- al Vy l2y/3 F2 CC) CC) @ = gl! x1 V2yp¥3 fe eh y!2 74/3 ae all a f/2y V3 4. al Vy V2y/3 ay a ell2ql13 Ff? @Y@)ENENA") = ey! 2y//3 4 yl! yf l/24 V3 L a VpV2y/3 +7 fof 24h 113 ab ala! V2,¥3 frat) a) G09.) = 102 all ylltay1V3 oh oy Vy V2 7/3 SE gl yl /2y/113 “ft gll/ ql Vey¥3 4 yl y/2yl/8 PILI CRA GYEYVS al VV! 4 all 2g lB 4. gl ah VO VS 4 Val 2/3 4 all ylll2epl VB which are evidently all equal. SECTION Lv. Pros. Let there be a function such, that more than two of its types are equal: find the equal values of this function. Solution. Let A, B, C, D, &c. denote the types, which, according to the hypothesis, are equal. In order ~ from hence to find the equal values, proceed as follows. 1. First permute the type A, or even any other, ac- cording to the rule of transposition, that A= B, as was shown in the foregoing §. If the period, which we obtain from this, consist of yx types, then we have at once jp equal types. 2. Then permute each of these yz types in particular by the rule 4=C. I shall assume that there are pi’ types; then we have generally ju’ types, which are all equal. + 3. Permute again each of the ju’ types obtained by the rule d= D, which may give yw” types; then we have in all ph 1 Ti e equal types. 4. In the same way we proceed, when we successively make use of the rules 4= KH, A=F, &. B=C, B=D, &e. C= D, C= E, &e. or, in short, when we put the given 103 types A, B, C, &c. taken two and two in all possible ways, equal to one another. 5. Let the number of types, obtained according to the directions in 1, 2, 3, 4,= ». ‘Take now from all the types, which arise from all the possible transpositions of a’jv,a’/’, &c. any other, which is not amongst those already found, and proceed with this according to the same directions, then we obtain v equal types. If we con- tinue this proceeding, till all the types are exhausted, we at length obtain a number of divisions, each consisting of v equal types. But that the types in every such division are different from the types in all the other divisions, is an immediate consequence of § 53. IV. Corollary. It follows from this solution, that the number of the different values which a function can con- tain, is always a sub-multiple of the number of all the values, which arise from all the transpositions of a’, 2’, a", &e. ExampreE I. Let there be a function such, that Lf (2’) (2/’) (7) aor) Satie (yet) (a//’) (a’) pms: =f: (x) @!”) (&) (0): required to find its equal values. From the equation A= B, or f: (a’) (a”) (a”) (a’") =f: («”) (4 (2’) (2’”), we obtain, in the first yess the period. Fe @) 2%) 0%) @%) =F: @) &” (9 oe =f: (a) (x) (x) (a) 104 From each of these three types we obtain, by the appli- cation of the equation A=C, or f: (x) («”) (2) (a’") = 3 5 oh (a/”) (a/") (2’), a period of four equal values, .*- in all twelve equal types, viz. Wp (2) eat, er) (ee) ye tes, (are) ida (2) ee pa Tr (en) (art) (2’) (a) =) Cagle ( %) fae} Cals —_— Fe (0) 0) @) = fi @) @) OY) (") = Fe 0) @!) (a) = fe @) OH) ©) = Fite) NEN) EAF CTO) Ge) Fe (a) (eC) (0) =F 0) () ) At length we obtain from these twelve types, by means of the:.equation, B= C, orf: (a) (a) (2/) = fi: (0) (&”) (2’") (#4), consequently by the permutation of the roots in the two last brackets, twelve other types, which, together with the former, give all the twenty-four types of the function f: (a’) (a”) (a/”) (2’”). Hence it follows, that a function of the supposed nature must necessarily be symmetrical. Examece. I]. Let a function be such, that f: A)” 0”) 0) =f: 8”) 0) @”) QL!) = yy oe (x’) CL) a) et} = a CAs (a2) (a’) Eta required to find its equal values. The equation 4= 8, or f: (7’) (2) G@”) @’”) = FS: @”) @’) (&”) (2’”) gives no more than these two equal types. If to these we apply the equation 4=C, orf: (a!) Cal”) @") Ca! afr) (W!) (0), we obtain the four following equal types: af (2’) (a’’) Can) (an) =k (ay) (2) (a) ae") = Fo) VO) O) =F: 0) W) @!) @!") = 105 From these we again obtain, by means of the equation A=D, or f: (x) (v) (@”) OM =H (2!) (x!) (v) (v”) the following eight equal types: ii (2) (oe) (ar) (2) ay: (a7) (x/") (a’) Gale —_— die (x’) (x) (ar?) (ae"h) ry (xi®) Geli) (x’) (a/’) — fe 2%) @) 0” @) =f: @”) 0) @) = 7: we) (x’) cath) ae} = Ay (a’”) GH) (ath) ae —_ The equations B=C, B=D, C=D always again give the same types. The type f: (2’) (2”) (2!) (2/%) con- sequently contains no more than seven equal types. If now we take any other type of the remaining six- teen, viz. f: (2) (a’) (a’") (@”) and proceed with it as we previously did with f: (2’) (a”) (a’”) (a/”), we again obtain eight equal types, viz. fi 2) a!) HY) (=f: @”) 2) @”) (0) = of: ir) (a’) ta) C745) = 8, (a76} (at) Cae) (x’) — FOE EIVMN=f: CV CONOCI= Ff: (@) @) (0) (= fs 0” 2!) (2) (0) = If from among the eight remaining types, we select any one, for instance, f: (a) (a/”) (a/") (a’), and proceed with it in the same way, we obtain them all. Hence .°. it follows, that the function of the supposed kind can have no more than three different values, viz. : (a!) (a) (0) 2”), f: 2”) (2) 2!) 2), fF?) (2) (a/") (a), and that consequently such a function will not lead to any equation higher than the third degree. Of this nature are the functions (a/ +a/—a/’—2/")?, ax!’ + 4/%/%, and innumerable others; the method to find which will be given hereafter. But if we wish to use 2 function of this kind to solve equations of the fourth P a 106 degree, it is not sufficient that the transformed equation should be of a lower degree than the given one ; but we must likewise be enabled, from the known values of such a function, to determine the roots 2’, a, w’/’, a/” by equations of a lower degree than the fourth, because otherwise the transformation of the equation will be of no use. In the functions a/v” + 2/2", (a/ +2 —a/”—2'")?*, this is actually the case, as we have already seen in § 41 and 42. In the sequel the conditions will be given, under which it is generally possible, from the known value of a function f: (a’) (2) (a””)......(a) to find the values of the roots x’, 2”, 2’, ... 2 by equations of a lower degree than the mth. SECTION LVI. Pros. Determine the degree of the equation on which a given function depends. Solution 1. If in a function f: (2’) (v”) (@”)...... (ax) there are all the roots of the given equation, and if this function be such, that in each transposition of its roots it changes its value, then the transformed equation is necessarily of the degree 1. 2. 3....... *. If the function be such that a number of types A, B,C, D, &c. are equal, and if the number of the equal types which can in general be derived from them, by the directions given in 1, 2, 3, 4, of the foregoing §, =v, then the number of the different values, of which such a function is capable, or the degree of the trans- formed equation, ‘ . : "i pee! wo Gs and 3 Uf the function remain the same, when m roots change their places in all possible ways, then the degree of the transformed equation A ha . ee entree Itt + I 4. If the function still remain the same, when m/ other roots, and again m’ other roots, and so on, change their places, then the degree of the transformed equation The ime lS 2. inl SC 1 C2 imix &e. 5. If the function be such, that each time its value remains unchanged, when m roots, and again m’ other roots, &c. change their places in all possible ways, and if, besides, a number of types 4, B, C, D, &c. are equal, then the degree of the transformed equation A ea) ereseeo ete OCF ee og e@eneooe @eaoaretoeevec pL 1.2...mx1.2...m’x1.2...m’ x &. xv when v retains the signification 1t does in 2. G. If all the roots of the given equation be not in the function f: (2’) (av) (a/”)...(@™), and if the equation be of the nth degree, then all the formule given in 4, 2, 3, 4, 5, must be multiplied by the factor. Ho — De eas n—ptt a i om 108 The reason of all this is sufficiently evident from what goes before. SECTION LVII. Pros. From the given equation ; gw" — Ax’ + Br’ ?— Cr’? + &e. = 0 find the equation on which the function f: (a) (0” ) (a’”)...(e) depends. Solution 1. Seek first all the different values which this function contains, both by substituting for its roots the other roots of the given equation, and by their permutation. Let these different values be denoted by 9’, y”, y', CARE y. . 2. Then form the equation Cre) Cy) ME) are lr aime and actually multiply the factors in the first part. Then 7 — At 4 Ber? — Cr 4+ &e. = 0 is the equation obtained from this operation, .°. Al = yf! ey! + y!! +y + &e. Bayyl tyy! typ" yy +8 Cc! ay yy te + yyy!” + yyy" - &e. &e. 3. The functions A’, B’, C’, &c. are symmetrical in relation to y’, y”, y'’, .... y™, and consequently no trans- position of these magnitudes can effect any change in the © values of the former functions. But the magnitudes 4’, Ys o's 4.4 are themselves again functions of the roots a’, a’, a’, &c. and such too as merely transform 109 into one another, when the roots are transformed and permuted in every possible way. Consequently no further change takes place by transforming and permuting the roots in the above expressions for 4’, B’, C’, &c. than that y’, y’, y/”, &c. change their places. Now, since this effects no change in the values of 4’, B’, C’, &c. con- sequently these values remain unchanged by the trans- formation and permutation of 2’, x’’, x”, &c. ‘Therefore the coefficients A’, B’, C’, &c. are necessarily symmetrical functions of the roots 2’, a’, a/’, &e. 4. In the two first sections, however, it was shown, that every symmetrical function of the roots of an equa- tion, may always be expressed rationally by the’ coeffi- cients of this equation. ‘Therefore also the coefficients A’, B’, C’, &c. may always be expressed rationally by A, B, C, &e. 5. Consequently an equation may always be found, on which depends a given function of the roots of another equation, and the coefficients of the former will always be rational functions of the coefficients of the latter. (> a0.) [V.—ON ELIMINATION, TOGETHER WITH ITS APPLI- CATION TO THE REDUCTION OF EQUATIONS. SECTION LVIII. Pros. Let there be equations of the first degree given, which contain as many unknown magnitudes : required to reduce their solution to the solution of n — 1 equations of the first degree only, which contain but n—1 of these unknown magnitudes. Solution 1. Let ax + by + cz +..... tkv + lw=A ae+ by + 2 +..... +hv + hw= A, ax + by + ce +..... + hv + lw = A, On U4 Dy YA CASH eee +k, 9+h_w=4,1 be the given equations; 2, y, 2 ...... 0, w, the » un- known magnitudes; a, b,c, ......k, 0; a,, by, ¢, ... ky, l,, 3Uidiep ay las anni es spe xc. ; | likewise (4,0 peels cd the given magnitudes, 2, Assume IT,, II, II;, ...... T,1 as n—1 magnitudes hitherto unknown, and multiply the second equation by I],, the third by I], the fourth by Tl,, and so on; lastly, the last by II,_;; then add all these products to the first equation ; bence arises the equation ie A+ ATT aaa aie +4,,0,,.= (a+ a], + af, + ...... Fi he bises) 2 +(b+ 56,01, + b,1T, + ...... +. bo Hayy +(c+ cf], + eII, + ...... +.'c¢3, ys +( +211, + 40, + ...... + U_, 1,_,) 3. Now we try to determine the factors T],, If, My, ..- iJ,-;, mm such a way, that the coefficients of all the un- known magnitudes 2, Y, 2, 0... v, w, those excepted which we wish to find vanish. If, for instance, we wished to find r, we put b aa TT, + b,[1, + esenos -}- Dati ma hE | Cc + cI, +- cl], + woeere + Coyle ses (a {+ 101, + “&0, + ...... de att Pama 4. Hence the equation in 2 is reduced to the follow- ing one : A+ All, + AI], + ...... + A, = (a+ afl, + aI, + ...... bY ag Fy) @ and this gives "LO ASS ®NAL, eee Sa lad 9 A a+ all, + aJI, + ...... eT! ete fd Pala and the determination of the assumed magnitudes [1,, T1,, ag, as etne II,,, depends upon the solution of the equations in 3. 5. In order .*. to find the value of x, we must solve the 1 equations in 3, from it determine the x—1 mag- 112 nitudes TI,, Iz, ITs, ...++. Ti,_,, and then substitute their values in the expression found for x. G. If both in 3 and 4 we substitute a for b, we shall find y, and in the same manner we find z, when in 3 and A we substitute a for c, and so on. Remarx. The reduction of equations here given, may sometimes be used with advantage, as will be seen by an example given further on in this work. But ifwe intend merely to solve the given equations, we shall by these means attain our object but very slowly ; in this case the ~ method in the following § is preferable. SECTION LIX. Pros. Let the following n equations of the first degree be given : axrctbhy toast... thvtlhw=m, Ant + by y + CyB H weaee. + k,v + lw =m, a; V + by 4- C32 a eecees 4- kv a. Lw — Ms 402+ by + G2 +... +hk,v +10 =m, in which there are x unknown magnitudes a, y, 2, ...... v, w: find the values of these magnitudes directly, and without any substitution or any other calculation. Solution 1. If we merely had the two’ equations with | two unknown magnitudes a2 = by = Mm, Oy Ci bey eee f ee 113 we should have found them in the usual way \ = ‘ Wyk mb, — m,b, ye hittln = asm eS its a,b, ae ab, ty a,b, —T* a.b, 2, If we had the three equations with the three un- known magnitudes ar +by+ ¢2 = m, av + by + og = m, asx + by + cz = m, we then should find m,b,c, — m,b,c¢, — m.b,c,; — m.b,c, + m,b,c, — m,b,c, a,b,c, — a,b,c, —a,b,c, + a,b,c, + a,b,c. —a,b.c, a,m,c; — a,MsC, — am,C, + A,m,C, + G,M,C, — a4,M,C, a,b,c; — a,b,c, — a.bycz + ayb3c, + a,b,c. — a3b,c, a,b,m;— a,bym, — a,b,m, + a,b,5m, + a;b\m, — ajb,m, a,b,c, — a,b,c. — a,b,c; + a,b,c, + asb,c, — a,b,c, 4 i. cx) 3. From the formule in 1 and 2, the rules for the solution of the above general equations follow by induc- tion. In order to abbreviate them, I shall call the num- bers which are affixed to the letters m, a, b, &c. symbo- lical numbers. (a) Take the product a,b, c,...... k,1l,; then per- mute the symbolical numbers in all possible ways, while the letters themselves are not changed ; the aggregate of all these 1.2.3..... . n products, then gives the common denominator in the values of (5) In order to find the signs of every one of the terms, Q ~~ er ep, 114 of which the denominator consists, try how often in such a term a lower symbolical number follows a higher one, mediately or immediately. If the num- ber of these successions be even or O, then the sign of the term is +; if it be odd, the sign is (c) If the common denominator be found, we obtain from it the numerator in the value of 2, merely by substituting m for a; the numerator in the value of y, by substituting m for b; the numerator in the value of z, by substituting m for c; and so in like manner with the other unknown magnitudes. Thus the denominator in the values of x, y, z, is merely the product a, b,c,, with the 1 . 2 . 3 permutations of the symbolical numbers ; and with respect to the signs, if, for instance, the term a,b;c,, has the sign +, because it contains two successions of a lower symbolical number to a higher, viz. 21, 31; but in the term a,b, c,, there are three such successions, viz. 32, 31, 21, and this term con- sequently has the sign —. Likewise the numerators are formed in the manner given in (c). Exame.e. From the four equations av+ by +e2+ du =m, au + by + 2 + du = m, asx + byt c2 + dju = m, av + by + gz + du = m, we obtain for the common denominator in the values of I, Y, 2, u, the following expression : ea * a - 9 7 | | | a,b,c,d, — a,b,c,d,; — a,b;c,d, + a,b,c,d, a,b,c,d; — a,b,c,d, — a,b\csd, + a,b,c,d, a,b,c,d, — a,b,c,d, — a.b,c,d; + ayb,cd, a,b,c,d, — a,b,c,d, — asb,c,d, + a,b,c,d, asb.c,d, — a,b,c.d, — a,b,c.d; + a,b,c,d, a,b,c,d; — a,b,c,d, — a,b,c,d, + a,b3c,d, +++ + SECTION LX. Since the values of the unknown magnitudes in the solution of the foregoing § always appear in the form of fractions, it may sometimes happen, that the common denominator = 0, as, for instance, in the two equations a,b,—a,b,=0, and the three equations a,b,c; — a,b,c, — a,b,c, + a,b,c, + a;b,c, — a,b,c, = 0. In this case, if also the numerator = 0, then we arrive at expres- 4 0 ee sions of the form—.. Such a form as this merely mdicates, that the conditions given in the equations are not such, that the values of the unknown magnitudes can be deter- mined by their means alone. ‘Thus, if we had the two equations 3x + 5y = 16, 6x + 10y = 32, then would a, = 3,b, = 5, a, = 6, b, = 10, m, = 16, m, = 32, and consequently from the formule in 1 of the foregoing §, 16.10 — 32.5 0 3.32 —6.16_0 Bsr 1O Ola M02": 7.38, 1052 6 Sao the values of 2 and y .*. would remain undetermined. ’ But we immediately see why they must continue to be undetermined. For if we divide the second equation by 2, we obtain the first; consequently the latter is contained in the former, and we have .*. in fact no more than one equation, from which neither x nor y can be determined. Panama 116 But if the given equations be such, that the denomi- nator vanishes in the value of an unknown magnitude, but not the numerator, consequently that we arrive at ° a . an expression of the form —= © ; then this result always 0 indicates, that the relations expressed by the equation are contradictory, and cannot obtain at the same time, while the unknown magnitudes, as is always assumed here, have only finite values. Thus, suppose we have the two equations 3v + 5y = 16, 6v + 10y = 20, we obtain shane 60 — 36 from 1 of the foregoing §, x = —, y = ——3 __ conse- 0 0 quently, as we are convinced that there can be no other values but these, there must be contradictory relations in the given equations. This, indeed, is really the case ; for if we multiply the first equation by 2, we get 6x + 10y = 32, whereas, from the second equation 67+ 10y= 20. SECTION LXI. The problems, § LVIII. and LXIX. contain all that relates to the elimination of equations of the first degree. I shall now direct my attention to the elimination of equations of higher degrees ; and I shall first assume, that there are no more than two equations given with two, or even more unknown magnitudes. Here two cases must be considered: first, when the first equation, in reference to the magnitudes to be eliminated, is of the first degree, and the second of a higher; secondly, when both equations are of higher degrees. There is no difficulty in the first case; for we only 117 require to find from the first equation the value of the magnitude to be eliminated, and substitute this value in the second, when we obtain an equation, in which this magnitude does not occur. In the second case, by multiplying by proper factors, and by the requisite combination of the results thus ob- tained, we always try to reduce the degree of the equations in reference to the magnitude to be eliminated, till we come to an equation, which contains only the first power of this magnitude. If from this equation we find the value of the magnitude to be eliminated, and substitute it in that equation in which it occurs in the lowest power, we shall obtain the required final equation. The following problems will sufficiently elucidate the foregoing. SECTION LXII. Pros. Let p, q, 7, p’, 9/57”; be functions of y ; further, let the two equations Lp gE Pras LE po vq’: “e230 _be given between x and y: find the equation, which arises | from the elimination of wv. Solution 1. Multiply the first equation by p’, the second by p, then subtract the results thus obtained from one another, and divide by w ; hence arises the equation PY — Pq + (pr’—p'r) x = 0 and this gives Bean Toe Pg = 74, pr — pr 118 2, Further, multiply the first equation by 7”, and the second by r, and subtract ; then we have pr — pr + (qr —/r)x =0 If in this equation we substitute for x its value obtained from 1, we get the equation Ch) «-. (pr’—p’r? + (p’'g—py’) Gr’ -q‘r) = 0 which only contains y, and which consequently is the final equation sought. 3. If we had immediately substituted the value of x from 1, in one of the given equations, for instance in I, we should have found GAP TPT) PI Pt Peek pr — pr =F (pr — pry o and if we multiply by (pr’ —p’r)*, and then divide by p, we get the same equation as in 2. SECTION LXIII. Pros. From the two equations Lip gre ar 16 ID. p’ + g'x + 7/2? + 8/2? = o eliminate 2, supposing that p, q, 7, p’, 4, 7’, 8’, ave such expressions as do not contain wv. Solution 1. Multiply the first equation by p’, the second by p, and subtract the results, and we get the equation py — p'g + (pr! — p’r) x + psa? = 0 2. If we combine the equation I. with this one, the 119 case in the preceding § enters here; only that pq’—p/q, pr’ —p’r, ps’, are here what p’, g’, 7’, were in the former. We only require .*. to.substitute the former values for the latter in the equation (w) of the foregoing §. If this be actually done, we obtain the equation (ps! + qrp’—prq’)’ + (pgs'— pre’ + rp’) x (pq7 — Fp’ — pr’ + prp’) = 0 3. If we solve this equation, and then divide by p, we obtain p's? + prr? + prq? + rp? — grp'd + C(P—2pr) (rp’r’ + pqs’) + (3pqr—@ p's! — pag’ —p'qr’s’ =0 an equation which does not contain x. SECTION LXIV. Pros. Again, let p, 9,7, s, p’, 9’, 7,’ 8’, be functions which do not contain w: find the result of the elimination of x from the two equations I. p + qv + ra? + sx II. p? + gr tere? +. 9/2? = 0 | i=) Solution 1. Multiply the first equation by p’, the second by p, and subtract the results; after dividing by x, this gives PY — gp’ + (pr’ — rp’) v + ( ps’ — sp’) x = 0 2. Further, multiply the first equation by s’, the second ‘by s, and again subtract ; this gives sp’ — ps’ + (sq! — qs’) x + (sr — rs‘) a? = 0. — 120 3. It is not necessary to continue the reduction further ; for since the equations found in 1 and 2 are similar to the equations I and II, in § 62, for which the result — of the elimination was there found, it is only necessary in the equation (i) of that section, to make the follow- ing substitutions : py —qW' for p, sp'—ps! for p’ pr’ —rp’ for g, sq’— qs’ for gq’ ps’ — sp! for r, sr’ — rs! for 1’ 4. By this substitution we obtain, after the proper solution, (pq! — gp’)? (sr! — rs/P — 2(pq/ — gp’) (ps' — sp’) 7 (sp! — ps’) (sri — rs’) + Cps’ — sp’)? (sp’ — ps’? : + (pr! — rp)? (sp — pst) (sr! — 78’) -— Cog’ — apie (pr? — rp’) (sq — 9s) (sr! — rs’) — (pr = rp (ps’ — sp’) (sp’ — ps’) (sq’ — 9s’) + (pq — ope (pe! — ROG ce qs teeta 5. The first part of this equation consists of seven — terms, of which five are divisible by sp’ — ps’. The — other two, viz. the first and fifth, taken together, give (pq — gp’) (sr — rs’) x [(pq! — gp’) (sr! — rs") — (pr! — rp’) (sq! — qs’)] | = (pq! — gp") (sr — rs’) (pqr’s! + rsp/q'—prq's! —qsp'r") © = (pq — 9p’) (sr — rs!) (sp'—ps’) (rq —qr"’) and consequently the sum of these two terms is also Y divisible by sp/—- ps’. 6. If .:. the equation in 4 be divided by sp/—ps’, we 7 at last obtain the equation 121 (pq — gp’) (sr! — rs’) (rq! — qr’) + 2 (pq! — 9p’) (sp’ — ps’) (sr’ — rs’)-+(sp’ — ps')?-+(pr’ — rp’)? (sr’ — rs’) + (pr’ — rp’) (sp’ — ps’) oe — gs’)— (py — gp’) (sq — 9s’)? = 0 SECTION LXV. Pros. From the two equations I. pt+aqrv+ ra? + sx? + tr4+=0 Il. p+ 7/etre’?t+ c+ tet =o. eliminate the magnitude x. Solution 1. Multiply the first equation by p’, the second by p, and subtract ; after dividing by «a, this gives py —gp! + Cpr’ — rp’) x + (ps! —sp') a + (pl/—tp’) # =o 2. Further, multiply the first equation by ¢’, the second by ¢, and again subtract ; this gives t/—tp! + (qt’—td’) v + (rt —tr’) x° pt'—tp’ + (qv —tq + (st’—ts’) a3 = 0 3. Since the equations in 1 and 2 are both of the third _ degree, in order to save the trouble of carrying on the operation, we need only use immediately the equation found in 6 of the foregoing §, when we make the fol oyaDg substitutions in it : py — gp’ for p, we — tp’ for p’ pr’ — rp’ for q, gt! — tq! for q/ ps’ — sp’ for r, rt’ — tr’ for x’ pt’ — tp’ for s, st’ — ts’ for s/ R 122 and the result of this substitution is the final equation sought. SECTION LXVI. Pros. From the two general equations UO ie a 2 ee ce OCH a ee + vr"= 0 TI. pl od’ vee 4 ka bia 0 eliminate 2. Soluiion 1. I shall assume, that m Ul [a//2— 9 (x 4. x!) a x/x/| aye 134 or, since x’-- x" = 14’, x’x’= B’ (v?—a! A’ + B’) (a!/2—2!' A! — B’) (2/2 —a/"A’ + BY) = 0 2. The actual multiplication of the three factors in the first part of this equation, gives (1°)?—(122).A/ + (2°) B’ + (122) A? — (12) AB’ + (2) B? — (15) 42 + (12) 42B! — (1) AB? + B® = 0 or, when for the numerical expressions their values are substituted from the annexed tables, C°— BCA’ + (B°—2AC)B'+ ACA?—(AB—30) AB’ + (42 — 2B) B?— CA® + BAPB — ANB? + BY = 0 which is the final equation sought. SECTION LXXV. Pros. From the given equations I. 2% — Ac” 4+ Ba”? — Cr” + &. = 0 II. a’ — Ae’ 4 Ba"? — Cr" 4+ &. = 0 eliminate x, by the method in § LX XII. Solution 1. Since x’, x”, x”, &e. are the roots of the -equation II, then (v—x’) (v1—x”) (1—x”).........= x” — Ala? + Bx’? — Cx? + &e. If in this for x we substitute a’, 2”, x’, &c, successively, then (2’—x’) (a’—x”) (a/ — x!) &e.= x!” — Ae" + &e. ( a —x/ ) ( al’ ue x” ) Cxi/ ‘rs x/// ) & c= x A’ glial + &e. | (yf — x/ ) ( gl yXl/ ) (all! xl! ) &e,e xl" — A! yl/"") ae &e. | 135 2. If these values be substituted in the equation (xp), § 72, the latter is transformed into (x _— Ay! 4 B'y’"-2 aad C'7""-3 -t. &c.) x (a! — AlylV ag. Blyl2 — Clx!*-3. 4 &e.) x (al — Aly Blyl-2— Clg!" &e.) a x &e. The first part of this equation is no other than the pro- duct of all the expressions, which arise from substituting in the equation II for x all the roots 2’, 2”, 2’, &. successively of the equation I. 3. But we immediately see, that the above product undergoes no change by any transformation of the roots x’, «!’, «/’, &c. as in a transformation of this kind one factor is merely changed into another. ‘The first part of the equation is .*. necessarily a symmetrical function of the above roots, which consequently may always be omitted according to known rules. In this way we _ obtain an equation, which no longer contains x, and which .*. is the final.equation sought. From the opera- | tion itself, it follows besides, that it is complete, and contains nothing extraneous. Remark. The problem from two given equations with two unknown magnitudes to eliminate one of these magni- _ tudes, is consequently now solved in its most general form. The actual calculation involves many difficulties, and amongst these chiefly are the solution of the product in the first part of the equation in 2, and its reduction to - numerical expressions. How these difficulties may be 136 removed by means of combinations, will be shown in the following §. SECTION LXXVI. Pros. Represent directly the result of the elimination of xv from the two equations I and II of the foregoing §, fully solved. Solution 1. To the equation II of the foregoing §, we can always, by dividing by its last term, give the following form : 1+(1)v+(2)x? +(3)a3+......+(2) x" = 0 in which the coefficients (1), (2), (3), «++ (n) denote given functions of y. This notation was chosen merely in order to facilitate the application of the combination- operations, and to make the law of the terms more _ evident. In order further to show, that two or more such coefficients are to be multiplied together, I shall merely put the numbers representing this operation in brackets near each other, and in these make use of repeating exponents. Thus .*. (123), (2456), (1°27), the first of these denotes the product of the coefficients (1), (2), (3); the second, the product of the coefficients (2), (4), (5), (6), and the third, the product of the third power of the coefficient (1) and the second power of the coefficient (2). 2. Put, as was done in the foregoing §, 2’, a”, x”, &e. successively for x, consequently the first part of the equation in 2 of that §, has the following form : 137 [1+(i)a’ + (2)? + (8)a8 + (4) + (5) 05 4-&e.] x [1+ (1)a” + (2) a’? + (3) a4 4 (40 + (5) 2’ + &e. | x [1 +(1)a!”! + (2)a/?2 + (8)a/3 + (4) a4 +-(5)a*+ &e. | &e. 7 The number of factors here is equal to the degree of the equation I, .-. = m. 3. First take the product of the two first factors, we then obtain V+ GH) + FP tx) + (BaP +ai%) a3 (1’) (er®) aE (1 2)(a/x’? + Dany a5 (4) (a + a4) 6 (5) (a? +2’) a &e. +- Qa 3) (x/2/8 x3 y/’) ap (14) (a/a!* + a4") 4. (2?) (artr/2) we (23) lah ltete bae ke at 4. Hence, if the equation I were of the second degree only, then this product must be represented by numerical expressions, as follow: 1+ (1)[1]+ (2) [2] + (3) [3] + (4 [43 + (5)[5] 4+ &e. + (17)[12] + (12)[12]+ (13)[13](4+ (14) [14] + (2%) [2°] + (23)[23] 5. Now if we multiply the product in 3 by the third | factor in 2, we then obtain, when the terms are pro- '_perly arranged, the product 1+(1)(a! $a 4a”) 4 (2)(a? 40/2 4.0/2) (12) (a0! ala! allel) + (3) (0/3 42/34 9/8) fopl!2 4. lB ull A. yl alli? + (12) eons i i) + (13) (ems 138 + (4) (als a!/4 4 g//4) al al be a/3yl/ 4 gl gl3 4 15 (13) sy alg l//3 " thy a (2?) ( al 2a lS Ae apf 2glI/2 Ae ap! /2yl1/2) a (122) (a! 2!/a!!? me Ae aga l’) 2 ( 5) (a’5 4 x5 4. all!) im (14) ea //4 4. alsa!’ 4- a! yl l4 ats : pl Bal3 he glBy/12 4 yl 2yll3. 4 + (23) erin A. gl By l13 4. al ie) 4 (123) (ala! a! + a! x! yl! 4 x!" l//) (122) (aa! 2a!!2 4/2/42 4 a2 pl /2y!/) &c. 6. If .*. the equation I be only of the third degree, © then the product in 2 may be represented as follows : | 1+()E114 2) [2] + (8) [8] + [4] +5] +80. +(1[] + (12) [121+ (13)[13] + (14) [14] + (1°) [15] + (22) [27] + (23) [23] + (122)[ 122] + (123) [123] © + (12°)[12°] 5 %. It is not necessary to continue the multiplication — further, as we may very easily perceive the law from the ¢ products already found. Thus we see, immediately, that the figures in the parentheses and brackets are always the t same, and compounded in the same way. The included numerical expressions merely denote this, ; that they are all the possible numerical divisors for sey” Hy “ I ae a ; 2 = oat ~~ oe 1. a) eo aad > a ae 7 > { fe Me j y tigen gee yr nd c ” 139 the numbers 1, 2, 3, 4, ...... mn. I say all the possible numerical divisions; because some vanish, as, for ex- ample, in the products in 4 and 6; this merely proceeds from this, that the numerical expressions for such divisions in the assumed degree of the equation I, are not possible, because more roots are required for their formation than this equation can possibly have. Gene- rally, when we have to do with particular equations, all those divisions vanish, for which either the numerical expressions, or the products of the coefficients do not obtain. 8. How all the possible divisions of the numbers may be easily found, without the chance of omitting one, may be seen by the combination-analysis. In order .*. to find the elimination of x from the two equations, I. «”—Ar™ + Br®? + Cr” + &.=0 Il. 14+(1)r+(2)a°+(3)a?4+ ... 4+ (n)2"=0 we must observe the following rules: (a) Analyse the numbers 1, 2, 3, 4, ...... mm in all possible ways. (b) From every such analysis a@y6... make a term of the form (ay0...) [aByo...]. (c) Then, if the sum of all these terms is repre- sented by S', consequently 1+S5=0 is the final equation sought. = i T 2 140 9. All the numerical expressions relate to the equation I, and may partly be taken from the annexed tables, and partly may be ealeulated by the methods given in the two first sections. Nothing now remains but to elucidate this operation by a few examples. SECTION LXXVII. Examp.e 1. Find the result of the elimination of x from the two equations I 2f?—Ar+B=0 II. Qxv> + rt + C2? + Di? + Er + F = O. First give the equation II the form 1 +(1)v+ (2)a? + (3) 2° + (4) 24 + (5)a°; consequently put (1) = va D ¢ 6 a ; (2) = ¥ (3) = F (4) = FP (5) = re Since the equation I is here only of the second degree, the divisions need not be continued further than the second class, because the numerical expressions for higher classes do not obtain (§ LXXVI, VII). The final equation .'. has the following form : O0=1+(1)[1]+(2)[2] +(3) [3] + (4) [4] + (5) [5] +(1°)[ 17] 4+ (12)[12]+ (13)[13] + (14)[14] + (2°) [2°] + (23) [23] + (15)[15]+ (25)|25] + (35)[35]+ (45)[45] + (5°) [57] + (24)[24] + (34)[34]+ (4°)[4°] + (3°) [37] Or, when we take the numerical expressions from the tables, for the numbers (1), (2), (3), (4), (5), substitute their values, and then multiply the whole equation by F°, 14] O=f* + €FA + DF(4°—2B) + CF(A?—3AB) +@¢°B + DEAB +BF(A*—44°B+2B*) + AF(4°—54B4+ 5AB*) +€€(4°B—2B*) + B€(4°B-3AB*) +D°B + DAB +9€(A4*B—44°B* + 2B) + AD C4°.B* — 3 AB?) +BD(A*b? —2 B?) + BC AB +@°B° +AC(A*B*—2B)498 AB + A? Be +%°B Examp.e II. Let the two equations Wee tea Ao Peat —n CO Il. Q@x? + %r* + Cr + D=O0 be given. If we reduce the equation II to the form ] + (1)xr + (2)x? + (3)r3; then (1) = = oe a, C= Since in this case the equation I is of the third degree, we do not require to continue the divisions beyond the third class. The final equation .-. has the form 0=1 + (11+ (2)[2]+(9)13]+ C318] + 28) 125] + (1°)[1°}+ (12)[12] + (2°)[2*] + (1%3)[1°3] + (2°)[ 1°] + (1%2)[1%2] + (12*)[127] + (3*) [3°] + (13°)[137] + (23°)[23°] + (3°)/3)] + (123)[123] + (2°3)[2°3] + (2°) [2°] or, when we substitute for the numerical expressions and for the numbers (1), (2), (3), their values, and then multiply by D® 142 0=D)+¢D°A + BD(42—2B)+8D2(4~34B+43C) +C°DB 4+ BED (AB—38C) +4¢D(A°B—2 BP— AC)+abD( A B*—2 A?C— BC) +B°D (B*—2AC) + 40° (4°C—2BC) +B6C°AC +%6°CBC +94°p(.BS—3ABC+4+3C*) + 80C(B°C—2 AC*) +ABC(ABC—s8C*) + 25° _4 C? + 163 C? + 4° BC? + A°C? | In order to show the use of this formula in a particular ease, I shall assume that the two equations 1 —2ax* + 4ayx—y? = 0 ax* + y*x—ay*?=0 are given. If we equate these with the equations I, II, we find A=2a, B= 4ay, C—7, A=0,% =<, Cy D=—ay’. Since here A=0, the foregoing equation is reduced to 0=D°+€D°4+ BD*( A*— 2B) 4+ BED(AB—30C) +C°DB +@°C +p (B?—2AC) + BCBC + BCP +6°CAC and this equation obtains for every case, in which one of the given equations is of the third, and the other of the second degree. If in these we make the requisite sub- stitutions, we obtain the required final equation y + ay’ + Cay’ — 12a4y’ — 12a°y4 = 0 or y + avy + Cay? — 12a'y — 120° = 0. 143 SECTION LXXVIII. The method to find the final equation by means of the symmetrical functions, originated with Euler, who first made use of it in the Memoirs of the Berlin Academy for the year 1748, and applied it to a few easy examples. A short time after this, Cramer, in the second Appendix to his ‘* Introduction 4 |’ Analyse des Lignes Courbes,” p. 660, &c. 1750, by a suitable method of notation (which I have partly adopted), made it more general, and the operation more easy. Both these great men chiefly en- deavoured to prove by it, that two lines, one of which is of the mth, the other of the nth order, can be cut in no more than mn points. The proof of this does not belong to this work, but to the higher branches of geo- metry. It is quite enough to know, that this solely de- pends on the following rule, and is an easy and imme- diate consequence of it. SECTION LXXIX. Rule. When in the two equations IT. v"+ Av + Av + Avr3 4. eooeee t-A=O0 II. 2*+ Ay + Aly? 4 Aa?3.4 ey ies + A=0 between two magnitudes x, ¥, the coefficients yy A, v}. Wa A. Al Al A A! ,are merely whole rational functions of | y, and that A and A’ are of the first degree, A and A’ of 3 3 | the second, A and A’ of the third, and so on; then the _ final equation in y, which arises from the elimination of | #, can never exceed the degree mn. 144 Proor 1. Let x’, 1”, a’, &e. be the roots of the 1 i 2 equation I, then is—[1]=A, [2]=4?—24,—[3]= 4—34A+ 3A, and so on; from which we see, that [1] contains no other power of y than y', [2] no higher one than y*,[3] no higher one than y°; aud it may also be satisfactorily proved with little trouble, from the nature of the formule, § VIII, that in the supposed nature of the 1 2 coeflicients -A, A, &c. the numerical expression (u) gene- rally contains no higher power of y than y. 2. Further, no expression of the form (a@yé ... ) can contain any higher power of y, than y’****7+°,... The accuracy of this assertion appears from 1, together with the remark in § XXIV. 3. From § LX XV the first part of the final equation (the other=o) is the product of the m factors 1 Np n pf oe Al gl, Aegis ag + Aa! eceee -+- A’ 1 ny n eS 4 A le Aa PE ces Seg 1 n—r n GE Se Ag OPA eee me “+. A! &e. after we have eliminated in it the roots 2’, x”, x’, &. by means of the coefficients of the equation I. | 4. The general term of this product is a—{A N—y —wT A! AoE Se cl ge 145 Now, since the product must be a symmetrical function of a’, a’, a’, &e. this term necessarily belongs to l—T A! aA rs bad foagl TAT ey | 5. Butby 2, the highest exponentof y in[uvz...]is equal to the sum of the radical exponents r+ v A oes further, by the hypothesis, n—j: is the highest exponent of y in Mh m—y nV—Tr A’, n—vy the highest in 4’, n—7 the highest in A’, &e, .. mu—(u + v + 7 + &e.) the highest in the product A’ A A’ &e. But from both it follows, that m7 is the highest exponent of y in the term 4’ dA’ A’ &c. [uva... |], and that consequently in this term there can be no higher power of y than y””. 6. Now, since what has been here proved of an indeter- minate term, obtains for every term in particular, it follows .*. that in the final equation there can be no higher power of y than y””. SECTION LXXxX. When more than two equations with more than two unknown magnitudes are given, then in general there is no other way but to combine these equations in the usual way, two and two, and thus get rid of one unknown mag- nitude after the other. But Bezout observes, with truth, in the above-mentioned work, that this method is very defective, because a number of useless factors enters into the successive eliminations, by which not only the opera- _ tion is lengthened, but likewise the degree of the final | U 146 equation becomes much higher than it ought to be, and what is much more objectionable is, that these factors do not show themselves till the calculation is completed. Since, however, these difficulties can be got rid of even by Bezout’s method in no other way than by considering a great number of single cases, (but neither the object nor the limits of this work allow of such a detail as this), I shall consequently not enter into these inquiries at pre- sent, but leave them for consideration till a future period. Tschirnhausen, in the ‘* Acta Eruditorum,” for the year 1683, has given a method for solving those equations which are founded solely on eliminations. ‘This method — only requires to transform the given equation, by means of an assumed auxiliary one, into another, wkich contains any number of indeterminate magnitudes, by the proper determination of which it is possible to remove as many terms as we please, and by that means give it the form of an equation of two terms, of a quadratic, of a cubic, or of any other equation, whose solution is already known, or may be considered as known. Its inventor considered it as general, and so it is indeed; only its application often requires the solution of higher equations than the given one itself. The following problems will elucidate what has been just said. ‘ SECTION LXXXI. Pros. Let the two equations I. a” +4+ar"1 4+ ba” + cx” > + &e. = 0 Il. y+ 44 Br+ Ca? + Da? + &. = o be given, in which the coefficients a, b, c, &c. 4, B, C, &e. | ¢ a ae 147 contain neither « nor y: determine the degree of the final equation, which is obtained by the elimination of «, im terms of y. Solution. Let 2’, x’, x/”, &c. be the roots of the equa- tion I; then, according to § LX XV, the first part of the final equation (the other = 0) is the preduct of the following m factors : y+ At Br + Cr? + Dr® + &e. yt A+ Br” + Cro’? + Dr’? + &e. y+ A+ Boll! + Cr/2 + Del’? + &e, &e. Now, since in these factors y occurs in no other term but the first, consequently in the product there can be no higher power of y than y”. The equation in terms of y — Is .*. necessarily of the mth degree, and consequently always of the same degree as the equation I, of which degree besides the equation II may also be. Corollary. If .*.an equation 2” Eee ax”! + by”? + cxr"3 -- &e. =——.0 be given, we can transform it into another of the same | degree in numberless ways. For this purpose, we only need assume any auxiliary equation of the form ¥Y¥+A4+ Be + Cr? + Dr® + &e. and eliminate x from both equations. Now, since both the degree and the coefficients of the auxiliary equation are undetermined, we can always determine both, in the way required, by the form which we have determined on giving to the transformed equation. ‘Thus, if we wish to 148 reduce the equation of the second degree 27 + ax + b=o to the form 7?+ K=o0, we assume the auxrliary equation y+A+a=o0. By eliminating x from this and the given ‘ equation, we get y? + (24 —a)y + A? —a44+b=0. Now, since the second term vanishes, we have 2A —a=o0, and A = 3a; and by this value of A, the auxiliary equation is transformed into y + 3a + x = 0, and the transformed one into y2?—1a2+b=0. The last gives: y=+ /(A@¢—)), and if this value be substituted in the first, we then obtain 7 = —3a 7} V (1a?—D), as required, SECTION LXXXII. Pros. Reduce the general equation of the third degree a+ av + br +c=0 to an equation of the form y° + K =o. Solution 1. Since the transformed equation is to have the form y®? + K =o, in which the second and. third term vanish, we must then assume an auxiliary equation with two indeterminate coefficients, in order, after having | performed the elimination, to determine this one, in such” i a way as this condition requires. Let .-. yt+A+t+ Br+av’=o be this auxiliary equation. | | 2. Now, in order to eliminate x from the two equations I. a + az? + bn + ¢=.0 Il, y 4+- A+ Be +a22=0 149 equate these with the equations I, II, in the second example, § LXXVII. This gives A=—a, B=), C0, APS 0, B= 1, Bed ss Eee Yh -f- A. Consequently the final equation in the above § is trans- formed into the following: (y + A)? — (cB — @ + 2b) (y + A? + (0B? + (3c —ab) B+ 8 —2ac) (y+ A) —c B* + ac B? —be B +c?=0 3. If in this equation we solve the powers of y + A, we then obtain an equation of the form y + Ay’ +ly+K =0 and H = 3A—aB+a@ — 2b IT =3A? —2A(aB—@ + 2b) + OB + (3c — ab) B + B — 2ac K = A?’ —a4?B + DAB? —cB? + (a2 — 2b)A? + (3c—ab) 4B + acb? + (#—2ac) d—beB + ¢ 4, If we wish to reduce this equation to 7° + K = 0, we must put H and J = 0; this gives the two equations 3A —aB + a — 2b=0 3A? — 24 (aB — a? + 2b) + bB* + (3¢ — ab) B + 6° — 2ac=0 Since the first of these equations is of the first, and the second of the second degree, in reference to 4 and B, we consequently can find these coefficients by the solution of an equation of the second degree, and the substitution of 150 these values in the expression for K gives K, and at the same time the reduced equation y° + K = o. Corollary. Having found the reduced equation, we are also enabled to find the roots of the given equation. Thus, from y° + K = 0, we obtain, when 1, a, 8, denote 3 3 the cube roots of unity, y= — / Ky, y=—av Kk, y=-—B8 v K. If these values of y, together with those of 4 and B, be substituted in the auxiliary equa- tion, we then obtain the required value of x by the solution of equations of the second degree. ReEMaARK. Since the values of 4 and B depend on equations of the second degree, strictly speaking, we get 2 x 2 corresponding values of these magnitudes. Now, since every two corresponding values may be com- bined with each ofthe three values of y, these substitutioris give six different equations of the second degree. Every one of these gives two values of x, and consequently we obtain generally twelve values of x, although the given equation can have no more than three roots. We must, however, keep in mind, that only those values of x may be assumed, which verify at the same time the two equations I, II. In order to find these values, we need only .-. seek the common divisor of 2° + ax? + br +¢ and 2? + Br + A + y in the usual way. The division of the first expression by the second, gives the quotient x + a— B, and the remainder . (B?— aB+b—A—y)x+(B—a) (444) +e 15] This remainder must vanish. We have consequently (Be—-aB+b—A—y)x+(B-a)(A+y)+c=0 and hence we obtain (B—a)(A+y)+c B—aB+b—A-y If in this equation we substitute for 4, B, their values _ from the equations in 4, also for y its three values succes- sively, — Yy K,—a v K,—8 V KX, we obtain three values of x, which are the roots of the given equation. Moreover, it matters not, in this case, which we use of the corres- ponding values of 4 and B, because they always give the same values of x ; of which we can easily persuade our- selves by actual calculation, SECTION LXXXII. Pros. Transform the general equation of the fourth degree | a + ax + bv? +cr +d=0 into another of the form y4+ Hy’?+K=o. Solution 1. Since the transformed equation is to have | the form y+ + Hy?+K=o0, in which two terms vanish, viz. the second and fourth, we must consequently assume -an auxiliary equation, with two arbitrary magnitudes. | Let .-. | yt+A+ Br+x=o0 | be this auxiliary equation. 2. In order from I. y+ 44+ Br+22=0 Il. y +a? + db? + cr +d=0 152 to climinate x, it is only necessary to compare these equations with those of the first example in § LX XVII; we then find A’/=o, B/=1, C/=a, D'=0,, =a F/=d, A=—B, B=y+A. If we make these substi- tutions in the final equation in the above mentioned §, it is by these means transformed into an equation of the form | (y+A)+P(y+ AP t+ Q(y+ Apr RGt At S= 0 and then P=—aB+@ — 2b Q = bB? + (38c—ab) B + BP — 2ac + 2d R= — cB + (ac—4d) B? + (8ad — be) B +c —2bd S = dB — adB’ + bd B® —cdB + @ 3. If we arrange this equation according to y, we obtain p+ (444+ Py + 64? + 3PA+ Q)Y + (44 + 3PA? + 2QA + RY +444 P4+Q44RhA+S=0 in which any two terms may be eliminated at pleasure, by | merely determining the letters A, B, conformably to it. 4, Now, in order, as the problem requires, to eliminate | the second and fourth terms, we put 4A + P=0 4A3 + 3PA® + 2QA + R=0 ! fe ; : 5. The first gives A= cont and if we introduce this 153 value into the other equation, by omitting the fractions, we have p> — 4PQ + 8k =0 If in this equation we make use of the. above values of P, Q, R, we then obtain for B an equation of the third degree, viz. (4ab —a? — 8c) B® + (3at—140°b + 20ac + 8b? — 32d) B? 4+(—3a + 160° — 16ab? — 20a°’c + 32ad + 16bc) B + a& — Gad + 8 a'c — 8 ad + 8a°b? — 16abe + 82 = 6 ‘Having determined’ B from this equation, we need only ‘substitute its value in the above expressions for P, Q, I iS, in order to find these coefficients also. 6. Further, the equation in 3, by putting —— for A is ‘transformed into def oe ee Us RCN Tage ec cae OS Sas6 Unie sea he Fae 3 or, when we substitute for R its values = _ — into LY ESO a OR ae A Gees is AO Bia yr A ker and this equation has the form y* + Hy? + K =o, as “required. + S=0; Corollary. Now from this transformed equation we ‘may find the roots of the equation 2* + a2? + ba?+ cx +d=0 ima similar way as in the foregoing § for the equation. of ‘the third degree. Thus, since the equation in 5 gives x 154 three values for B, and the substitution of each of these values in the transformed equation gives four values of y, we .*. obtain generally twelve values for y. Tach of these values of y, together with that of B when substituted in the auxiliary equation a? + Br + A +y=0, or x? + fe : Br — i +y=0, gives two values for x, and we .*. obtain generally twenty-four values of . Now, in order to learn which of these values are at the same time roots of the given equation, we must seek the common divisors of the two expressions a4 + ax> + br? + cx + d, 2? + Br —= + y. With this view, we divide the first expres- sion by the last, until we come to a remainder, which contains x in the first power only ; this remainder must .*. be =o. In this way we obtain the equation |B? — aB? + bB —c— (a — 2B) (y-=)]e +(B* —aB +8) (y—2) —(y—4)2—d=0 and hence d— (Be — aB +3) (y—*) + y—2) oe BP —aB? + bB—e~(a—2B) (y—*) | Now, if we substitute in this expression of x for y its four values from the transformed equation, we thus obtain the roots of the given equation, and indeed we shall always find the same four values for x, whichever value of B we make use of. 155 Remark. From this and the two foregoing sections we deduce at least this much, that T'schirnhausen’s method leads to the actual solution of equations of the second, third, and fourth degree, although in a very laborious way. Whether, and how far this method is also applicable to higher degrees, will be the subject of inquiry hereafter. V.—ON_THE ROOTS OF THE EQUATION 2” —1=0, AND ITS APPLICATION TO THE ELIMINATION OF SURDS FROM EQUATIONS. A METHOD, BY WHICH TO FIND SOLVABLE EQUATIONS, AND SOME OTHER SUBJECTS CONNECTED WITH IT. SECTION LXXXIV, Prog. Find an equation, which merely contains the imaginary roots of the equation x’—1=0. Solution. Here two cases must be taken into considera- tion, viz. first, when m is an odd; secondly, when x is an even number. 1. Let be an odd number. In this case there is no more than one real root, viz. + 1, and consequently r—1 must be a divisor of x’—1. If.:. we divide the equation x*—1=0 by x—1, and make the quotient =o, we obtain an equation which only contains the imaginary roots, and this is © Gp eee eo etek dt 8, +L a 2. Let m bean even number, .*. the given equation is of the form x*”—1=o0. In this case it has necessarily two real roots, viz. + 1 and — 1, and no more. Conse- quently both »—1 and x+1 must be divisors of «*~*, 157 . .*. also the product (r—1) (v+1)=2°—1. If .°. we divide the equation a”—1=o0 by 2°—1, we thus obtain an equation, which only contains the imaginary roots, and this is Mich 2 fe beens fe Bee? Lie Corollary. In order .*. to find all the roots of the “equation x”—1=0, we must, when z is an odd number, endeavour to solve the equation a”! + v-* + ... +x + 1=o, and when m is an even number, the equation a tet... +2°?-+1=0. The latter, because it only contains even powers of x, may always be reduced to an n—2 a) ys - by substituting y for x°. equation of the degree Exampter I. The equation 2°— 1=0 divided by x—1, gives | v+r+i1l=o0 —1+/—3 and when solved, 7 = The three roots of _ this equation are consequently | f -1+/—-3 —1—/—3 _ Exampte II. The equation at—1=0 divided by a?—1, gives are bya O whence we obtain v= +./—1. The four roots of the equation 1*—1=o are consequently +1, —-1,4+/—1, —/-1 158 Exampte III. The equation «°—1=0 divided by w—1, gives a+e+taere+evr+1=o. This equation may be analyzed into two quadratic equa- tions e+(egt+t/5\xr+1=0 we+(s—4V/5)xr+1=0 and the solution of these two equations gives the four following imaginary rooots : t[—-1-—/5+ / (10— antsy on [-1—- / 5 — J/ (10-25) J — 1] [-14+ V5 + / (1042/5) /—1] [-1+ V5— J (1042075) / — 1] Bie Ble Ale Exampte IV. The equation 25—1=0 divided by a—1, gives +x? +1=0 and the solution of this equation gives hs FO asthe The six roots of the equation «®°—1=0 are .° i + 1 +f nai’ oats yates hy ee A faa ae SECTION LXXXV. / Pros. Reduce the equation 2"—k=o to an equation of the form y"—1=o. I 159 Solution. Put r=y/k, and substitute this value in the equation a*—k=o, then this equation is transformed into ky’—k=o0, or when divided by k, into ¥’—1=o. Corollary. If .-. we have in any way already solved the equation y"—1=o, and denote by 1, a, 6, 7, 6, & &e. its n roots, or the value of y, we then obtain from P=: Jk the n following roots of the equation a*—k=o: Jk, ale ark, iis SA/k, Ey ie &e. SECTION LXXXVI. Pros. Reduce the equation a*7—1=o to an equation of the form y’—1=o. Solution. Put «?=y, then a=y’. If this value be substituted in the given equation, it is transformed into y—1=o0. Corollary. If we denote the roots of the equation x’—1=0 by 1, a, 8, y, 6, «, &c., then the roots of the equation 2’—y=0 (foregoing §) are b P P P P P VYs avy, BAY, Vy OV, EVYs &e. -Now, since we can substitute for y each of the roots of the equation 7’—1=o0, we obtain by these substitutions all the pq roots, of the equation a*’—1=o. _ Examp ce. To find the roots of the equation x!?—1=o, . put p=4, p=3. We consequently have the two equa- tions 160 ai— y=o, y—-1=0. Now, the roots of the equation «+-1=0 (§ LXXXIV, - example II) are + 1, —1,+ /—-1, —-V7 —1,-- the roots of the equation «1—y=0 (foregoing §). 4 4 4 4 VIpp—VptV—-t.Vy—-V m1. VAY Further, the roots of the equation y3—1=0(§ LXXXIV, example I). —14+/—-3 —1—/vV/-—3 4 2 4 2 If we successively substitute these values for y, we obtain the twelve following roots of the equation 2!’—1=o0: 1, — v—1, —V—1 4—l+/-3 4-14 /-3' 4-l4 /-3 4-1+ /-3 I gp et 1. yy ee 2 2 2 2 pV) Cea iye dues Cae han | ae eee} 4-1- /-3 Pty ee J ——) v=]. y—-——,- v=). y-— — SECTION LXXXVII. Pros. Under the supposition that 2 is a prime num- ber, from any one of the imaginary roots of the equation a” —1=0, find all the remaining ones. Solution 1. Let a denote one of the imaginary roots of the equation 2*— 1=0, so that a* — J =o0, or ' a= 1, 2. Since a® = 1, then also (a) " = (o") ™ = lene .*. a is a root of the equation 2* — 1 = 0, then must also \ ; 161 a” be one of its roots. Therefore the equation 2*—1=0 has, besides a, the roots a’, a®, a*, a®, &c. 3. But since in this way weshould find an infinite number of roots, and the equation «* — 1 =o can only have n roots, we may safely presume, that im the series of powers a, a’, a®, at, a®, &c. there must be an infinite number of equal roots. ‘This likewise is really the case: for we find a*t! = a®.a=a, a? =a". a? = 0", a3 —a’. a? = a’, &e. 4. Generally, when we exceed the nth power, we shall find only one of the n following roots ene ts Cis sess tdess Op os Oe of which the last = 1. For let a” be any power of a, andm > n. Further, let g denote the quotient, which we obtain after dividing m by n, and r the remainder, consequently r < 2; thenm=ng+r. We have.:. Gee Oe = a7. a) (a.) og, = 1). a a ut a’, since r < m, is necessarily one of the powers a, a?, a’, a‘, @enae eee a 5. The conclusions drawn hitherto obtain, whether x be a prime number or not. In the particular case, when m is a prime number, according to the supposition in the problem, it may be proved, that the roots a, a®, ao eee are all different from one another. For we suppose two of these roots a“, a’, to be equal, and » > p. Then we divide the equation a’ = a“ by a, and obtain a’ = 1; but it may be shown, as follows, that this equation is impossible, Y Dem amie 162 6. Thus, since n is a prime number, and v — p < 2, the numbers v — jp and 7 are .*. prime to one another. Consequently, as is already known from indeterminate analysis, two whole positive numbers t, “, may always be found, such that (vy —p)t=nu+1. If.-.a*=1, then also must a%-“”!= 1, and consequently also a™*) = 1, or a”. a = 1, or a= 1; which is impossible (1). 4%. Since .*. the roots a, a, a®, a‘, ...... a®, aS tar as the number 7, are all different from one another, then these are the 2 roots of the equation a”—1=o0. If.. an imaginary root a be given, we then have likewise all the remaining ones. Corollary. If .*. we denote the imaginary roots of the equation x — 1 =o bya, 8, y, 6, &c., then, when x is a prime number, all the roots of this equation may be represented in one or other of the following ways : either by a, a”5\ a? ...<+00 oe ne _ or by te hist cyan Oe rete bas < atier ae 9 or by Vy one darecaryes Ve a cyt &e. or, which is the same, we can substitute in the series of TOOLS Gye G5 C's, sense a"), a”, for a each imaginary root a’, a®, a*......a” 4, and then we shall always obtain the same 7” roots. Exampie. When x = 3, the roots are a, a’, a. If for the root a, we substitute the following one a®, we 163 obtain a2, a‘, a®. But since a? = 1, then at = a, and a® = a®, and we have .*. here a2, a, a®, as before. When n = 5, the roots are a, a’, a®, a*, a. If we put a? for a, then, on the contrary, we have a’, a*, a®°, a’, a", or, since a® = 1, a’, a‘, a, a®, a’; consequently the same roots as before. In‘ like manner, when a’ is put for a, we find a®, a°, a°, al, al, or a®, a, a*, a’, a, and when a’ is put for a, a‘, a®, a, al, a®, or a‘, a®, a”, a, a”; consequently always the same roots, only in a different order. SECTION LXXXVIII. Rules. I. When is divisible by m, then all the roots of the equation x” — 1 = 0, must also be roots of the equation 7 —1=o0. yh si n . Proof. Since nis divisible by m, then— =q is a whole m number, and n=qm. The equation x” — 1=0 is consequently a”—1=0, and if we put a”=y, y—1=o0. Now, y?—1= 01s divisible by y — 1; consequently also, if again xv” is put for y, 27” —1 is divisible by «*—1; .°. the roots of the equation x” — 1 = 0 are also roots of the equation a’” — 1 =0, or v”—1=o0. Q. E. D. II. When a root (unity excepted) of the equation x” —1=0, which is also a root of the equation x” —1=o, is of a low degree, and that of the very lowest possible, then must be divisible by m. 164 Proof. Let a be the common root, consequently a’ —1=0,anda”—1=0. Now, if xz be not divisible by m, then x divided by m gives the quotient g and the remainder r, so that n= gm-+r, andr ae If .:. we divide the equation r+—1=o0 by «*—1, we obtain the equation Cipt ca t 8 EY whose roots -- ./—1 and —»/—1 are such, that they do not become +1 till raised to the fourth power. 167 _ Examp.e If. Let 2'?—1=0 be the given equation, consequently »= 12. This number has two simple ffactors, viz. 2 and 3. We have .‘.p=2, q=3, and | it] . consequently » = = Opie <4... NOWenewe divide x” —1=0 by x®—1 and a‘—1, we obtain the two equations a®4+1=0 a®+a4#+1 =o. ‘Their greatest common divisor is ae— er+iso. Hence we find 1+ /—383 /3+/—1 , and these four roots are peculiar to the equation v!’— 1 =o, , because they do not become +1 till raised to the twelfth ~ power. In order to find the above roots, it is only necessary to solve the equations #®°—1=0, #t—1=0, and to take the common roots only once. The roots of the equations ‘—1=0, c%—1=0 arein§ LXXXIV. In this way we obtain the following eight roots. —1+ /—8 rane AU pat /—3 +1, + Ce i, +\/ 2) ‘ 2 which together with the four preceding, give the twelve roots of z”’—1=o. ‘This mode of expressing them, is, as we see, much more simple than that in § LXXXVI. / Remarx. A root which is peculiar to the equation «"—1=0, and which consequently belongs tono equation of 168 a lower degree of this form, is termed a primitive root of this equation. SECTION XC. Pros. Let » be a compound number, and a a given primitive root: find all the roots of this equation. Solution 1. In § LXX XVII it has been proved, that for every , though a may be any imaginary root, the powers a, a’, a®...a”, are always roots of the equation x*—1==0. 2. I affirm, then, that when, as has been here supposed, a is a primitive root, in the series of magnitudes a, TANT aN io ge a”, there are no two, which are equal to one another. For if a“=a’, then a“’=1, consequently a 1s a root of the equation *’—1=0; therefore the root of an equation of the form #*—1=0, of a lower degree than’ n, and consequently no primitive root, which is contrary to the hypothesis. 3. Since .-. the magnitudes a, a2, a3, a‘,......a" are all roots of the equation z”—1=o0, and all different from one another, they are the n roots of this equation, which were sought. SECTION XCI. Pros. Let n be a compound number, and a a pri- mitive root of the equation x” ~1=0, .*. a, a’, a, a‘,....4 169 all the roots, of this equation (§ XC): find a eriterion by which to distinguish the primitive roots of this equation from the others. Solution 1. If two whole numbers m, n, have a common measure, there may always be found a whole number ¢, which is less than », and such, that mé is divisible by n ; on the other hand, if the numbers m, n, are prime to each other, then ¢ cannot be less than n, if mé is divisible by x. 2. Now let a” be any one of the magnitudes a, a’, a’, a. If this be a root of an equation 2° —1 =o, _ we then must have a” — 1 = 0, ora” = 13 .*. mt must _ be divisible by n. 8. From this condition and from 1 it follows, that when ithe numbers m, n, have a common measure, there can | always be found an equation 2° —1=o0 of a lower degree than the nth, of which a” is a root; but that no such equation can be found, when m, n, are prime to each other. a‘, ...... a’, all those, without exception, are primitive roots of the equation 2” — 1 = 0, whose exponents have no common measure with n; and this .*. is the criterion by which the primitive numbers may be distinguished from the others. ' ExameP.e. Amongst all the roots a, a2, a®, at, a®, a° Z 4. But hence it follows, that of the powers a, a’, a®, 170 a’, a®, a, a’, a, al®, of the equation 2’ — 1 = 0, there are no more than the four a, a’, a’, al, whose exponents have no common measure with 12, and consequently which are primitive roots of this equation ; and these roots can be no other than the four which were found in the second example, § LXXXIX; viz. | 473+ 4/-—1, /J38—-4V—1 —1/38+232/—1,-4V8—3V7—1 In order to be convinced of this, assume one of them, viz, 3. /3+4/—1, for a; by actually raising this root i the fifth, seventh, and eleventh powers, we find: Hecate ie Vans Sate Sick es case a” TMA E NE MUR —$/3—4V/-—1 jie tee Si at ht al eal and these are the same as the foregoing. We should have Ye NH =} ~J obtained the same result, if we had put every other of the four above-mentioned roots for a. ‘That this must be the case, may, besides, be seen without actually completing the Sopa for if in a, a®, a’, a'', we substitute a®, a’, a, successively for a, and omit in the exponent the multiples of 12, we then obtain a’, a®, a, a®, Oras. aang ent ee es OY al, haven mea, abe Ga has OL ass was: Gs .'. always the same roots, only in a different order. SECTION X€CII, If we compare the equation 27— 1 =o with the general equation a” — Ax" 4+ Ba? — Cr"? + ....0 171 Por + 7 =.0,.4:3 0B = 6, "Coy hic... 0, /’= +1. If.+. we denote the roots of this equation by a, By y, 0, &e. we have atBt+y+eo+ &.=0 aB tay + &. + By + 6d 4+ &.&. = 0 aby + aBd + &. + Byd + &e. &. =o and so on to the product of all the roots, which is = — 1, or = + 1, according as m is even or odd. Since in the two first chapters, the letters, a, @, y,° 6, &c. are used to denote the radical exponents, in order to prevent mistakes, I shall once for all remind my readers, that these letters, when they are in the brackets | [ ], always denote, as heretofore, radical exponents, but ° in every other case the roots themselves. Further, in order to indicate, that. a numerical expression relates exclusively to the roots of an equation of the form a” — 1 = 0, I shall place a dash over the left side of the bracket thus ‘[a@yé......«] im reference to the equation x” — 1= 0, denotes a numerical expression for the radi- cal exponents a, £, 7, 6, «+++ K. SECTION XCIII. Pros. Find the sums of the powers of the roots of the equation a” —1 =o. Solution 1. If we compare this equation with the general one a* + Ag 4+ Bi? 4+... + Pr+Q=0, me.find.4 — op B=. 6,5 C= 0, &. P= ont — 1. Consequently, by means of the equations in 7, § VIII, we obtain 172 ‘{1] = 0, [2] = 0, [3] =0,...... ‘{n — 1] =o on the other hand ‘[n] = 2. In like manner we find ‘[n+1]=0, ‘[n+2]=0, ‘[n+3]=0,...... ‘[2n—1]=0 on the other hand ‘[2n]=n. Generally, all those sums of powers, whose radical exponents are divisible by , are equal to n, all the remaining ones, on the contrary, =0. 2. If we put cr = att the equation a” —1=0, Is ) Y ; 1 transformed into —— 1 = 0, or y” —1=0, whose roots consequently are the reciprocals of the roots of the former equation (§ X). But since the equations x” — 1 = 0, y” — 1 = 0, are similar to one another, we also have Mian Lil =O, eS [= On pa siane ‘[—n+1]=0, ‘[—n]=n and generally, all those negative sums of powers, whose exponents are divisible by , =, all the remaining ones, on the contrary, = 0. SECTION XCIV. Pros. Find the value of ‘[a@}. Solution. Since, generally, for every equation [af] =— [a] [2] — fat Bl, and when a = #, 2[a*| = [a’|—- [2a], so also in particular for the equation «” — 1 = o. ‘[ag] = “[a] ‘[e] — ‘fa + 2] ‘Tal. } | QA | 178 should also have obtained, if we had squared both parts of the equation v= 4/k. j é , Further, let x=\/ k. If we denote the three roots of the equation 27—1=o0 by a, 8, y, then the irrational 3 3 magnitude \/ k has the three values oN /, k, e\/ k, wW k, and these consequently must be the roots of the required equation. It is.°. 3 3 3 (w—aN/k) (w—@N/k) (v—y/k) = 0. By actual multiplication, we obtain 28—[1]\/ k.a2+'[12P/2.2—"[1°] kao: or, since by the preceding §, [1]=0, [1?]=o0, ‘[1°]= Li wk x—k=0; a rational equation, which we should also have found, by ’ 3 raising both parts of the equation x=N/k to the third power. 4 . I shall now put r=N/k. Since +1, —1, +/—1, — /—1, are the four roots of the equation 2‘—1=o, then PEN IRONY Teton ad NY Kee 1 N/ Te ae : 4 the four values of / k, and the required equation is .*. (e—A/ ky (@ N/a — V —-1A/ i+ /—1A/k)=0 or by actually performing the multiplication | a+—k=0, as was required. If for \/ k we had only taken the two values +N/ ky | 179 EA / k, and from these had formed the equation (x—N/ k) (x +h/ k=0) we might have foreseen, at once, that no rational equation could be found. And this is actually the case; thus we obtain 22— ./k—=o. Each of the two values of x contains the irrational magnitude ./k, and since this has a two-fold value, we consequently obtain the four values of x. Hence we perceive, at least, how we are to proceed with equations of the first degree, in order to render them rational. ‘Thus, if =A be an equation of this kind, and we denote by 4’, A”, A’, &c. the different values which the irrational expression 4 contains, by reason of the many significations which its irrational magnitudes have, then . (a— A’) (x— A”) (0X ~~ A”) wees 0 is always the equation free from irrational magnitudes, which was sought. The following problems will throw more light on this subject. SECTION XCVIII. Pros. Make the equation r= /p + 4/q rational. Solution. The irrational expression »/p + /q may here have four different values, according as we give the roots /p, »/q, the sign + or —, and these values are +J/p+0/q,—-Vp—Vq, + /p—Vq, —Vp + V9. The equation free from irrational magnitudes is conse- quently 180 (c— /p— Vq) @4+ Vpt V9) (x— V/pt+ V9) (@+ Vp— V9) =o or, when we multiply the first and second factors together, as also the third and fourth (2° —p—q—2 /pq) (PW —p—qt+2/pq) = o; or lastly, by completing the multiplication the second time v+—2(p+qxe+ (p—g)=o SECTION XCIX. Pros. Make the equation r=a/p+ may have three values viz. 18] 3 3 3 o\/p, e/ P; Vp when a, 6, y denote the three roots of the equation 3 a°—1=0, unity included. These three values of \/ p 3 correspond to the three following values of its square \/ Ties 3 3 3 a\/p’, en/p°, pV. Consequently x can contain no more than these three values : aah/ p+ 2N/ p?, ga\/p + @0N/p°, yaX/ p+ yth/ p*. The rational equation is .*. of the third degree. 2. It is represented by v— Pre+ Qr—R=o0 then P= (aa\/ joes ab\/ P+ (ga\/ pt INA P*) + (yap +b Vp") = [Jap + [2]0\/p° Q= (aa\/p + a%b\/p*) (@a\/p + @0N/p°) + (aa\/p +a°\/p*) (yaX/p + yh/ p’) + (ga\/ pt eron/ P*) (ya\/ pt yo/ p*) - ‘[12|aN/ p?+ [12 ]abp + [27]. bp\/ Pp eros (aa\/, pt a°tN/, Pp’) (gaN/ P +6°0N/p°) (ya\/p+ y p*) = [1°Jatp + [122 ]Ja'bp\/ p+ [12%]ab pV p’ ne [2°] b*p? 182 3. Here, then, the numerical expressions {1], [2], [12]> [2%], [122], [127], [12], ‘[1°], [2°] occur, the first six of which vanish, by 2, § XCV. Further, by the same §, since here n=3, [12]=—3, ‘[1°]=1, ‘[25]=1. By the substitution of these values in the expressions for P, Q, R, we obtain P=o0, Q= —Sabp, R=a*p+b*p*, and hence the required rational equation x — 3abpxr — ap — bp? = o. Remark. We could essentially have shortened the calculation for determining the values of P, Q, R, by omitting at once all those terms in which p is included under the radical sign, because it might have been fore- seen that they vanish in the results, as the required equation must contain no irrational magnitudes. SECTION CI. The problem in the preceding § leads immediately to the solution of equations of the third degree. For since 3 3 3 3 the supposed roots aa\/, ptat\/ Ps ga\/ pt e20\/ ps 3 3 aN p 4 y20N/. p® led to the equation «* — 3abpr— a®p — b’p?=o, it may be inferred conversely, that every equa- tion of this form must have these three roots. If we put p=1, then this equation is transformed into a — 3abr — a — hb? = 0 and the three roots of this equation are consequently aa + a’b, Ba + 6°, ya + yb. Since one of the three roots a, @, y must be equal to 183 unity, we may .*. put y=1; farther, if a’=@, 6°=a; the three roots of the equation «°— 3abu — a®—b*=0 con- sequently assume the following form : aa +b, fat+ab, a+b. We obtain also precisely the same result from Cardan’s - Formula, which, as is already known, tries to reduce the given equation to the form «°— 3abr—a®—b*?=0. SECTION CII. The problem, § C, may also be solved by another _ method. Thus put N/p=y, then \/ pay’, +. cay _ by? : the value of y determines that of x. But y has three values, viz. a\/ Ps er/ Ps y/ Pp, which are all com- . prehended in the equation 7°’—p=0; consequently the _ values of x must be the result of the elimination of y in _ the two equations | I. »y—p=o Il. x — ay — by? =0 Inorder to perform this elimination, we give, by § LX XVI, _ to the equation II. the form 1+(1)y+ (2)y°=0, so that SS (1)= — (2)= -<, and we then obtain the following equation : | o=1+ (1) [1] + (2) [2] + (2) [12] + @) [2°] + (1?)[1?]+ (2°) [1°] + (2°2)[1°2] + (12?) [127] + (2%) [2°] - The numerical expressions may be taken from the annexed tables, if we put d=o, B=o, C=p. If after this we 184 : : : a b substitute again for (1), (2), their values——, — =» we £ then obtain the equation 2°— 3abpx —b*p—a’p*=0, as in § Cc. SECTION CIIlI. 3 Pros. Make the equation n=aN/, p+0N/ q rational. 3 Solution 1. The cubic irrational magnitude »/p has 3 three values, viz. a\/ P> e\/ P> y/ p. In like manner \/ q contains the values a\/ q a\/ q) J q. Each of the first may be combined with each of the latter, and this gives nine values of a, viz. aa\/ p+abN/ 4, gaX/p +.abN/q, y/ p+ ab\/q aa\/'p 8 ab\/ q, ga/p Tr BiN/q, yWJ/p Kl BdN/q x2 3 3 3 3 3 AN p 4 yh/ q; ga\/ p+ytr/ qs ya/ Pp +700/ q Hence we can find the rational equation in the usual way. But we can also attam this object by elimination as in” the preceding §. 3 $s . 2. With this view, put a\/ p=y, oN / q=2; then | w=y+z2. Now, since y has the values aa\/ ps eaX/ Ps : 3 3 vii! p, and z the values ab\/ q> ab\/ q yoN/ q, which are all included in the two equations y* — a’p =o, z’—b°q=0, it merely amounts to this, from the three equations 185 Livan sig ite Il. y—a’p=o0 IT. 3 — bg =0 to eliminate the magnitudes y and 2. 3. Raise the equation I. to the third power, and put for y°, 2°, their values a°p, b°q from II. and III., also x for y+2z, we then obtain a = a’p + b'¢q + 3yzr or 2° — a®p — b'g = 3 yzr 4, Raise this equation again to the third power, then we get : (23 — Bp — bg) = 27y3 2 2° and if for y*, 2° we substitute their values (a? — ap — bq) = 2703 b? pq. 5. If this equation be solved and properly arranged, we obtain xu — 3(a%p+b°q)a® + [3(a%p + b°9)? — 270°b* pq] a° — (a’p + b°q)’ = 0, and since this is of the ninth degree, it consequently is, as appears from 1, the most simple rational equation — which can be deduced from x=aN/ p+oN/ q: SECTION ClIv. ' Pros. Make the equation xr = /p + /qg + Vr rational. _ Solution 1. Since the quadratic irrational magnitudes 28 186 may be assumed either positive or negative, their combina- tion gives for v the following eight roots of the required equation : J/p+ V/qt+vr, —-Vp—-Vv/q-— vr J/p+ V/qg—- Vr, — Vp—- vat vr J/p— V/qt+V/r, —Vp+ Vqa-— vt J/p— /q—V/r, — /pt Vat vr 2. Since here every two roots which are opposite each other only differ in these signs, the equation can only eontain even powers of x, and it has .°., when we put a= y, the following form : yi — Ay + By — Cy+ D=0 and the roots of this equation are (/p + /q+ Vr), (/p + J/q — J/ry (/p — J/q + Vr, (/p — V9 — vr 3. In order to determine from hence the coefficients A, B, C, D, we only need take the sum of these roots, the sum of every two of them, and so on. The following treatment, which has been frequently made use of already in the preceding part of this work, leads to the object in a shorter way. Let $1, S2, 83, S4, denote the sum of these roots, the sum of their squares, cubes, and fourth: powers ; then, when in SIX, — A and —C are put for A and C, and the symbol § for the one [ ] there used, | A= S81 pa Asi 2 qc BS! — AS2 + 83 3 CSI — BS2 + AS3 — S4 4 D= 187 4. The expressions S11, $2, $3, 84, must necessarily be rational, because otherwise the coefficients 4, B, C, D, could not be rational. Consequently the irrational mag- nitudes in the solution must alternately be left out, and they may .*. be entirely omitted in the calculation. With reference to this remark, and since we treat the trnomial /p+ /q+ /r as a binomial /p + (/q + vr), the calculation stands thus : | (Vp+ /qt+ V/rP=p +q4+rt&e. (pt Vqt+ Vryap?t 6p(/gt Vr? + (q+ V7) &e. =p? + 6p(q+r)+q?+ Ogr +r? + &e. | =p + PF +1? 4 6(pqt prt qr) + &e. (Vp t+ Vg+ r= p + 1ap?(V gt Vr)? + 5p (9+ | Jr)t + (/ q+ /1r)84+ &e. =p’ + 15p*(q+r)+ 15p( @ + 6gr +7") +9? + 15¢'r + 15qr? + 7° + &e. =P +P+r + 15(p'g+ pg + pr + pr + gr + qr’) + 99pqr + &e. (/p+ /q+ Vry=p* + 28p3( /q+ Vr)? + 70p? (/ q+ Sr) + 28p( / q+ V/ryP+( 4 q+ Jr)? + &e. =p* + 28p*(q-+r) + T0p? (gq? + Ggr + r?) +28p(q? + 15q¢r+ 15qr*+r°) +° + 28q°r + 70q?r* + 28qr? + r* &e, =pi t+ gt + rt + 28(p'q + pg? + pr + pr’ + gr + qr’) + 70( pg? + per’ + qr") + 420( pg’ + par + p'qr) + &e. 188 5. It is easily seen, that if, stead of /p+ /qg+ V7, we had raised every alternate one of the expressions J/pt+/q— Vr, Jp—V9g+ V7, /p— V/q— 4/r to the same powers, the rational parts would have been the same. Now, since in $11, $2, $3, $4, the rational terms must be left out, we obtain S1=4 (p+qt+r) S2=4 [p°+e+r?+6(pq+pr+qr)] S8a4 [PtP tr + 15(p 9 + pe + pr + pre + gn + qr°) + 90pqr] S4=4 [pt + q' + r4+28 (p'q + pe +p'r + prt gr + qr’) + 70 (p*q? + per? + q°r°) + 420 (pqr* + pg’ +p°qr)| or more briefly, when the brackets [ | refer to the mag- nitudes p, q, 7; S1=4 [1] s2=4 ((2] + 6 [1°)) S3=4 ([3] + 15 [12] + 90 [1°)) S4=4 ((4] + 28[13] + 70 [2°] + 420 [122]) 6. If these values be substituted in the equations in 3, we then obtain the coefficients 4, B, C, D, expressed by the given magnitudes p, q, r- | SECTION CV. ’ Pros. Make the equation v= /p+ /qg+ V/r+ V/s rational. | | | Solution 1. It may be shown by inferences, asis in | and 2 of the preceding §, that, when we put v® = y, the rational equation is of the form 189 y> — Ay’ + By — Cy + Dy — ky + Fy — Gy + H=0 and has the following expressions for roots : (Jp + Vqg + V/r + vs)? (/p + Vq + /r — +/s)* (Jp + /q — Vr + V/s)? (/p + /q — Vr — Vs)? (Jp — Vg + Vr + Vs)? CV pg Tn Ta 8) (/p — V/q— vr + “WV/s)? (/p — vq — Vr — vs)? 2. Give the symbol S the meaning which it had in the preceding §, and first of all try to find the expressions S1, S2, +83, ...... S8. Since the calculation is managed in the same way as in the preceding §, I shall not detain my readers with it, but only remind them, that in the involution the expression /p+ /qg+ /r+ /s may be considered as a binomial, whose two parts are »/s, and \W/p t/q + Vr. 7 | 3. Consequently we have (Jp+/qt+ vr+ vVsyP= s+ (/pt+ /qt vr) + &e. (Jp + /¢a4+ Vr + Vs) = s+ 638(./pt+ Ag+ V/rPe(V/pt VW9t Vr) + &e. (Jp + V/qt+ vVr+ Vs) = (8+ Sas Sq + J/rP+15s(/pt+ /q+ Vr) + (/p+ /q+ V7r)°+ &e. &c. 190 or, when the developement of the powers of p+ /q+ /r in 4 of the preceding §&, are used, (/p+ Vqat /r+ Vs = ptatrtst &. (J/p+ Vqt+ Vrt+ Vs = P+P+r4+24 6(pqt+prt+pst+gqr+9gstrs)+&e. (Jp + /qg+ V¥r4+ vs)? = P+tPtrP4sH+15 (p'qtpP+prt+pr t+ pst ps +g r+qrt+_qst+qs+r'%s+rs’) +90 (pgr+pqst+ prs+qrs + &e. &e. in which only the irrational terms have been omitted. 4. For the same reasons as in 5 of the preceding §, we obtain from hence S1 = 8 [1] S2 = 8 ((2] + 6[1°]) S3 = 8 ({3] + 15 [12] + 90 [1°]) &e. a and the substitution of these values in the formule in 3 of the preceding §, which must be extended for this pur- pose, gives the coefficients 4, B, C, &e. SECTION CVI. Pros. Make the following equation of the first degree, with an indeterminate number of quadratic irrational magnitudes, rational, viz. a= Spt Sgt Vr + ASH ceseee + Sw. Solution 1, By the two preceding §§ it is easily — 19] inferred, that when » is the number of the irrational magnitudes ./p, /q, ... /w, the degree of the rational equation is equal to the power 2”. But since the different values of x are such, that two of them are always similar, but with different signs, the equation consequently is only of the 2”’th degree, when we put 2? = y. 2. The conclusions in the two preceding §§, when ex- tended, give the following results : Ne org S2 = 2" ([2] + 6 (1°) S3 = 2" ([3] + 15[12] + 90 [1°}) S4 = 2" ([4] + 28 [13] + 70 [27] + 420[172] + 2520 [1°*]) : S5 = 21 ([5] + 45 [14] +210[23]+1260(173] + 3150 [127] + 18900[1°2] + 113400 (1°) &e, 3. Hence the law of the formation is easily perceived. As an example, I will take S5. The number 5, and its divisions into combinations of two, three, &c, give the numerical expressions [5], [14], [23], [1°3], [127], | [182], [1°]. The coefficients are no other than the number of transpositions of different things, whose re~ peating exponents are twice as great as the radical expo- nents of the numerical expression; consequently the coefficients of [5], [14], [23], [123], [127], '[1°2], [1°], the number of transpositions of the different things a'°, @b®, a'b®, a°b?c°, a®b'ct, a®b?c?d*, a?b?c2d?e®, or 1, 45, 210, 1260, 3150, 18900, 113400. Those of my readers, who ‘understand the polynomial theorem, will not have the 192 least difficulty in comprehending the reason of this. For by the two preceding sections, the expressions $1, $2, 83, &c., when 2" is left out, are no other than the developments of the second, fourth, sixth, eighth, &c. powers of Ap + J/g + S71 + wseeee + ./w, with the’ omission of all those terms which contain irrational magnitudes, or, which is the same, the developments of the even powers of p+q+r+.....: + w with the omission of all those terms in which there are odd exponents, and by dividing the exponents in the remaining ones by two. 4. Now, if we put 2”=m, then the required rational equation a” — Ay”? + Br®-* — Caz”> + &. = and the coefficients 4, B, C, &c. are determined by the following equations : | A=S1 2B = AS'1 — S2 8C = BS1 — AS2 + S3 &e. Remark. ‘To this belongs the celebrated problem which Fermat proposed to the analysts of his time, and to the solution of which he more particularly challenged Descartes. It is this from the equation ab = »/(ab — a?) + J (@ + ad + 2*)+ /ma + / (d — a’) — VJ (ar + a’) to take away the irrational magnitudes. It is only | ! necessary to substitute v for ab, and for the compound — magnitudes under the radical signs to put the monomials— Ps % 7) 8,3 then it only remains to make the equation _ v= J/pt J/qt /r+ Vs + 4/t rational, and in the” 193 equation thus obtained, for x to substitute again its values p, q, 7; 8, t. SECTION CVII. 4 4 4 Pros. Make the equation r= a\/p ++ b\/p? bs ASP? rational. 4 e ° Solution 1. Put \/p= y, then this equation is x — ay — by’? — cy? = 0. Now, since y contains four values, viz. + y¥, —¥Y, +y/—1, —y/—1, we get the four following equa- tions, all of which obtain at the same time. v— ay — by? — cy’ ==.0 x + ay i Ges =0 x — ay/—1+ by? + cy?/—1=0 © + ay/—1+4+ by — cy®/—-1= 0 or (x — by?) — (ay + cy’) 0 (v — by?) + (ay + cy’) = 0 (vx + by’) — (ay — cy?) /—1= 0 (x + by?) + (ay — cy?) /—-1=0 The equation sought must .*. be the product of these. g.y itethe two first and the two last be multiplied together, we obtain x? — 2by?x + byt — a’y? — 2acy* — cy® = 0 x + 2by’x + by* + ay? — 2acy* + cy’ = o A 194 or if in these equations we substitute p for 44, [a2 + (b? — 2ac)p| — br + @& + py’? = 0 [a® + (b? — 2ac)p] + (2br + a? + p)y? = 0 3. If we multiply these equations, and then put p for y', we obtain the required rational equation of the fourth degree, a* — 2(b? + 2ac)px? — 4(a? + &p)bpxr + (6? — 2ac)’p® — (a + ep)p = 0 and the four roots of this equation are ap + OV/ph+eV/p" —a\/p + W\/p—eN/p? a\/p _f/—1l— oA /p2—c\/p?. /—I _ a\/p J —1— B\/pe-tc\/p?. /—1 Corollary When .*. an equation of the fourth degree has the form just found, then its four roots may always be determined without any further calculation. I shall now show, that, presupposing the solution of cubic equations to be known, every equation of the fourth degree can have this form. SECTION CVIII. Let -* x — Ax? — Br —C=0 ee be the equation to be solved: it is general, because in ~ every equation the second term, if there be such, may — be omitted. If this equation be identical with that in 3_ 195 of the preceding §, we then must have I. 2p(0?+2ac) = A II. 4bp(2+cCp)= B IIT. (a? +¢*p)*p...(B—2ac)¢p? = C The two first equations give (0? + 2ac)p = WD rit Ga eer and if we make use of these values in the equation ITI, after having previously given it the form | (a? +c*p)?p — (6? + 2ac)*p? + 8ab cp? = C wwe then obtain Be T6Rp 7 From the equation I we also obtain | IV. 4acp = A — 2b’p and the substitution of this value in the equation just found, gives RB? 160° Since the three equations I, II, III, contain four ndeterminate magnitudes a, 6, c, p, we can .*. assume my one of them. Put 6 = 1, then, after getting rid of he denominator, B? — 4A4’p + 32Ap? — 64p® = 16Cp 2 2 + 8ab*cp? = C A ~— + 2dbip — abip? = C | ‘ or, when arranged according to p, Vip — 4 Ap? + i(C+ aap —~B =o } 196 an equation of the third degree, which merely contains the unknown magnitude p. From the equations II and IV we obtain, when we put 6 = 1, and divide the latter by 2 /p, A—2p 2/p and when we add the second to the first, and also subtract the one from the other, then again extract the square root from the sum and the remainder A e+ep= = 2ac /p = ae een oD. A a—c/p= VA Fe Wp Vp) But the four roots in 3 of the preceding §, when we put 6=1, has the following form Vp + (a + cp) \/p Jp — (a + evp) \/p iin ula ex/ BY NY oe eee : SR phic (eet) Ny piten/ aeogll | and when in this we substitute for a + c/p, a = cr/p | their values, we obtain the following roots of the equation | vi — Aa? — Br — ‘i 0: ng) Vp + We Vv (By/p + 24p — 4p’) ae — /p ren ad Coe Jp + 2Ap — 4p’) — vp betes Sp AS Jp + 2A4p — 4p”) 197 Having .*. already determined the value of p from the equation V, we also obtain the roots of the given equa- tion. Besides, it is exactly the same which of the three values of p we make use of, because in each case we must necessarily always get the same roots. SECTION CIX. Pros. Make the equation v= a\/p + b\/p? + \/p? + dV/p rational 5 Solution 1. When we put \/p = Ys 6 — p=, then the equation is transformed into | x — ay — by? — cy® — dy* =o. Now y has five values, viz. ay, Gy, yy, dy, ey, when a, By Y> 8) € denote the five roots of the equation y’ — 1 =0 (unity included) ; we have .:. the five fellow- ing distinct equations : II | one oO r— aay — aby — acy? — atdy! v— Pay — phy’ — Bey? — Bidy* = v— yay — yby’ — rey’ — yidy* = x — day — S&by — Scy> — dtdyt = ex—eay— eby — &cy> — &dyt =o and their product will give the required rational equation, 5 if we again put \/p for y. 2. In fact, this implies no more than to eliminate x from the two equations x — ay — by? — cy’ — dy* =o, _ ¥y—p=o, and consequently in this case all the methods of elimination in the preceding chapter are applicable. 198 If we make use of Cramer’s method as the easiest, we shall arrive at numerical expressions, which exceed the limits of the annexed tables, and .-. must be calculated. But we arrive at this object in a much shorter way by managing the calculation in such a way, that the nume- rical expressions refer only to the roots of unity, because these are more easily calculated. 3. For this purpose we only require from the two equations I. 2—1l=o0 Il. x — ayz — by?2* — cys? — dy*z* = 0 to eliminate the magnitude z ; for if we substitute in IT the five values a, , y, 0, ¢, of 2 from the first, we then obtain the same equations as in 1. 4. In order to be able to apply Cramer’s method of elimination (§ LX-XVI), wegivethe equation IT the form 1 + (1) 2 + (2) 2? + (3) & + (4) 24 =0 by? thn (1)= —2, 2) = -%, (8) = - 2, 4) 5. Since the numerical expressions in the equation, § LVIII, LIX, (c), in the present case relate to the roots of the equation 2° — 1 = 0, then all those in which the sum of the radical exponents is not divisible by five vanish, by 2, § XCV. With reference to this remark, we obtain the following final equation : ee . " a ae SS ae Ee ee 199 o= 1+ (14) 14]+( 247) 247] + (34°) [349] + (4°) [47] + (23) 23 ]+( 374 YL 324 J+ (1248) [124°] + (123)‘[123]-+( 124° )'[ 124? | + (1374?) [13747] + (127)'[12*] + (1234) [1234] + (27347) [27347] + (1°2)1°2]+( 13° )[ 13° J+ (23°4)[23°4] + (15) J+( 2% YL 2% J+ (3) [3] +-( 273? )'[ 2°38? ] +( 1°34) 1°34 ] + (172°4) ‘[172°4] + (17238) ‘[1°237] +( 12°3 )'[12°3] +( 2 )T 2°] ae Co es Se eR ee ee 6. The numerical expressions in this equation may _ also be calculated by § XCV, which, indeed, is not diffi- cult for the present case. If after this again, we put for the symbols (1), (2), (3), (4), their values from 4, likewise p for y°, and multiply the equation by x°, we then obtain o= a? —5ad \ pa’ — 5bd? px —Sdcd°p*x —d*p* ’ —5be ) =o + 5 (ab?d + abe? + ac?d’?p + b’cd?p) p* SECTION CX. The equation v= a\/p + B\ /p? ae A/p? -- n/p! led to an equation of the fifth degree, of the form in 7 of the preceding §; and the five roots of this last equation are consequently aa\/p + a°b\/p? + ac\/p? + atd\/p' r= ea\/p + eb\/p + BN) Ete gan /p' c= ya\/p + yb\/p? Le ye\/p ag fd\/p' 1 = 8a\/p + SO\/p? + S'c\/p? + SdN/p! r= ca\/p a b\/p? + &c\/p? + dd\/p! Therefore, conversely, if an equation of the fifth degree x has the given form, we have its roots in its stead. If. we could reduce every given equation of the fifth degree to this form, we should then have the general solution of equations of this degree. To effect this, it is indispen- 201 sably necessary, from the given coefficients 4, B, C, D, by means of the equations in 7 of the preceding §, to be able to determine the magnitudes a, b, c, d, ps, one of which is arbitrary, in a similar way with that in § CVIII in the case of equations of the fourth degree, and also in § C, where the transformed equation had Cardan’s form. But all the endeavours of the greatest Analysts to attain this object have been fruitless, and we shall see in the sequel, why it must be the case. However, a treatise by Euler on the general solution of equations, and parti- cularly those of the fifth degree, may always be read with _ pleasure and instruction ; it is to be found in the ninth | part of the new Petersburg Commentaries, and also in | the third Part of Michelsen’s Translation of Euler’s | Introduction. | SECTION CXI. Although, however, we cannot obtain the general solution of equations of the fifth degree by the method in \the preceding §, yet there are several particular equations, to which this solution is applicable, of which I shall, with Euler, only adduce those which do not lead to very com- plicated forms. | I. If in the equations in 7, § CIX, we put c= 0, d =o, we then have ; A =o, B= 5abp, C = 5a°bp, D —_ ap +. b°p. From the second and third of these equations we obtain } 5 ae a, Oe ee B? | 0 An ie are hence C? 3 D=— 5B Pa 25C Also 5 5/ C2 5 5/ B3 4 — pees b 2=— Sp =/S, Wy SE, if .:. the equation C? BS eee 6 oie ie ate! Oxy, pee ead ek ares 5 i A a 5B 25C be given, then C? BP aalts diy fae abe aA aa +a ioe G is one of its roots, and the remaining roots are obtained by substituting @, y, 6, « successively for a. We should in like manner have found the same equa- tion and the same roots, if we had put a and 8, or a and c, orb andd=o. Thus if we put =o, andd=o, we have A=0, B= 5ecp,.C = dac*p* D= a’p + cp’. From the second and third equation we obtain Ie ay alle Cp a TP 35C 5B and these give B> C2 Die ee 250+ 5B 5 o 5 4 Bs 5 tee 5/€2 | /p ne V oe Mp To hye | We have .:, again the equation B C Pe ta abe Sih fou dg ay | TT pataca a OaraS gaa 203 and one of its roots = aa\/p +a p* = 2 oN S The remaining ones are obtained by substituting 8, 7; O5 &> successively for a. Moreover, that the five roots, which we find by these means, are not different from those already found, we may easily convince ourselves by putting a’, a’, a*, a ( = 1) for, y, 0, « (§ LX XXVII). II. If in the equations in 7, § CVIII, we put b=0, and c=o, we obtain A=5adp, B=0, C= — 50d’ p’, D —_— ap + d°p* The first and third of these equations give pte at 5 fas the fourth gives = (ap + dp)? = (a'p — dp')? + Aaid’p* A 5 sir — dp) 2 nt = (ap — d’p*) + 4(4) " consequently A 5 a’p — dp* = v[p- 4 (=) ] Now since ap + dp = D then A 5 ép=4D+ v[D'— (=) ] Ci 1D— V[ 4D? _ (Ahi 204 and .°. oN/p = Nat vu -(4)'T] dN/ p= \/[yD- J pp—(4)'] Consequently, if the equation (9) v— At +2 —D=o then each of its roots is expressed by 5 5 o/ [D+ VED (2) ]]+ 5 5 ANS 4 hy ype Fe) ee en/an= van (4) This root resembles very much, as we see, that which Cardan’s formula gives for equations of the third degree. Besides this equation belongs to a peculiar class of par- ticular equations of all degrees, the solution of which was first taught by Moivre, and of which we shall treat hereafter. 3 SECTION CXII. In the same way as in § CX the equation v= aN/p + BN/p° 3 oN/p! + dX/p' was made rational, every other equation of the form v=aN/p+bN/ p2-+cN/ p? + PP H 13 “+ XJ pt! may generally be made rational, and the degree of the rational equation will always be equal to the radical index. In this there is no other difficulty than the trouble of the calculation. Hauber has omitted this operation when n= 6 205 [See the Second Collection of Combination Analytical ‘Treatises, p. 248]. It would be desirable, if it could also be done with other values of , because from them, as has been already shown by a few examples, the solutions of a great number of particular equations might be derived, which are so much the more worthy of observation, be- cause they cannot be analyzed; for otherwise there could not be in the roots any radicals of the same degree as the equations themselves. Yet a great number of par- ticular cases of the same kind may be found without such complicated calculations, by omitting at the very beginning, several terms in the general irrational expres- sion aN] p+ 0N/ p+ 0N/ p94 oss tIN/ ps as the fol- lowing problems will show. SECTION CXIII. Pros. Make the equation « = aN/ p + bx/ p* Ya- tional. Solution. Here the two cases, in which n is an even, and the other where nis an odd number, must be dis- tinguished. First Case. 2m 2m 1. Let x = aX/ p + bN/ p’?; further, let a, Bs ¥> 6, &c. be the roots of the equation a2"”—1=—o0, If y : : | 2m ‘be substituted for \/ p> then the different values, which & has in relation to these irrational magnitudes, are . - ae. - oe —_. 206 aay + a? by” Bay + Bby? 2h 9/2 yay + by : &e. ‘ as far as 2m. ‘The required rational equation is repre- sented by i f2m—1 af otha NGS Smet ¥) oO Zn ~~ ade Qn— i a2” — Ax + Ax Ax ita ao \ ™ m+t i Age Ax as AGT ib degsasber eens: ‘ 2n—1 2m . | — Ar + A=o0 the upper signs obtain when m is even, and the lower 5 } 2 3 when m is odd. The coefficients A, A, A, &c. are then: the sums of the former values of x, taken singly, two and ~ two, three and three, &c. 2, The developement of these combinations gives A="[1ay + “[2) oy A="1"| ay? 4+ [12] aby® =|. 127] b? y* I A 3 A= [Iay® + [1°2] @by* + [1227] abey> + [2°] b°y? ‘ 8. But it is evident from 2, § XCVI, that all the 1 2 3 m | coefficients 4 A A, as far as A &c. vanish, because the sum of the radical exponents in each numerical expres- sion, is always < 2m, and consequently cannot be divisible by 2m; this was evident before, from this cons sideration, that in the required equation there are only such powers of y, as are divisible by 2m, for otherwise 207 it could not be rational. It only remains to find the m m+1 m+2 Qm cocflicients A A, 4,...... A. 4. When in the terms of which these coefficients con- sist, we omit all those, which by 2, § XCVI, = 0, we then find yj — 127 b”y°” A — { erat a 6"! yn A _ ‘ [1 4 oe | a’ OF a= A — ey ed eoray A — fa orig A a? by” 17a eel any” + [2%] ll 5. But (§ XCVI). BSc ayer De mtg amie te EB To Mer Sais #b eh a4 roto sia a re 4 ea ‘ heya: Lr ee yO UE 2 Baia ti 4 5 eT ea te 4 i ee Sr so eoarterteontete - m+3 Ue leg a ier =! Vals IRCCS te A RA pha cL | Pts G89: naa ee —_m.m—1.m?—4.m°>—9 Ce CO. Las © e e e e ° ° . ° e e ° e. e 9 @ 1, See ok Im—2 a7 42-29] — 41. -__ QIN [1 2] 5 awe is dactoue Q2m—2 nm — ee m? —m— 2” = , 2m 1 e 2 ry 3 e 4, esceceecvesceen aeeo 2m—2 Li, 2 Oo eaeee. PP erate I9nm— 1 Mh bes a LE RMI EE McA, SEK a RNS LY ag [ ] Tt Bare eee cee ct vas ent (1.2.3...m—1)? 1.2.3...2m—1 ‘(2°"] = +3 (2m)? — 2 ? J=+G2..mpx1ae™ 128 eon =+2—-1=>4+1 in which the upper signs obtain when m is an even, and the lower when x is an odd number. 6. If we substitute these values in the expressions for m m+1 m+2 A, A, A, &c., and then again put p for y’”, we obtain the required rational equation 0 — 2b” px” — —; Bm va. 0°e pres m.m — 1 ET FORME HA m.a' _—s gered TRBT ST aie m.em*—1.m?—4 paces fe Se Le RE) a 6 Jm—3 M—3 1288 456 tO pe m.em—1.m—4.m—9 a Ss EON Pr eh at 0, 8 Jym—4 m—$3 [ico WSR BO 6 eons ee ht 2 m.m—1.m—4.m?—9...m’—m—2 eng his Sal Eon Wee EEE Gy a Fe Qt? \ 112 OLS BE TU oe bpa yet ap b?"p? pete 209 Second Case. 2m-+1 2m+1 ‘ 7. Now, let r=a p+iN/ p’, or y be substituted 2m-+4-1 for \/ p, e=ay+by*. Further, let 1 2 3 pat! Ay 4 Ag — Ag occas __ mrt Mm+2 __ m+3 e.-e e068 + Ax” + Ay! + Ax”? + eee eee Qm Qn+1 ; eevee + Av—A=0 be the required equation, in which the upper signs obtain, when m is even, and the lower when it is odd. Then, as 1 2 3 m+ before, all the coefficients 4, 4, A, &c. as far as A vanish, and we have the numerical expressions referred to the equation v*”"t!—1=0, m+ aie | 00; Yo m+-2 A _— Hleeaan | rad Cocos Fg m+3 A — [beeing be ate m+4 A= a | 1 om abe yaa i = ieee SI ae byt! rae ee Gaye ef. (2a oe yet 8. But De) =k ea et eta ‘eo some ang Er Sa RSE ARP me [uz Biers S ; Al m—1 ee amt = Fut em+l SE 1 6 BeBe tebe tiees- m+2 ma of ee ae |= tT is ot ees ie m.m—l1 -m+2 TERE a we pT a RE os “nin Sap Pe bovees m+ 3 ‘(172m |= .2m+1 [ | 173.3. nae x 1.2.3.4.5.6.7 s; —m.m—1.m—4.m+3 AS nh i BOR CaM PTA + Ting was SUSE ee Le aee ae Veer eine tee Im—1 \ lmn—* Sate oe! ti tain a CERs tee S es tabi Dep eid aN I . ] [1 2] ye ed) Pe Re 2m— 1 sh dae 2 m.m=n1.m—4.m7—9...m—m—-2 .2m—1 = .2m-4-1 ] eo Bi tohs hee testes Q2m—1 Pe he PO ess fee Fares 11) 17°?) erie 1= +1 i Se a 2 e 3 eeeseee eereeaee8 2m+1 dd oi [ae —_ alg 9. If these values be regularly substituted, we then obtain the required equation amt? — (2m +1) ab”px m.m+1 aaa Ages 1 ° le art 1, 2ropee age mre m.m—1.m+2 si Fa Tinh Sat “IS PY 1. a&b"—px - PCT UREN A TE ee m.m—1.m—4.m+3 “2 ; 1.a7h" ny" ipa wR BRUAE MOLLY ch einen rs m.m—1.m?—4.m—9.m-+ 4 Fes. 2 1: 9},m—4,,, .m—4 1 aS RUaM RVG OHT NBUTOS cutie: Mae e e e e ——2 mm=—1m—A.m=9,...m—m=2.2m—1 alas 1] 2.3% A he Ae ead eae one 7 arty — 5: itty —0 211 SECTION CXIVe Conversely, if equations of the form in 6 and 9 of the preceding § be given, we may always find their roots. As examples, and for the sake of their use, I shall here give a few equations of this kind. I. When m=2, we obtain from 9 of the preceding §, the equation 2° — 5ab’pa? — 5a°bpx — ap — b’p? = 0 and each of its roots aa\/p + 20N/p* when a is a root of the equation 2°...1 = 0. Moreover, this equation is the same as that in I, § CXI, which was derived from the general equation of the fifth degree in § CX. II. When m=3, we obtain from 6 of the preceding § the ae «°— 2b°px* — ga*b*px? — 6atbpx —a®p + b°p? = o the roots of which are expressed by aaX/ p+ ab\/ p? ‘when a denotes a root of the equation 2°— 1 =o. } f Further, for the same value of m we obtain from 9 of the preceding §, the equation — Tab* px? — 140°b* px? — Ta’ bpx — a’p — bly? = 0 and each of its roots aa\/ pt ab\/ p? _when a denotes a root of the equation 27—1=o. 212 III. When m=4, we obtain from 6 of the preceding § the equation a8 — 2b4px* — 16a°%b*® px? — 20a*b?px? - — 8a%bpa — a®p + bp? = o and for each of its roots aa/ p+ a\/p° Further, for the same value of m, we obtain from 9 the equation a? — Qgab‘pat — 30a°b*pa® — 27a°b?px® — ga‘bpx — ap — bp? = 0 and for each of its roots 9 9 aa\/ p + abN/ p? and so on. SECTION CXV. Pros. Make the equation x = a\/ pt b\/ pt | rational. Solution 1. If we denote the roots of the equation | a"—1=0 by a, 8, 7, 6, &c. then the roots of the required. rational equation are aaX/p+a'0N/ gaN/ p +2" 0N/ p> ya\/ pyN/ p> &e. : | | / 213 or since a” = @" = y" = &. = 1, aaX/ p+ -0N/ p gaX/ p+ e-WN/ Vim y\/ p+ y7WN/ p> &e. Hence we could derive this equation in the same way as in § CXIII; the following method, however, which has been often used already, leads to the object in a shorter way. 2. Denote the sum of the first, second, third, and so on, powers of these roots by S11, $2, $3, &c.; then St = ‘(1Ja\/ p+ [—1\/ p> 52 = [2]aN/p? + 2'[olabp + —2]8-N/ p> 83 = \[3]oX/p?+3'[1]a°%p\/ p+ oN Farr Tab pN/ py" + [3] VV p> S4 = ‘[4] a\/ p+ 4°[2]a°%bpN/ p? 4-6 [o]a*b?p? +4 [—2[ab%p/ Pei A}btp'\/ pa &e. 3. From the form of these values, and from § XCVI, it follews, that the expressions S11, 3, S5, &c. = 0, and that generally cach expression Su=o, when y is an odd number and less than ». Further, since o] = 7, we | have S2 = “nabp S4 = di na??? io eZ S6 = 6.5. * nap? 7 ee i" PR Ay NS See eee and generally -2u-1. mm Zc ceee S2u = Ebb fA dle eit Rena Let. Bite B ceecccveee fl whilst 2u — abp — 5p = 0 and each of its roots is 6 ad \/ p+eon/p Remark. Compare Michelsen’s Translation of Euler’s Introduction, third Book, pp. 10, 11, with this §. Euler finds the same equation, in a shorter, but less analytical way ; what in his method is\/ @ and «, in nine is abp and a’p + b’p"-’. Compare also with it Tuguenin’s Mathematical Contributions for the further ‘mprovement of the young Geometrician, p. 181, and } 0 on. | } If we put nabp= 4, a'p+ bp = 7 t ‘hen the above general equation, when z is an odd number, | Qe 218 is transformed into n.n—3 7 : | | the first part of this equation being continued till we come to a coefficient = 0. But from the two equations nabp = A, ap + b’p”' = T, we obtain | (a’p — Bp? = (ap + Dip") — 4a°bip" | =— 7? — 4A i n” | consequently 4A” a'p — bp = V(T?— ~5) | If we combine this equation with the one a’p + bp = T, we obtain, by addition and subtraction, ap. 4T 4 pyres) oy sii: Fl Sk — and when we extract the nth root : n n i 4.A" A/p =N/ T+ iV (T?- DI : : 4A” N/ p= x/ [47-37 hha. cape Therefore ‘ Hing: ease a ely 4A. A/arey v(t N TAY To is the general expression for every root of the abot equation 2"— 4x"~?+ &c, = 0. / fy 219 From the resemblance of this formula to Cardan’s, it follows, that the equation of Moivre is only an extension of Cardan’s, and that both may be deduced in the same way, as is actually shown in the two above-mentioned works. SECTION CXVI. Pros. Make the equation x= a\/ pt tN/ p= rational, in the case in which n is an odd number, and not divi- sible by three. Solution 1. Let 1 2 3 n wa Ae & Ag + Ar + cee oe A= ye the required equation, whose roots consequently are aa\/, P +a ON / ON: aaN/ pt = tp\/ = n R n 1 % 1 gaN/ p + e—0\/ p- or ga\/p + RN &e. &e. Further, let the symbols $i, $2, S3, &c. have the same ignification in reference to these roots, as in the preced- og §. 2. Any undetermined power k of the first of the above oots contains, when a only is considered, the following erms : ; h—3 M6 kg —(U—3) ~ _—2t 9 @ 9 Sa pestcve a a a‘, a de same terms contain also, with reference to @, and so on, ae power k of the second root. Consequently $k, when 220 a, By yy &c. only are considered, consists of the following terms : ‘{k], ‘(k—3], ‘[k+6], «..... ‘{—2k+3], [—2k] 3. Now, if k = B. SECTION CXVII. In the same way as in § CXIII, § CXV, and § CXVI, we can derive innumerable other general forms of solvible equations from the binomial \/ p+ V/; p’ by eliminating the irrational magnitudes. However, since otherwise this subject possesses no interest, I shall here content myself with giving an equation, which Waring, one of the most celebrated Analysts that England ever possessed, gives” in a treatise on the general solution of equations, by assuming oN pt o\/ p® (Philosophical Transactions for the year 1779, p. 92). When n is odd, this equa- tion is 225 x” — p[na’*bx? n.n—T.n— Saba + 9.n—10.%—11 | Re A hee CIEE es Te + nm.n—11.n—12.n—13.n—14 5 n—15}5 +10 &e. od lah 3. Yai gee at ra + Sig n—3 ntl 1 n n 5 PA n+3 SGN es UR i arnt Die remener ye : — a?5*z 1 n.n—T.n—9 .n—11.n—13 ee ee CS ERY, ENT eh POET + 1 2.n—9 .n—-11 .m—13.n—15.n—17.n—19 = nar) ea tap ee eee ae: H,® oF Re —a"p — b’p®? = o. gt '2h378 + The factor p? has the sign +, when "is a whole number, but in other cases the sign —. SECTION CXVIII. The method by which equations of the first degree are made rational, which hitherto we have chiefly used for finding solvable equations, can also be used with advantage when we merely wish to clear the unknown magnitudes in ‘an equation of the radical sign. I shall elucidate this by _two examples. Let the equation PEN GEN tN oe tVV teen =O ibe given, and let p, g, r, s, &c. be rational functions of | ‘the unknown magnitudes y, z, &c.: it is required to make ‘this equation rational. 226 Put—p = v; then we have in reference to x the equa tion of the first degree, viz. =N/g thf rt 0/40 bun We try to make this equation rational, and then again put — p for x, we then have the required equation clear of irrational magnitudes. If the equation NY Ara llr RIN ERK ANG 3 Pe: =0 be given, in which there is not even one rational term, , 9 we put — \/ p=a, and then make the equation ZUMA q. 7 NA + X/ s + ..,... rational. Having done this, we ; substitute in the obtained equation for x successively its values PONY, Ds EN) Ps Le / p, &c.; then there arise p equations. If we multiply these together, we then — obtain the required equation clear of irrational magnitudes. On the subject of clearing equations of irrational magni- tudes, there is a very able treatise by Fischer, in the Hindenburg Archives of Mathematics, Number VIII, written with that clearness and perspicuity peculiar to the author. | The rational equation which is obtained from the equa- tion v= N/ p+8/q+N/r-+N/ 5 + ...so long as tha irrational magnitudes \/», \/ 4, N/r, Nee &c. have no particular relation to each other, always rises to the degree fLviro-.-., because the uw values of Vp. the v vales of Vai the w values of NY, and so on, may be combined MS 27 together exactly this number of times. But if the above irrational magnitudes have any relation to each other, so that if one or other is determined, the remaining ones are either all, or only in part determined, then the rational equation is always of a lower degree. Here follow a few examples by way of elucidation. _ The rational equation for «= x/ pt \/ peeN/ Pp? +N/ ptt oe +N/ p* only rises to the nth degree, because Mp? = (N/p)% N/p? = (N/), N/ pt = (N/ py, Be. and consequently x contains no more values than the 1 irrational magnitudes / Dp. m 12 3 5 The rational equation for 7 = N/n+-'/p+/¢ only rises, but necessarily, to the sixtieth degree, because 2 12 12 V p=(V/ p)*, and the twelve values of s/ p, combined 5 with the five values of / q, gives sixty different values of x. 1 9 8 _ The rational equation for x = \/p + \/pi te \/p 5 7 8 =p + p® + p’ only rises to the seventy-second degree. For if we reduce the small fractional Ca nat to the 56 a least common denominator, we have x = pee poe+p. [f ...« be a root of the equation y’?—1=0, then is 30 56 FTF 4 56 y7F 4 ip? Sor avo /n5 54 a5\ /pt 4 a2 /pi the corresponding value of x, and_ there are ag of these values. 228. BECTION CXIX. Pros. Find a factor, by which the given irrational expression PEN GAN PANS 8 # scene must be multi- _ plied, in order to make it rational. Solution. Let a, b, c, d, &c. denote the different values - ‘which this expression has, when the irrational magnitudes — are taken in all possible ways. Now, if we form the equation «= p + \/q + Vr +/+ Wes , then the rational equation derived from it, is (vx—a) (r—b) (x—c) (w—d) ...... = 0 and its last term = + abcd...... Now, since this product must be rational, it follows that the given expression is rational, when we multiply it by the product _ of all the remaining expressions which its different values give ; and this product consequently is the factor sought, Exampie I. Let the expression pt be given, If then 1, a, 6, be the three roots of the equation y° — 1 =o, then the required factor = (p+oX/4) (p+@\/4) or, sinceea + B= —1, a= 1, = p?—pr\/qtv/¢ Exampre IJ. For the expression p + /q + sr, the required factor 229 = (pt /q—vr) (p—Vqt Vr) (p— V9—- V7) =p —pqa—pr—-(’—qtnvq- (pe +9q-—-7) /r+ 2p J/ gr which is already known. ’ 3 4 Exampce III. For the expression pt+\/q-V75 when 1, «, , are the roots of the equation y’—1=0, we obtain the following factor : (pt qt) (p ta\/q+/r) (pt \/q+0/r) (p+\qeVr. /—1) (pta\/q+N/r- /—1) (te\/at\/r. V—1) (pt\q—-V/r V1) (pta\/q—V/r . V1) (p+8\/q-N/r. V1) (pta\/q—-V/) (p+ 8\/q—N/r) Remark. When p=o, a more simple factor may often be found, by which the object to make the given expres- sion rational may be attamed. Thus, if the expression ) V/. gM) r be already rational, when we multiply it only | by the factor (\/q 4s / 1) (M4 4 e\/r) im WA 2 | or +-\/rs and the expression /q+ /r+ 4/s is 'so by multiplying by the factor (Vg + /r — 4/s) _case, when the indices of the roots have a common divisor, or the irrational magnitudes have a certain relation to each other. } Corollary. From what has been already said, it follows, ‘that it is always possible, by the multiplication of the (Vq— Vrs Vs) (V/q— V/r— V/s). This is always the 230 numerator and denominator of a given fraction by a | proper factor, to clear the denominator of irrational mag- | nitudes. Consequently also an equation of the form i Xe, VatNn sa Vrty is Vs +o we may always be reduced | p FE / $N/r NJ! ries to an equation of the form ACU PETER iH and as one of this kind can always be made rational by the preceding §, so in like manner the former may always be made rational. SECTION CXX. Prox. Make the equation v = /p + vq rational, when the magnitudes p and g are not immediately given, but only assumed to be the roots of an equation of the second degree y* — dy + B=o. Solution. The values which x has by the different. determination of its irrational magnitudes, are J/p+/q—-Vpt+v/q DW Dean) DN G p The mere inspection of these values shows, that when we substitute p and q for each other, they undergo no further — change, than that one is transformed into the other. Consequently in the rational equation derived from x= /p + /q there is no change, when we substitute p for q, and it must .*. necessarily be a symmetrical function of these magnitudes, and consequently may be expressed rationally by the coefficients 4, B. Wf .:. we eliminate 251 p and q by means of the given coefficients 4, 2, we obtain the required rational equation. But from vx = /p + /q we obtain the equation (§ XCVIII.) 4 —2(pt qa? + (p+q)? =0 or vi —2(p + 4)x* + (p+) —4pq=0 If in the latter equation we substitute for p+qand pq their values 4 and B, we obtain the required equation a*—2 Av? + 427—4B=0 SECTION CXxXI. Pros. Make the equation repeat never ‘rational ; when the m magnitudes p, q, 1; 5, eo, are not immediately given, but only assumed to be the roots of a given equation of the mth degree y” — Ay" + By"? — Cy” + &s = 0 Solution. If we endeavour to find all the possible values of x, which arise from the different combinations of the values of the roots, and then in these put the jmagnitudes p, q, r, s, &c. for one another in any way, but et it be the same one in all the values, the consequence will only be this, that these values either undergo no shange, or merely that one is transformed into another. Por let a, 8, y, 6, &c. be the roots of the equation be = 1 =o, and a\/p+BV/q+y\/r+9\/s+ &e. any value of x. Now, if it be possible, that from this expres- 232 sion, by any substitution of the magnitudes p, 9, 7, s, &c. for one another, viz. by that of p for g, another expression a\/q-+B\/} Dy) r+ 3\/s-+ &e. is generated, which does not belong to the values of 2; then there must be, contrary to the supposition, a combination of the values of the roots, which is not included in the values of z. Since .*. the values of x remain the same, however the ‘magnitudes p, g, 7, s, &e. are subtituted for one another, then must the rational equation derived from x = Vi p tn /qtr—/rtn/s+ be. be a symmetrical function of these magnitudes, and consequently may be expressed rationally by 4A, B, C, &c. the coefficients of the given equation. If .*. we eliminate the magnitudes p, g, 7; s, &c. by means of these coefficients, we obtain the required equation. Corollary. "The equation, which we obtain under the | condition of the problem, is consequently always of the | i i same degree as the equation derived from x = V/) p+ \/q + \/r + v/s + &c. when the magnitudes p, q, 7, s, &e.| are independent of each other. In the latter case, how-. ever, the rational equation is of the degree nnnn... = n”, when m denotes the number of the magnitudes p, gq, rs s, &. (§ CXVIIT), consequently also in the former. SECTION CXXII. Rute The rational equation for PaVpt Vt V/rt Vet be. 233 in the case in which the m magnitudes p, q, 7, s, &¢. are either wholly independent of each other, or the roots of an equation of the mth degree, can only contain such powers of x, whose exponents are divisible by n. Proor. Let 1 2 cot, Lge Oo Poe ee At Se be the equation, which arises from the multiplication of all the positive distinct equations of the form | t—a\ /p—B\/g—y\/r—S\/o— eee Cet O ‘where a, £, y, 8, &c. denote the roots of the equation ‘v’—1=0; now it is evident, that the undetermined coefti- s ic . ° ° cient A can contain no other numerical expressions of these ‘roots, but those in which the sum of the radical expo- nents =p. Now, since these always vanish, when | p. 1s not divisible by » (§ XCVI. 2), then likewise must the term 41“ always vanish. But when p is not divisible by x, then alsok—p cannot be divisible by m, because k=n”; .*. all those terms vanish which contain k—np, Ie ccctly the exponent of x is not divisible by n. Therefore the rational equation contains those terms only, in which the exponent of x is divisible by n. | 'Q. E. D. SECTION CXxXIII. Pros. Let the m magnitudes y, z, t, u, &e. be given by the m equations oon 234 oe ll ° y* + Ay + By + Cyt + &e, 2 + Alert + Ber? 4 Cv + &. = 0 AM + Bl 4+ Clr + &e. = 0 &e. consequently irrational : required to make the equation w=yte2et+t+ut &. rational. Solution. Since y is given by an equation of the uth degree, z by an equation of the rth degree, and so on, we have p values for y, v values for z, and so on. If we combine all these values in as many ways as possible, to the number y + 2 + ¢-+u-+ &c., we find all the values” of x, and consequently by the multiplication of all the distinct equations of the form © —(y+2 ott+ut kop -= o the required equation, which necessarily is of the ree uva..-, because the different values of y, 2, t u, &c. may be combined in this number of ways. Now, I assert, that with respect to the different values of which may be 7, y”, y/”, &c. this equation is symme+ trical. For since the values of x in the substitution of these magnitudes, undergo no further change, than that one is transformed into another, consequently, also the equation itself must be such, that it ae a no change in the substitution of the magnitudes, y’, ys al", &e, Therefore, the magnitudes y’, y’, y/’, &c. may be eliminated by means of the coefficients A, B, C, &e, What has been said here of y and its different values 4 y’, y//, &e. applies also to 2, t, &c. and its values z, 1, 2", &e. v, t/, t, &e. &e.; and consequently these 235 magnitudes may also be eliminated by means of the coeflicients 4’, B’, C’, &c. 4”, B”’, OC”, &e.&c. In this way we likewise obtain a rational equation for x, which only contains known magnitudes ; and this is the equation sought. Examp.e. Let c=y+z2; let the magnitudes y and z be given by the two equations y— 4f + By—C=0 2— Az+ B’=0 required to find the rational equation for x. The different values of x are yf tay tal. yl! ts z, yl! bs race’ Yb fy yl & al! ‘Denote the sum of the first, second, third, &c. powers of these roots by S1, $2, 83, &c. we then have Si = 2(y/+y"+y) + 3(2/4+2") = 24 + 8A’ i §o = 2 (y+ y/2 ty!) + 2 (4 ty ty”) +2) 43 (22 42/2) = 2 (4° — 2B) 4+ 244 4+ 3(A? — 2B’) 83 = 2(y?+y? +9) 4 3(y2 ty? ty”) (/ +2/) 4 3(y ty! ty”) (2422) + 82% + 2) = 2 (4°— 3AB + 3C) + 3 (42 — 2B) A’ + 3d (A? — 2B’) + (84° — 38A/B’) &e. Thus having calculated the values of S1, S2, 83, &c. when the required equation is represented by 1 2 3 4 5 6 x —~ 42° + 4at*— da + de?— Ax + A=o 236 we then obtain its coefficients, by means of the equations: 1 A= S51 1 dA i AS! — §2 2 &e. SECTION CXXIV. Pros. Let f: (y) (z) @) @)...--- denote any rational — function of the magnitudes y, z, t, u, &c., which are - represented by the same number of equations yt + Ay"! + By + Cyr + &. = 0 | 3 + Bo" + Bo? + Ce + &.=0 * + dit + Bir ?* + Cir + &. = 0 find an equation for the values of this function. Solution. Put x = f: (y) (2) (t) (u)..., find all the possible values of this function, and from these form the equation for x: then eliminate the values y’, y/, 9/, &¢. | of y, by means of the coefficients 4, B, C, &c. of that equation by which this magnitude is given; which may always be done, because the equation for x must necessarily be a symmetrical function of the magnitudes y/, y, y”, &e. If we proceed with z, t, u, &c. in the same way as we did with y, we obtain an equation for x, whose coefficients are all known; and this is the required equa- tion. Corollary. In order to find all the values of z, we must combine the j values of y, the v values of z, and so on, in all possible ways in the function f: (y) (z) () (uw)... But it is evident, that the number of these com- binations = v7... ; consequently also this product gives the number of the values of x, and .-. the degree of the transformed equation. If the function only contains the magnitude y, or if «=f: (y), then the equation for y is only of the uth degree, consequently in this case, the transformed equa- tion is of the same degree, as the equation by which y is _given. ; | SECTION CXXvV. > Pros. The unknown magnitude x is given by the equation of the nth degree, viz. a* + Pot + Qr? 4+ Ro + &. =o the coefficients P, Q, &c. however, are not given imme- diately, but merely all assumed to be known functions of a magnitude y, which depends on an equation of the nth y* + Ay + Bye? + Cyr? + &e. = 0 required to find an equation for x, which only contains \known magnitudes. t + Solution. Denote the roots of the equation y* + Ay" + &e. = 0 by 7, y’, y”, &e., and introduce these values into the functions P, Q, &e. Now if we denote that, into which these functions are transformed, when we substitute in them y’, y”, y”, &e. successively for y, by P’, Q’, &., P’, Q”, &e. P”’, Q”, &. &e. ; we then 238 obtain the following j: equations, all of which must obtain at the same time : . go + Pla! + Qe? + Ba + &. = 0 at Plgt + QM"? + RYMx" + &. = 0 a + PU 4+ Qa? 4+ RY + &e. = 0 &e. The product of these .*. gives the required equation. But since these equations are such, that in the trans-_ position of the magnitudes y’, y”, y/”, &c. no other: change takes place, but that one is merely transformed to another ; consequently their product suffers no change in their transposition, and .-. with reference to y’, 9/ ayia . &e. they are symmetrical. Therefore these magnitudes may always be eliminated by means of the given coeffi- cients A, B, C, &e. Corollary. But conversely, every equation of the nut degree, which can be considered as arising from the elimi: nation of y in the two equations 2” + Px" + &. = 9; y* + Ay# + &c. = 0, may always be solved, if the solution of equations of the uth and pth degrees be pre- supposed ; for the second equation gives the value of and when we substitute this value in the first equation, then the latter gives the value of x. : SECTION CXXVI. equation of the nth degree, viz. x” +. Py +f Oi an Rat a &e. =— 0 239 It is assumed, that the coefficients P, Q, R, &c. are functions of the magnitudes y, 2, t, u, &c., and that these magnitudes are given by the equations | Os 6. 16 y* + Ay + Bye ?* + Cy+ + &e. 3 + A’ 4+ Bor? + Cer + Se. tt Je A’ po -f By + CCR a4 &e. &e. Find an equation for x, which contains known magnitudes l only. Solution. In the first place we consider P, Q, R, &c. ‘merely as functions of 7, and eliminate these magnitudes iby the method in the preceding §; then we obtain an equation for x, of the muth degree, which only contains the unknown magnitudes z, ¢, u, &c. If in the same way we also eliminate the magnitudes gz, t, u, &e. successively, we shall at length get an equation of the nuyz...th degree, which only contains x and the known magnitudes 4, B, C, &. A’, B’, C4, &e.,: AW, BY, C’, &e. &c. and which consequently is the equation sought. ( 240 ) VI.—GENERAL INQUIRIES RESPECTING THE TRANS- FORMATION OF EQUATIONS. SECTION CXXVII : AT the end of the fourth chapter, the transformation of equations and 'T’schirnhausen’s method were mentioned, and its application to the solution of equations of the third and fourth degrees. This is now the proper place to give some deeper inquiries respecting it, in order that we may ascertain, what may be expected of this method in its | application to equations of higher degrees. SECTION CXXVIII. Pros. Let a +: Age 4+ Br + Cr’? + &. = oO be the given equation, and et ar + be. the + ly the auxiliary one; consequently the transformed equation for y is of the nth degree (§ LX X XI), viz. + Gy’ + By’? + Cy’ + &. =0 Show in ee dimension the coefficients a, b, c, &c. of thee | assumed equation enter into the coefficients A, 38, €. 241 Solution. If a’, a, a/”, &e. denote the roots of the given equation, then the ‘natin equation y” + Ay’ + &c. = 0 has the following roots : a al ba HL oceans “bh + yl! a alin ae ax" J by//"—? + a | Kis + l UA pln a qyzllin-t al by!" ae Sirah bs an heal! 4 l &e. Now, since the coefficients @, 38, €, &c. are the sums of these roots taken singly, two and two, three and three, and soon, we may .*. conclude, that the letters a, b, c, &. occur in @ in the first dimension, in 38 in the second, i in ¢ in the third, and, generally, 1 in the pth coefficient in the pth power. “ SECTION CXXIxX. Pros. Required to find of what degree the auxiliary | equation | a™ + ar™! + by" + pees ko +lay umust be, when it is possible to transform the general equa- tion of the nth degree s+ Ax + Br’? + Cr*? + &.=0 ‘into an equation of two terms of the form y —V=o0 Solution. In the equation y’ — V=o0, n—1 terms are wanting; .*. the auxiliary equation contains as many undetermined magnitudes a, b, c, &¢c. by determining ‘which, we are enabled to eliminate these terms ; it must ppeequently be of the n— 1th degree, and .°. en: 21 —————— = 242 SECTION CXXxX. Pros. To transform the given equation 1° — Aa? +- Bx — C = 0 into one of two terms y® — V=o, we must assume the auxiliary equation a? + ar + b= y (preceding §) : determining, a priori, the degree of the equations, on which the coefficients a, b, depend. Solution.1. If we denote one of the two primitive roots of the equation y>—1=0 by a, and put / V=y9 then ’, ay’, «%y/ are the three roots of the equation y3 —V=o. Each of the three roots 2’, a’, x!"’, of the given equation corresponds to one of the values of y:_ which ? It remains undetermined. 2. If we combine the values of z in all possible ways with the values of y, we then obtain the six following — combinations : (3 ay’ s “s (" ay’, =) a, a’, yl J? v’, all, alt | / yin e-y / / 2,,/ of be, aY yg ay ( » AY, wy : a, x’, wae w!, xl”, A i / Das es / / . ale Hy Y's a4 ‘ a, x, hii]? gl! all, at e 3. If the values of x and y, which correspond to each — other in the first combination, be substituted in the auxiliary equation, we obtain the three equations : af? +ar +be y' oy all? + aa’ + b= ay’ , oy V2. gall 4. b = aly! | | 24 as and by means of these equations the values of a and } may be determined from a’, x”, 2”. In the first place, in order to determine a, multiply the second equation by #, and the third by 2°, and then add it to the first; thus, since a4 = a3,a =a, andl +a + a? — Tl] = 0; vA eal 4 2yll2 4 a(a! + aa 4 ay!) — 9 we obtain a’? a ay//2 ae ary!//2 der oer + an! + gigi 4, From the remaining five combinations in 2, there may be found five other values ofa. To effect this, however, it is not necessary to begin the calculation anew : for since the above combinations only differ in this, that the roots v, «’, 2/” are transposed, we can even take the trans- dosition found in the expression for a. By these means ve obtain the six following values. a? Pee From these functions the equations for p and q may be /actually found by the method given in the third chapter. SECTION CXXXIV. The results in the two foregoing Sections may also be ‘immediately derived from considering the equations (d) an 2, § CXXXII. Thus, in § CX XXII, in order to ‘combine the four values of x with the four values of y mm all possible ways, instead of pre-supposing a transforma- tion of the former, as was there done, we assume a trans- formation of the latter. The equations (¢), by the first ‘as well as the other transformation, undergo twenty-four ‘changes; and since each such change gives a value of a, We obtain twenty-four values of a, of which, as we have already seen, no more than six are different. This conclusion, however, might have been foreseen, a priori, without knowing the value of a. Amongst the ‘twenty-four combinations of the roots a/, 27 , ve’, x!% with Yo~y’, y//—1,—7/./—1, there are also the following 256 x8 4 a72 +b +o= yf / — I a/3 4 aa’? + ba” +0 = —y' V1 P34. al. bell +o zy! 34 a/@t br tco= and these four equations might have been obtained from those in 2, § CX XXII, by substituting throughout yy JI for y’. However, by such a substitution as this, the value of a can undergo no change ; for after we have eliminated y, it matters not in the least what we sub- stitute for it. Hence it follows, that these equations must give the same value of a, as the former ; and since the former might also have been obtained from the latter by substituting the roots a’, 2”, a’, x", for LE Sa consequently it follows, that the expression for a must be such, that it suffers no change by the above substitution 5 it must .*. necessarily be of the form ve (re) a: Oe (res) Gio) SF ie (2/") (a4) CB which coincides with 6, § CX XXIT. If in the equations (f) in 2, § CXXXII, we cull stitute —./—1, for ./—1, we again obtain a new set of equations, which only differ from the equations (¢) in this, that in the former 2’” is combined with y/—1, and a” with —y/./—1; whereas in the latter the reverse of this is the case ; we might .*. have obtained them also merely by substituting 2/”” for 2”. But hence it follows, that the expression for a must 3 such, that we obtain Fe) (") (al”?) from f= (0). (0)! when we merely put —./—1 for /—1, which agrees with § CX XXIII, Corollary. % Further, since the functions p, q of the preceding §, 257 as sum and product of the two functions f: (x”) (v”) er’) (at") 5 fo (a), (2) (r"") () byathecsubstiention of aes a for ./—1 suffer no change, so likewise they undergo no change by the substitution of a” for wv’ ; and they are consequently functions of the form p: (a/) (<4) (a fe) (c tal = o: (a’) Catt) (x Ay) Casal and since they are also functions of the form p: Cia 9: (e7>) ey — p : Cut) Cae) Crt} Cary because they are compounded of these ; it follows then, as in: the foregoing §, that they can have three different values only, and they consequently depend on equations of the third degree. SECTION CXXXV. Pros. To transform the equation of the fifth degree e+ Art + Br? + Cr? + Dr + FE =0 into one of two terms y° — V = 0, assume the auxiliary equation : 3 . a4+ar?+ be? +cer+d=y: required to find the degree of the equations, which ' must be solved, in order to determine the coefficients aa, 0; c, d. Solution 1. If « denote one of the imaginary roots of the equation 7°,— 1 =o, and we put / V=y’, then Vy, ay’, «y’, ay’, ay’, as we know already, are the five roots of the equation y>—V=o. If we introduce these es of y, together with the values of x, into the 2. 258 auxiliary equation, we then obtain the five following equations . a + axr® 4 br? + er’ +d= y' v4 +a 4 bv? ter! +d= ay! VA qyll3 4. by? 4 ox! 4 d = ay! e484 axl¥9 4 bx! 4 ox!” +d = aby! a4 + av’ + bx”? + ex” +d = aby! 2. By these equations, which, with reference to a, Bs c, d, are only of the first degree, we can express each of these coefficients by the roots a’, 2”, a/’, «/", x”, and then, as was done in the foregoing § in equations of the third and fourth degrees, permute in the expressions thus found the above roots as often as possible; in the present instance 1.2.3.4.5= 120 times; the number of different values, which we obtain by these means, will then determine the degree of the equations on which the magnitudes a, b, c, d depend. But let it not be supposed, that the elimination, and also the comparison of the 120 results, are very laborious in themselves: another particular inquiry will always be required, if the | equations obtained last cannot be reduced lower. We wish .*. to try whether, from the nature of the above equations themselves, there are not certain indications by which we may more easily attain the desired object. 4 3. Since the number of the values of each of the cocfiicients a, b, c, d, say a, (the same conclusions may be made for the others) has its foundation merely in this, that the values of x and y may be combined in more ways a \ ) | { | ee 259 than one, it only remains to examine the results of these different combinations. But we obtain all the possible combinations of the values of x and y, when in the above five equations we either permute the values of x, or the values of y in every way. If we make choice of the first method, and, according to Hindenburg’s Rule of Permu- tation, let x’ retain its place in the first equation, when we transpose the roots xv”, x’, 2/”, aw”, we then obtain twenty-four sets, each consisting of five equations. If after this we introduce the roots a”, 2’, a’, x’, succes- sively into the first equation, when we permute the four remaining roots, we then obtain 120 sets in all, each.consisting of five equations, and consequently all the combinations of the values of x with the values of y. Each such set gives a value of a; .*. collectively 120 values of a. | 4. Now I assert, that amongst these 120 values of a, there are no more than twenty-four different ones, and that these different values are obtained from the first _ twenty-four sets. For, in the first place, it is easily seen, 'that it is quite immaterial whether we introduce the roots a”, a’, 2/", a”, successively in the first equation, and permute the other roots, or whether we substitute in all the five equations ay’, a%y’ ay’, ay’, successively for y’, and after each such substitution permute the roots x’, x’, x/”, x”, when 2’ retains its place in the first equation. Now, since / occurs only once in the value of a, because it was eliminated at the very beginning in the _ above equations, so with respect to this value it matters ' not what we substitute for y/; and .-., by the above-men- 260 tioned substitutions, we shall find no other values than those which we obtain, when we let 2” retain its place, and merely permute the roots 2”, 2/””, a/”, x”. 5. Now, we are certain that the expression for a, which we obtain by actual calculation, is such, that amongst the 120 values which arise from the transpo- sition of all the five roots, 2’, 2’, x”, a/”, x”, there are no more than twenty-four different ones, and that these last values are those which are obtained exclusively from the transposition of the four roots a’, a/”, «’”, 2”. The equation for a .*. rises no higher than the twenty-fourth degree. We shall now see whether this equation cannot be reduced to others of lower degrees. 6. Since we do not know the expression for a, we shall .*. immediately assume the transposition of the roots 2”, al’, x”, x”, 1n the above five equations, while in this we set the first completely aside, because a’ may be considered as a constant magnitude. For this purpose, according to — Hindenburg’s Rule of Transposition, we again let a/ — retain its place, and only permute the roots a”, 2”, x’; we then obtain six sets, each consisting of five equations. If, after this, we put a”, 2’”, x”, successively for 2” in the” second equation, and after every such substitution per- mute the other three roots, we then obtain in all the twenty-four sets of equations, which give the twenty-four different values of a. | 7. Instead of the above method, we can also make use of the following one. First put in the above five equa- 261 tions «?, 2°, a* successively for «, we then obtain, since —%, a’=2*, a=, &c., the four following combina- tions of the values of x and y: | a, vw, o, a”, yf, Hy’, wy’, wy’, wy! yy By/, ay’, aby’, a2y! y'; aty/’, aby! ary’, ay! and if we permute in each of these combinations the three roots v’”’, 2/”, x”, we then obtain the twenty-four combinations of the values of x and y, which are possible only under the condition that 2’ continues to be combined with y’. - 8. If .:. we had expressed the coefficient a, from the five equations in, -by; the reots ana4f, .at(G far Caenl sat would only have been necessary, 1n order to find its twenty- four different values, to have transformed « into 7, 23, 24, in each of these four values merely to have permuted the three roots 2’, v/”, x”. 9, Now, if we assume that the values of a, which give the four combinations in 7, are the roots of an equation of the fourth degree a* — pa® + qa’? — ra 4+-s = 0 and if we denote these roots by a’, a’, a”, a”, then Pp — a’ -f a’ ae al’ fs a/’ g=dd! + aa” + da” + ad! + &e. Xe. & ; 262 Now, since the functions a’, a’, a/’, a’/”, are such, that by the substitution of « for a, 2°, «*, they merely are transformed into one another, consequently p, g, 7, s, are symmetrical functions of the roots of the equation y’—1=0, and .*. they cannot contain «. The functions P> % > 8, consequently contain neither y’ nor «; and .°. in all the transpositions of the roots a’, a’, a’, a/”, xv", have no more than six different values, viz. only those which arise from the permutation of the roots a/””,.v/’, x”. 10. Consequently the coefficients p, 9, 7, s, all depend on equations of the sixth degree ; and if we do not mind _the trouble, these equations may be actually found by the method given in the third chapter. 4 But whether they are capable of any further reduction or not, will be seen in the sequel. I shall now only observe, that it is quite sufficient to solve one of these equations, viz, that for p, because then, as will be seen in the following chapter, the coefficients q, 75 8, may be found directly. SECTION CXXXVI. Pros. To transform the given equation x” + Av’ + Br * + Cr’ + &. = 0 into one of two terms, viz. y” — V = 0, we assume the auxiliary equation ; a + av + br"? 4+... the tloy with n—1 undetermined coefficients a, b,c, ......k, 2: on the supposition that 7 is a prime number, required to determine the degrees of the equations, which must be solved, in order to find the assumed coefficients. 263 Solution 1. If we put \/ V =’, and « denote an imaginary root of the equation y”— 1 = 0, then y/, ay’, Pe see oe ek a"—ly’, are the n roots of the equation xy’—V=o. If we combine these values with the roots of the given equation, we obtain the following 7 equations : ge AN: 5 dome a Coll iii alee Sa + kr’ + l= y/ BN a8 bal det Ue ay! i fe Nigel. al HP 4 Gal ores ee = kal’ + l= a*y/ &e. from which we first eliminate y’, and from the n— 1 equations obtained by this elimination, the values a, 3, c, d, &c, may be determined by the roots a’, x’, x/’, x3 ' &e. 2. The coefficient @ (and this, as well as what follows, | obtains also of b,c, d, &c.) in general contains as many | values as the » values of 2 may be combined with the n values of y, in sets of m equations, as those in 1, on the condition, that every such set is different from the others in the same set. Of these combinations, however, there | are exactly as many as there are transpositions in the x roots 'in the above set of m equations; consequently the number of the values which the coefficient a can have = 1.2.3 bo tec n. _ Consequently also the equation on which a depends, must be of the 1.2 .3...... nth degree, supposing that amongst these values there are no equal ones. _ 8. When in the above equations we substitute o7/, wy’, i’, veer.. e"'y’ successively for y’, we then 264 obtain z sets of equations, in which the values of w and y are arranged according to the following scheme: a ry tai! f° Sad VS AEN Be ae eee a” / 2, // 3 4, )/ ee. Y 9 ay’, ay 9 ay’, ay 9 Perec oes a Y ah / 5, ,/ ay s aYs «Y's ay’ y GY’ ssecovece Yy a?’ ay! ay! ay’ ay! 9 tteeeeee ay! 3p ,/ 4,,/ 5,,/ 6,,/ / vet ays ays ay aYs a! 9 eeerseoee ay e e e e e ° se e e e e ° ° e e e es Eb iy =i, / / 2,,/ -3,,/ Bie Mie Cor Ys Yo AY, Y'g veevereee TY ee Byam Tk 4, oY’ y ay’ oy’, qeteieeceer ae Yy’- Here we find, as may be seen on inspection, v’ always united with another value of y, while at the same time the remaining n—1 values of x are combined with the remaining n—1 values of y. If now, in each of these n combinations we permute the roots a”, 2/”, a/" ...0, we then always obtan 1.2.3 ...... n—1 combinations, and consequently from the whole together all the 1.2. 3 . n combinations, in which the values of x can enter with the values of y. 4. In order .*. to find the results of all the possible combinations of the values of x and y, it is only necessary, in the above equations, tosubstitute ay’, ay’, «°y’,...07—!y/ successively for y’, then transpose the roots 2”, 2//’, x’, --. 2 in all possible ways, and from each set thus ob- tained find a value of a; or, which is here the same, find first the expression for a, and then make use of the above substitution and transposition. But since y’ has totally vanished in the expression for a (1), so by the substitu- tion of ay’, a°y’, ay’, ......... ay for y’, this expression 265 suffers no change, and .°. there ‘are no more different values of this magnitude, than arise from the transposi- tion of the x — 1 roots 2”, a’, a’%, ...... u™. 5. Consequently the magnitude a depends on an equa- tion of the 1. 2.3...... »—1th degree, and its roots are the values of the expression for this magnitude, which arises from the transposition of the roots 2”, a/”, ... v. Hence, then, this equation may actually be found, by the method given in the third chapter, and may be expressed in known magnitudes. 6. Since the unequal values of a belong exclusively to peed 2.8 ..... . m—1 sets, which arise from the trans- position of then — 1 roots xv”, x’, a’¥, ...... 0, in the above equations, it is allowable to consider the first equa- tion, together with its roots 2’, y’, as constant and inva- riable, and consequently we only require to take into consideration the n — 1 equations ri Ligeratiee MW i pidconeipe AN Sanne” + ka’ +1 = ay’ 2 ii emt ON 9 Lacon VI EAE + kx!’ + l= ay’ If in these equations we transpose the roots 2”, x/”, ...«, in all possible ways, we obtain all the combina- tions between these roots and the roots ay’, ay’, ay’, ... ay, But these combinations may also be found as follows. 7. In § LX XXVII, Cor., it was shown that, when in \ the series of roots ~, a”, a°,..."-' we successively substi- 2M 266 tute a, a, at, ...... a7! for a, the same roots, but in a different order, always present themselves. Hence it follows, that when in the equations in 6, we substitute successively 2”, a3, at, ...... a! for a, the nm — 1 sets of equations thus obtained, only differ from one another in the combination of the values of x with the values of y. Now since also every set has also another of the magni- tudes ay’, a®y’, ay’, ...... ay’ in the first place, it follows, that we obtain all the’ combinations of the values of x with the values of y, when in each set we permute the n — 2 roots 2’, 2/”, ...... a” in the n — 2 last equa- tions in all possible ways. 8. The changes which we have effected with the equa- tions themselves, we can also effect with their result, the expression for a. Thus, if by the equations in 1 we have expressed the magnitude a by a’, 2”, x//”,...2, we then substitute in it successively a”, a°, a‘,...... a"! for a, and permute in each of the values thus obtained, merely the roots v//’, a/", a”,......u, while we let x’ and 2” retain their places. 9. Now, if we denote the »—1 values of a, which arise from the substitution of 2%, a3, a4, ...... a", for a, by a’, a’, a’”,......a\, and assume that they are the‘ roots of the following equation of the n— 1th degree: a’ — pa’? a qa’ ais ra’— ae vers then the coefficients p, gq, r, &c. merely as functions of / // a ° . TEN Ree GEE. iy a”-, in like manner can have no more different values than the 1. 2. 3......2—1, which | 267 arise from the substitution of @ for a%, a, at,......a", and from the transposition of the roots a/”, 2”, 2”,...2. But since these coefficients are also at the same time symmetrical functions of a, 2%, a3,......a°-, consequently of the roots of the equation y* —1=0; .°., by the fore- going chapter, they must be rational, and consequently do not contain a Wherefore these coefficients have no more than the 1 . 2 . 3...n—2 different values, which arise from the transposition of the roots a/”, x’, 2”, ...©, and .*. they all depend on equations of the 1.2.3......2—2th degree. 10. Let 1.2. 3.......—1=p, and let pe Ap + Bip Clpt~ + &e. = "0 be the equation for p; then, as is already known from the third chapter, the coefficients A’, B’, C’, &c. may always be found, and expressed rationally by the coefti- cients A, B, C, &c. of the given equation a+ Ax’—'+ &e. —o. If, then, we can solve this equation, and from it determine the values of p, then also we may, as will be . shown in the following chapter, find the coefficients q, 7, s, | &c. directly, and without solving any otherequation. Now, \ if we denote the values of p, q, 7, &c. which we thus obtain, \ by p's, 7’, &e., ps gq’, 1’, &e, p”, i’, rl’, &e. &e. we then obtain the following 1 . 2. 3....—2 equations: a’ + pas” =} Dore -. 7’ a"—* + &e. —o0o qa} =o par vi gan at ra" a8 &e. =o an ee par + far + yl gr-4 4. &e. =0¢o &e. | into which the equation for a in 5 may be analyzed. But 268 whether or not the equation for p is capable of any reduc- tion, this is not yet the proper place to inquire. Remark. Hence it follows, that an equation of the nth degree, when 7 is a prime number, leads, according to this mode of transformation, to an equation of the 1.2.3...n—2th degree; consequently an equation of the fifth degree to an equation of the sixth degree, and an equation of the seventh degree leads even to one of the 120th degree ; and so on. SECTION CXXXVII. Pros, All that has been said in the problem in the foregoing § obtains, only m is a compound number: determine the degree of the equations on which the assumed coefficients a, b, c, d, &c. depend. Solution 1. When we suppose a, not to be, as in the preceding §, any arbitrary imaginary root of the equation y"—1=0, but only a primitive root of it, then indeed all the conclusions made in 1, 2, 3, 4, 5, 6, of the pre- ceding § are applicable to this case; on the other hand, the following solutions, (7, 8, 9, 10), on account of this, very circumstance, must undergo some alterations. Thus, if we substitute here, as in 7 of the preceding §, in the series of roots a, a”, a’,,..,..0""! the powers a”, a, a4,......2*-) indiscriminately for a, we shall not always — find again the same roots, but this will only be the case ‘for those powers amongst them, a’, a*, at, &c. whose | f | exponents v, 7, 9, &c. have no common measure with n, — 269 because these only are primitive roots of the equation y"—1=0 (§ XCI). Thus, if be a primitive root of the equation y©—1=0, when in a, a’, a’, a‘, a®, we succes- sively substitute 2”, 2°, a4, a, for a, we obtain the follow- ing results: a%, at, 1, a, at; a®, 1, a, 1, a®5 at, 2%, 1, at, a2; a, a4, a3, a®, a, of which only the last con- tains again the same roots. 2. Hence it follows, that what has been said in the preceding § in 7, and the following solutions, respecting the substitution of the roots a’, a°, a4,......a"-' for a, must be limited to @”, a”, at, &c. whose exponents v, 7, 0, &c. have no common measure with x. If .*. we assume, that A is the number of the primitives 2’, a”, a, &c. and that the values of a, which we obtain by the substi- tution of these roots for a, are the roots of the following equations : a + pa + ga* + ra’ + &. = 0 consequently p (and the same obtains also of g, 1, &c.) is such a function, as by the substitution of the root a for a’, a, at, &e., or, which is the same, by the substi- tution of the root 2” for «™, 2, a, &e. it remains unchanged. Now, since by these means all the 1.2.3 «.a—1 values of p, taken A and X together, are equal, it follows, that this magnitude depends on an equation, whose degree ie Aatetsaaees TL aa aaa _ Moreover, in order actually to find the equation a + | pas + &¢.=0, it is only required in the expression for a, \ which we obtain from 1 of the foregoing §, to eliminate 270 the root « by means of that equation which only contains the primitive roots, the method to find which was given in § LKXXIX. Remark. Therefore the equation of the nth degree, when 2 is a compound number, leads to an equation of 1 e CARE SAY | Rea | . e the SaaS) eT th degree, in which A denotes the number of the primitive roots of the equation 27—1=0; consequently an equation of the fourth degree leads to an equation of the third degree, because the equation 2* — 1 = o has two primitive roots, a and a; an equation of the sixth degree to an equation of the sixtieth degree, because the equation x® — 1 = 0 has also two primitive roots, viz. @ and 2°; an equation of the eighth degree to an equation of the 1260th degree, because the equation a® —1=0 has four primitive roots, viz. a, a, a°, a’; and so on. From this and the foregoing §, it follows, that the reduction of the equation 2” + Ax’ + &e. = 0 to one of the form y” — V = 0, always leads to a higher equa- tion than the given one itself, whenever the given equation exceeds the fourth degree. However, I shall not enter here into the proof of other methods of transformations, because the whole of this subject will be considered here- after in a higher pomt of view, to which this is only preparatory. SECTION CXXXVIII. When we make the equation « = a\/ pt \/ p2 271 + c\/ p? ates spans + kv/. p’"', or more generally the equation x = a + b\/p + oN/ pt + dv/p + seseee +k \/ p” rational, we arrive, as we know from the foregoing chapter, at an equation of the nth degree, which I shall represent by 2” + Qa"! +. x"? + Ca’ 4 &e. =o, in which the coefficients A, 3, €, 3, &c. are cer- tain rational functions of the magnitudes a, b, c, d, &c. and p. Conversely, if an equation of the nth degree + Ax" + Bar? + Cr + Se. = 6 be given, and we assume that the roots have the above form, we then have, for the determination of a, b,c, d, &c. the n con- ditional equations €@ = A, B= B,€@=C, &e. If we solve these equations, and from them determine a, b, c, &e. _we then have at once the m roots of the given equation, 7 _ when we substitute successively for Aye p its nm values. | | | } | ; Consequently all depends on the solution of these con- ditional equations. How difficult and troublesome this solution must be for equations of rather high degrees; may be perceived from the form of those equations, which were found for the fifth degree in '7, § CIX..” Waring | and Euler thought, that in this way we must arrive at — the general solution of equations, by properly arranging and finishing the calculation, without regard to the trouble. But this trouble may be spared by subjecting a priori (as M. Lagrange does in the third volume of the New Memoirs of the Berlin Academy) the method to a preli- | minary proof. 272 SECTION CXXXIX. Pros. Let it be assumed that r=atbN/ p+eN/ p+d\/p+ igs 4hN/ pi is a root of the given equation wm Ar’ + Br’ * — Cr + &e. = on the supposition that is a prime number, required to determine the degrees of the equations on which the coeffi- cients a, b, c, d, &e. depend. Solution 1. Put \/ p=y:; then y, ay, By, yy, oy, &e. are the n values of \/ p> when 1, a, B, 'y, 0, &c. are the » roots of the equation a*7—1=o0. If .*. we denote the roots of the given equation by 2’, 2’, x’, &e. we have the following m equations : a’ =at byt cy? + dyP+..uns +- ky’ a” =a+aby-+ acy? + aidy®?+...... +a ky’ x/ at Bby ab Rey? coin ePdy? 3 SSR “f- speed Tyr a” =at+yby+y'cy? + y*dy? + ...... +" "ky" &e. from which the » unknown magnitudes a, b, ¢, d, ..... k, i must now be determined. 2. If we multiply these equations, in the order in | which they stand vertically, first by the powers 1, 2”, 6", y", &c., then by the powers 1, @”"', g’"', y”, &c., after this by the powers 1, 2’, p"*, y**, &c., and so on; lastly by 1, a, 8, y, &c., and add the m results ob- tained by every such multiplication together; we then f 973 get, since ‘[n] =m, and every numerical expression _ ‘[p], whose radical exponent p is not divisible by n=o0, nas Cathe! IL ove ERG: nyb = al atl 4 gt lg 4 ttyl” + &e. myrcmal al 2x! 4 gy! 4 ry 29/V 4 Be, n. yrd= a’ 1 aryl “+ Bru! + oa ft &e. | * ° ° . ° . ° e e e ° e ny tal fatal 4+ Bea’ 4 2a’ + &e. Ny kas al! 4. Ba!!! 4. ya’” + &e. __ 3. Hence now we may immediately determine a ; for, ) since a’ + a” + a// + 2/” + &. = [1] = A, we then . obtain from the first equation Cee n ' Consequently a has only one single value; on the other hand, all the remaining magnitudes 3, c, d, ...... a, ke, have more values than one, which may be obtained from » the n—1 following equations, by transposing their roots al, x, x’, ...... a in all possible ways. ‘The equation on which each of them depends, is .-. generally assumed | to be of the 1.2.3...... mth degree. But if amongst ' these values there should be any equal ones, or any re- _Jation amongst them, then the degree of the equations can be reduced. 4. Since one of the n-+1 magnitudes a, 6, c, d,...k, p, | may be assumed to be arbitrary, we shall put p = 1; _..alsoy=1. At the same time, in order to make the | formule more simple, we shall put | 2N ah eal ql! ; a) k — ——=9 a —_———> ———>5 ecceccrnee = 7 nN n nN By these means, if we omit the first, and set down the others in an inverted order, the equations in 2 are trans- formed into the following ones : a =a! + aa! 4 Br + 2” + &e. Glee a oh ta BAe yal ot Sie. al! = x! 4 ay 4 Bix! 4 yin!” + &e. qr) —_ a! an a" ql a es gl!’ 4+ y A/V of &e. 5. Since the roots of the equation x” — 1 = o (because by the hypothesis 2 is a prime number) may be expressed by 1, a, a, a3, ......a@”', let a denote any imaginary root whatever ; then we can put a2, 2°, at, &c. for 8, y, os &c.; by these means we obtain the first equation al = af coal! bal Byll a teil oft) and thejwalues offa%, dja’. s 3. a”—) are derived from this value of a’, when we substitute a, a°, a4, ...... a" for «, consequently when we substitute for a every imagi- nary root of the equation y"—1=o0. If.*. undetermined t denote each of the magnitudes a’, a’, a”, ...... a5 then tas aa pata bal cece bia ia™ 6. Now, in order to find all the values of which ¢ is capable, it is only necessary to permute the rootsa’, a/,a/”’, weeeee 2”, in all possible ways. For our purpose, how-— ever, it is more convenient, in this case, to proceed as follows: Ist, we only transpose the x — 2 roots 2”, 2’”, 275 vy .eeee. 0, while we let x’, 2” retain their places, consequently we obtain 1.2.3 ......%—2 results; 2ndly, we put in each of the results thus obtained, first a*, afterwards a, then a4, and so on, and lastly 2" for a; then generally we have 1.2.3 ......7—1 results, which we should also have obtained, if we had let 2’ retain its place, and merely transposed the roots a’, a’, w” ..ceceeee 03 Srdly. and lastly, we multiply the 1.2.3..,...%— 1 results so found, successively by a, a, a, ...... a", and we obtain, together with the former, the 1. 2.3...... ~ results, which arise from the — transposition of all the roots a’, v’, v/’, ...... c and at the same time also all the values of ¢. 4. If. we denote the 1.2.3 ...... 2— 1 values of t, which are obtained by the two first operations, by met. 4 Uv.. &e,, then alk the tya:24218 jag «207. Values of t may be expressed in the following way : Beets RU Sy UN cotta daaste. Cetra gies 1h cece, ce Wbe ot Leh aS Lowi, ov otiace E LIT ERTS LOSE Ly GEM og OPE RCC OL 4 PLLC &e. Beowenput, 2% 0/5 tsi) 0/4r/ i OG) &ee.5> then? the values in the first horizontal series are the roots of the equation ¢” — &/ = 0, those in the second series are the roots of the equation t” — 0” = o, &c.; consequently the equation for ¢ is the product of the equations f— 0 =0, 0-0 =0,f-0/=0 &e. 276 8. Hence it follows, that the equation for ¢ only con- tains such powers as are divisible by x. If .*. we put t”=0, so that Osea cer! + abel! 4+ aba!” + occ fb amy we then obtain an equation for M of the 1.2. 3...n—1th degree, whose roots are 6/, 0’, 6’, &c. which we obtain from 0, by transposing the n—1, roots a”, a/” x’”,.....0 and allowing v to retain its place. 9. Instead of transposing the roots 1”, 2”, 2’"......0 in the expression 8, it will be quite sufficient, as in ¢, merely to permute the n—2 roots w/”, a/", vy «1-606 Us then substitute a%, a3, a, ......a@" for a, and in the nm — 1 values of @ thus obtained, transpose the roots 2”, WV igak (7) 9 Uy wcccee LO, a 10. We now assume, that the x—1 values of 0, which arise from the substitution of a?, 2°, a+, ...... a"! for a, are roots of the following equation : (A) ...... 0 — pp? 4+ gf — rl 4+ &e. =0; then the coefficients p, q, r, &c. are functions of these values, and can .:., as is the case also with these last, suffer no change by the transposition of xv’. But since they are also symmetrical with respect to these values, consequently they can likewise undergo no change by the substitution of a”, a3, at, ...... a” for a, because the only consequence arising from this substitution is, that of the n — 1 values of 0, one is merely transformed into the other. Hence, however, it follows, that these coeffi- cients can have no more. unequal values, than those | 277 which arise exclusively from the transposition of the mp Bi TOOTS Bs Be ap ade se a™, and that consequently they all depend on equations of the 1.2. 3......n—2th degree. 11. Therefore the equation for 0, which, as we have seen, is of the 1.2.3 ...... m—1th degree, may always, when is a prime number, be analyzed into1 .2.3... ... 2 — 2 equations of the n — 1thdegree. Now, if (4) be one of these equations, and 0’, 0”, 0/” ...... 9°” n n n n its n — 1 roots, then \//, \/0”, \/0, 202 \/O°, are the corresponding values of t, or of a’, a’, a/”;...... . a”—, and if we substitute these values in any order im the expressions in 4, we then obtain Ke Ve! VATS Wan n Bat since we have put p = * r= 44 ~~ (+ -/ 0 +. ney ses But that for @ only those of the 1.2.3......%—1 values may be assumed, which belong to one and the same equation, it matters not which, 6"~' — p 0”-’+ &e. = 0, appears from this circumstance, that the »—1 values of t, consequently the corresponding values of 6 must so depend upon one another, that we obtain them all, when in one of them we substitute a’, a’, for a, 5. at, eeeere a! 12. If we wish actually to find the equation (4), first of all we solve the function 278 O-= (al + ar!’ 4 tr” + 0. a) according to the powers of «, which may very easily be done by means of the polynomial theorem, But since a", a", a't?, &, are no other than 1, a, a*, &c. conse- quently this development, after the proper reduction, takes the following form : &/ EV G+ EU gg? a8 har dee ne ee + EMgr—! and &/, &”, &”, ...... §&-" are mere functions of 2’, wv”, a!/, .e0e4. 0° without a Ifin this value of 0 we put ' successively for a, or, which is the ee aCe sth same, f, y, 0, &e. for a, wé then obtain the values of 0’, ip ee e@eeesee 7] Oaks VIZ. 5 6 — &/ a Eq Ak EUL 2 Be ape EMeg—! 0’ &/ -/- £9 7 E/// 2 - nee ae ay ae Ema! O/// — E/ oo Ely AE EM? ae by ate, ae Eryn! &e. : from which the equation (4) may be compounded in the usual way. ‘The coefficients p, q, r, &c. are then still functions of the roots 2’, a’, a/’’, ......v™, but such, that they only change when the roots a’, xv”, x”, ...... 1 are transposed. ‘he equations for these functions may be found by the method given in the third chapter. Moreover it is quite sufficient, as will be shown in the ' following chapter, to find one of these equations, for instance, that for p, because from the known value of p, the values of q, r, &c. may be found directly, and without the solution of any other equation. It is likewise suffi- cient, as follows from 11, to find only a single value for each of the coeflicients p, q, 7, &e. ~es 279 - 13. From all that has hitherto been said, it follows, that the solution of an equation of the nth degree, when n Is a prime number, depends on the solution of an equation for p of the1.2.3...1— 2th degree; and this result coincides with that ee has been found by another method in § CXXXVI. With the view to further elucidation, I shall now apply it to an equation of the fifth degree. Remark. When n is a compound number, then the conclusions drawn with respect to the substitution of the Boot. «x for a, a3; a*,...0:. a", suffer the same changes, as the conclusions, § CX X XVI, § CX X XVII, must un- dergo in the same case, and we shall then find, as we did there, that an equation of the nth degree leads to an . Le), 2, 3s--R—1 equation of the Cnty an th degree, when A denotes the number of the primitive roots of the equation y” — 1 — O, SECTION CXL. Pros. From the given equation of the fifth degree g— Ar + Be — Cr? + Dr — E= 0 find actually the reduced equation for the magnitude p of the foregoing §. Solution 1. The equation (4) in 10 of the foregoing §, when n= 5, is here 0+ — p® + gO? —r0+s=0 | | | | ) 280 and the roots of this equation are, when a, 8, y, 6, denote the imaginary roots of the equation y° —1=o, / —_ E/ + EM a + E/// 42 Je E a3 + E’ at I wm El 4 Elgg Ellige 4. EMgd 4. Eves Ome Ep EM ay EM ay 4. Eo 4. EM yA Om EF 4 ENS 4 ESE 4 IVS 4. EKA 2. If we add these equations together, we then obtain, when +0’ 4+0/+460/"=p, anda+e+y+é=[1]-1 =-1,0+@4+7+8%=‘2]—-1=—1, 8+ 8+7 += [3]—1=—1, a+ pitry4+t ot= [4] —-l=—1; p= 7h sie (E// 4 EM 4 EIV 4 EV) me EY (EC EMG BU EP ger) 3. The second part of the expression for p, viz. &/ + EM 4 E// 4 EV 4 EY, may be found immediately. For from 12 of the foregoing § it follows, that the develop- ment of | (a’ eh ax’ 2 ary! ae aey/¥ ais aye assumes the following forms: | Era Blaha eget EVES EV At and this form of the development is always correct, which- _ ever root of the equation 2°—1=o0 we substitute for a; | consequently also, when we put a=1. If this is actually done, we find Barhyo etree ert creed bie (a’ + ql/ “t ql 4+ al? + HHO: =[1}'= 4 We have .:. also p = 5&! — AP. 281 - > 4. In order .*. to determine p, it is only necessary to find &’, consequently that term in the development of (2% + an! 4 aa!’ + aa/” + atx’)®, which does not contain &. For this purpose we can give this expression the following form : a (cal + aa! + aby!” + ote!” + adx”)? or, since «°=1, the following one (av + ax! + aba! + ate!” 4+ abr) from which we derive this advantage, that the dashes over x coincide with the exponents of a. For now we have nothing more to do, as is shown in the polynomial theorem, ‘but to combine. the roots 2’, x’, 2’, 2”, x’, in all possible ways, in such a way that the sum of the dashes Pim. i li. == 20, <5 25,.becalse 2a aoa” =a°=1. In this way we find, when [5], [1°], are sub- stituted for 4/5 + y// ue gl5 + l/h J et, Ge Ee Cee, etd banc Wl ae Dak le - Bal y/Y ol a !3y/// -+- wi al! gf 98 4. 20 Jalal 3yY 4 a! By!VanY gl glBy/¥ 4 gl pl VV ella ¥3 4. alll g/VEyV yf 2xl/2y!/V + al2yl gl? ae gly! fe gf Ay lV2gV 4 4.302 axle pally IVE 4 gl2y/lOggV 4. gf Byll!y/ V2 pala V2g V2 4. g/I/2y/V V2 or when, for shortness’ sake, in the value of &’, we denote by ¢ that which is not to be found in the crotchets we get S= [5] +01 +2 p= 50 + 5[5] + 5[] — 4 5. Amongst the 120 values, which the function Z con- tains by the transposition of the roots a’, a, 2// by Ae hs 20 282 we find no more than six unequal ones, and they will be exactly those which arise exclusively from the transpo- | sition of the three roots 2’, a’/", x”. If we denote these — values by 2, ”, 2”, &/”, 2”, £”, and the corresponding — valle of p by p’, p’, p', p’”, p’s p’’, we then obtain ~ p=se’ +5f5o +5] -L£ p= 50" 4-5 (5) + 5 [1] — & p= 5 CLs 3 [5] ae [uF] fe A p's 50" + 5[5] + 5 [1] — 4 p= 50" + 5 [5] + 5[1°] — & p’= 62" 4+ 5[5]4 6[1}— & ] and these six values of p are the roots of the required equation. We already know, from the third chapter, — how to proceed further, in order to find this equation — itself. It would be better, however, instead of the equa- — tion for p to find that for ¢; for if we have Z, we have also p. | ( 283 ) VII.—A GENERAL METHOD BY WHICH, FROM THE KNOWN VALUE OF A GIVEN FUNCTION OF THE ~ ROOTS OF AN EQUATION, TO FIND THE VALUE OF EVERY OTHER FUNCTION OF THESE ROOTS. SECTION CXLI. ALL the methods which we have hitherto applied to the solution of equations, are founded either on analysis or transformation. The first, from its very nature, cannot be general, because every equation will not admit of being analyzed into others of lower degrees. Consequently, in the general solution of equations, we have no other mode left us but to transform the given equations into others, which in themselves are either solvible by the methods already known, or may be made so by analysis. Now, let it be assumed that we have transformed in any way, no matter which, the given equation | | n+ Ax’ + Bu’? + Cr’? 4+ &. = 0 into another w+ A’ + Bt" + Ct" + &. =o then the roots of the last equation must stand in some one relation to the roots of the first, or, in other words, t must admit of being expressed by some function of the roots 2’, 2”, 2”, &. Now I affirm, that it is always 284 allowable to assume ¢ to be a rational function of these roots. For, let Fis (2’) (2) a”) ... (2) betany irrational function of these roots, and let t= /": («’) (2”’) (a/”)...(x) 5 then this equation, as has been already shown in the fifth chapter, can always be made rational by removing the irrational magnitudes. We shall thus get an equation th + AN + Bip + Cp + &e. =o in whieh the coefficients 4’, B’, C’, &c. are all rational functions of 2’, wv’, a’, &. Now if we eliminate from this equation and the equation t” + 4/t7'4 Bu"? + &c.==0 all the powers of ¢, as fur as the first, we then obtain for ¢ a rational function only. In the first place .*. it is only necessary, in the trans- formation of the equations, to find such rational functions of a’, 2’, a’, &c., for which the transformed equation is either immediately solvible, or at least may be made to — depend on solvible equations. But this 1s not all; it is not sufficient to know the values of the assumed function ; we must also be able, from these values, to find the roots a’, a’, a//’, &c. I shall first handle the second subject, — and, according to Mr. Lagrange, in the third volume of the New Berlin Memoirs, show the method by which, from the known values of a given function, the value of every other function may be found, consequently also the roots themselves. Here two cases must be taken into consideration, viz. first, the case in which the given and the required function are homogeneous ; secondly, the ease in which they are not so. For the sake of greater perspicuity, when I treat of the 285 values of a function, I shall sometimes distinguish the values of forms from numerical values; the first are the different forms themselves, which arise from the trans- position of the roots a’, 2”, a’, &c.; the latter, the actual values of these forms expressed by given magnitudes. SECTION CXLII. Pros. Let it be assumed that the given equation LT) a? ede! 4 Br? + Cr? + &e. = 0 by the introduction of a new magnitude ¢t = f: (2) (x) (#’”)...(x™), according to the method in the third chapter, is transformed into an equation Ble + Pit 4 Or Ri oe EO 6 which is completely solvible, consequently all of whose roots may be found: from these known numerical values of the function ¢, it is required to find the numerical values of any other function y = @: (a’) (2”) (a@”) ...(x) respecting which it is assumed that it is homo- _ geneous to the former. Solution 1. Since the functions ¢, y, according to the hypothesis, are homogencous, then, by the transposition of the roots 2’, a”, a’, &c. the former must contain exactly as many unequal values as the latter. ‘The func- tion t, however, has 7 values, because the equation II, by which it is represented, has been assumed to be of the wth degree, consequently the second function has also z values. I shall denote the values of forms of ¢ by t/, ¢/’, w”, ...t™, and the values of forms of y by 4/5 y/.y/5..y > 286 and assume besides, that those which have the same number of dashes, arise from the same transpositions. 2. Since ¢ and y are homogeneous functions, conse- quently any expression whatever, which is compounded of these functions, can have no more unequal values of forms than these functions themselves. Consequently, also, such an expression as ¢‘y can have no more than z different values, and these are ty’, ty, ta wuceee “ (iy). If we take the sum of all these values, we obtain the function (b).ceeet yl bP yl UP YM” 4 eb HOY and this function has the property of remaining the same, however we transpose the roots a’, a’, w/”, &c.; it is.*. symmetrical with respect to the former roots, and con- sequently, let X be any number whatever, may always be expressed rationally by the coefficients 4, B, C, &c. of the given equation. 3. If we denote:the numerical values, which the func- tion (W) contains, when we substitute 0, 1, 2, 3,...r—1 sucessively for A; by 2) 21) Zo) 23)+++ ++ 2,1, we obtain the following 7 equations : . Yt Pt fl Hae + yl) = Bp» Byl Byf! RUOY Hire aces + try) = s; ty! Vy! PY! A evens + (MPy™ = zy t/y/ aq eh ad mite UB y ll! ae ate Oe + Ci") Py (7) — =e /t— whe ef my! ee phar vali whee tindy (ty — ne in which 2), 2,, 22) 25+++++-%,—15 are all known magnitudes expressed by the coefficients of the equation I. 287 . me INGW,: tebe e 2 teres . t, instead of the values of forms of the function, denote its numerical values, then these are no other than the roots of the equation IT, consequently, by the hypothesis, are all known. 'There- fore in the foregoing 7 equations there are no other un- known magnitudes but y’, y/”, y/”,...y™ ; and since their number is 7, consequently we have exactly the same number as of equations; they may .°., with a few excep- tions (which will be inquired into hereafter), always be calculated and expressed rationally by the magnitudes ¢’, ee, i!” ...... t® and 25 21, 22, 33)... 2,19 consequently also | by the magnitudes ¢’, t’”, t/” ...... t‘*), and the coefficients _ A, B, C, &e. of the given equation. ExampLe. When 7 = I, we only have y aes which must also be the case, because then ¢ and, y are \ symmetrical functions of a’, a, 2//’, &c. and .*. y no longer depends on ¢t, but only on the coefficients 4, B, C, &c. When z = 2, we have the two equations vy a y” = 2 ty + Uy! = 9 and hence ) Vane, L685 Bi ER | TS AE Ta sl RR SAG When 7 = 3, we have the three equations ft! y! a y! » yl! = 2% ti ty! + tly! 4 ty! = 2 t/2y’ -}. ail ol ob haat LL = 2) ‘ — o 288 and hence we obtain 1 8a (7 + + 2, + t/t! Fe ef ae t!’) (785 //) HANES (t/ + MEY + a ee Chl Sieg ial — (t/ ue t’’)z, +. Mlle. Y 7 ees t’) (7 — Ae In the same way, wee aw = 4, we find the following values for 9’, 9”, y/”, y/” set eg UA AT EE a U1) v —t”% —0) Z, — (+t 4"). UE UU 7) 2, 0” & (ts fhe. t’) Cie Tex vis) (t/ io t’”) 2, — (i +t”) 2, + (0 + vt F 4 ah ade al pit a (Ee a t’) (in pots ve) GH Sis i") 2— (t4+t/+t”)s2, + (tt + ttl’! wv he MAY 9 io vere’'sn Te pi eC Re) G7 a from which the law of the progression may be very easily prog y y ya inn seen. A SECTION CXLITi. Pros. All that has been said in the problem in the foregoing § holds, “with this single difference, that all the roots of the equation II, as was there assumed, are not known, but ‘merely one of them: required now to find the numerical value of the function y corresponding to this numerical value of the function ¢. Solution 1, Let t/ be the known root of the equation — II. If we divide this equation by t —@/, we obtain another equation GA - 289 ETL. a +e ptt Ve + Sate in which Pos f+ Qt? PY ++ Q R= 134+ Pt?+ Q4R Soa t4 + Pi? 4.:Q: 4 Re +S &e. and the roots of this equation are t,t”, /, 2. But since in this case the single root t/ was assumed to be known, we must merely endeavour to express 4’ by t’ ; and this object is most easily attained in the follow- ing way by means of the method of elimination given in § LVIII. Multiply the equations in 3 of the fore- going §, beginning with the last but one, and proceeding upwards, by P’, Q’, R’, &c. viz. the last but one bye Be. the one preceding it by Q’, and so on to the first; which is multiplied by U’, and then add the results thus ob | tained to the last equation; by these means we obtain SO ee Eg nb OY ete tree ens + U's, rt met ChE ae eae ‘ U’) $y! (i Se PAP) OV pllnmBane Aone. a U’) fey! (Th og Pilla? QUIS, + U%) &e. me -oommince fF t™, are the roots of the equation IIT, then all that which has been multiplied by RA EA ces y™ in the second part of the equation just found, = 0. We only .-. retain Wey Pe eri OS: Bae + U's, ty (6° Pir CV eS + UY QP 290 and hence it follows Sey + Pep + Oss) H+ Ux y= v4 P= 4 WT... + Ul 4. Ifin this we substitute for ¢/ every other root of the equation II, we then obtain the numerical values of — YY, Yl yy oveeee Y™. Tf .*.5t and y which are unde- termined, denote two corresponding values of the functions just given, we then have generally Bea Pla g + Qe + oo. +) OR, OES BERR QUEER BT EE and it is then Pe go Pe Qa PUP Pt + R=0 + PP +Q+Hh &e. Exampie. In § XX XIX we find, that when x* — 42° + Bx2?—Cr+D=o is the given equation, the function — t= v/a!’ + x!zx’” depends on the following equation of the third degree: Pian -- (AC — 4D) t— (C? a 4 BD + A?D)=0. ‘ T shall now assume, that we have so far solved this equa- tion, that we have found one of its roots, and that we now wished to determine from it the value of another function y = (aa — a/x/"), which 1s homogeneous to the former. Since here 7 = 3, we then have LQ + P's, + Q’ 2, 12 4 Pt + Ql. 4 Mi ‘ in \\ \ 291 Further, since P = — B, Q = AC — 4D, we have P=t+P=t-8 QVY=PW4+ Pt+Q= — Bt+AC— 4D It only remains now to determine the values of z,, z,, z, But fo maale! 4 al a!”, yo =(ale’ — wll/y/V 2 tl ala! pale”, yl! (ala!!! —a"/a/VP Wale! 4 allyl, yl = (ala —allal)? which, when we take the numerical expressions from the annexed 'Tables, give the following values : Qa yl ty’ + yl" = [2] — 6[14] = BF —-2AC—4D , a, Uy! + tly! 4 My = [32] — [1227] = BP — 3ABC + 3C + 34°D — 4BD gy = ty! $y! 4 t/Py// = [42] — 6[24] = BY— 44B°C + 24°C? + 4BC? + 44BD —4B°D —8ACD - | If we substitute the values of P’, Q’, 2, 2:5 2 here found, we obtain (B?—2AC—4D)? —(ABC— 3C?—3A?D)t ( +16.D?—42° D+ BC? + ABD—4ACD ) ae 32 — 2Bt+ AC — 4D ‘and by means of this expression we are now enabled, for each numerical value of the function ¢, to find a numerical value of the function 7. Remarx. By means of the differential calculus, we can give the denominator of the general expression for y in 4 | amore simple form. Thus, since _ me. (tv) (0 PAT VOD Hien. + U’) | tap Pe Ph) Qirrtiorad.¢ + U) 292 we then have, when we differentiate both sides, in reference to t, and divide by the differential dt, (t—t’) [w—1)t” °+ (x#—2) Pt" 4+ (7-38) Vt + &e.] 4 t7 1 Pitt? 4+ Qt" + Rt" * + &e. at™ + (w—1) Pt" + (7 —2) QU"? + (7 — 3) Ri” + &e. If in these we substitute ¢’ for ¢, we obtain the equation ge 4 Pies 4 Qe Ri + &e. =a" + (r—1) Pt + (w—2) QU™™ + &. and since this equation must be correct, whatever root we assume for ¢/, we then have generally eat Pet eS YES &e. at™—' + (7—1) Pt"? + (7—2) QU? + (w#—3) Rt" + &e. Consequently the value of y may also be expressed in the following way ; id Bea tchy eh ia tly So ay sti ewe dee te temas + Uz, Y= Tet Gal) Pr + 2) PF eT SECTION CXLIV. | If the formula of the foregoing § be generally appli- : cable, we are enabled, from the given value of any function | F: (0%) ”) «0... (@) to find the value of every | other function @: (a) (2) (#”)...... (x), homo-~_ geneous to it, and that immediately merely by a rational. expression. But it is also actually applicable in all ima-. ginable cases, with the single exception of the one in| which the value of ¢ is such, that the denominator of the) expression for y= 0; a case which was mentioned in § LX. In order to see how the ¢ase is here, J shall consider the denominator t/"—) + P/U" + Q/t’™ +4 &e. ) ' i} | . proportional } al 1 ¥ 293 m the expression for y in 3 of the foregoing §. It is, from its origin, no other than the product of the factors Ba Oe est A GL geese it vanishes, then one or other of these factors = 0, and t¢/ .*. must be equal to one, or even more of the roots 0”, BAU ee. oh t™, Hence it follows, that the case in which the denominator in the expression for y vanishes, can only obtain, when the equation FI has equal roots. But now it may likewise be seen, why this expression cannot give the value of y’. For so long as a number of BOOS bis BU oe ‘ are different from one another, v’ gives the value of y’, t” the value of y”, &. But if they are equal to one another, then the single root ¢/ must at once give the v values y’, y”, 7, ...... y 3 but since the expression found for y is rational, this is impossible. Hence it may be further concluded, that the v values 7’, eS os 550 y must be given by a single irrational expression, which contains exactly v values, or, which is the same, that they must depend on an equation of the ath degree, whose coefficients are all rational. How this equation may be found, will be seen immediately. SECTION CXLV. Auxiliary Rule. Pros. Let I denote any function of x, and let the equation y= (x — a)” II be given: required to find the value of the differential d” aes for the case, where w = a, L d ; pe Pi “al ays tiate this equation three times im succession, we then obtain successively, 294 Solution 1. Let m= 1; 2. y= (va) I. Hf we” differentiate this equation, we find dy = (w—a) dil + Ldx If in this equation we put x = a, then the first term of the second part vanishes, and we consequently have, when TI’ denotes what II becomes when we put v = gq, d dy = II/dr, and - mal TS 2. Letem=2; .. y= (w— alll. If we differen- tiate this equation twice successively, we find og dy = (x—a) dll + 2 (w~a) Lda 3 dy = (wa)? @il + 4 («@—a) dildy + 1. 2 Tdi? If we put + =a im the second equation, the two first terms of the second part vanish, and we then have d?y = 1 .211’dx?; consequently ae oe Lee es dx 8. Letm= 33 «0. y=(~—aP th. If we differen- dy = («—a)*d 11+ 3 (v—a)*Mdv Py = (x—a)wW ll +6 («—ayditde + 2.3 (v—a)Md2? / dy = (v—a)’a°S1 + 9. (a —a)*d* Idx + 18 (v—a)dlda? +1.2. 3 Idr* and When? we phe Sap te 8 TV aa, 8 ay — / ek a oe 2 dx 4. Generally, as is easily seen from the continuation of the calculation, we find for d’ y, after differentiating the 295 ‘equation y==(v—a)"I1 m times, a differential expression, whose last term is 1.2.3... mlIldx”, and in which all the remaining terms contain the factor w—a. If .:. we put v=a, we then obtain d’y = 1.2.83.,.... mII‘dx”, and consequently fi fuley Pari dy” SECTION CXLVI. Pros. When ¢ and y denote two homogeneous func- tions of the roots 2’, wv’, a/”, &c. of the given equation Bea? + Ax’? + Ba’? + Cr®?. + &e = @ ‘from the known value of the function ¢ it is required to find the value of the function y in the ease where the equation | ee Pp Oe ne a Or =O on which the first depends, contains equal roots, amongst ‘which is the known value of t. Solution 1. In the remark in § CXLIII, we find the following expression for abe 1 ag Ms Oey Af Pe See sp Oz, in which 2’ =t+ P, W@=t?}4+ Pt+Q, &c.; and from this general expression we obtain the particular values of y, ¥, y”, Se, when we substitute i’, t”, t’”, &e. for t. Now we wish, in the first place, to give this expression a ‘form which will be more convenient for our purpose. lt il ne BE tere Oe e 4, 296 9. Since t/, t”’, tv’, &c. are the roots of the equation | II, then w+ Pr + QU? + &. = (t—t’) (¢—t”) (t—t (tt)... (t— &) If we differentiate this equation in reference to ¢, we obtain, after dividing by de, at? + (r—1) Pi™ + (r—2) QU“? + &e. = (¢—t’) (tt) (€—-0") cece seen eee (t—t™) + (t—t/) ((-t/) 0") coe cec eee ees (t—t™) + (t—t/) (tt) (0) creceeeeeeee (¢—t)™) &e. If in this we substitute t/, t’, t/”, &c. successively for ¢, we obtain at!" + (7—1) Pt'" + (w—2) QU 4+ &e. = (/—t’) Y—t”) (/—t”) .....- vee (t/—E'7’) at!" + (¢—1) Pt" + (7-2) QU + &e. = (tl) (tf —) (Ua) cece eeee (” —t™) art! ae (7 — 1) Pil + (7—2) Qi" ae &e. = (1) (e!/ —¢’) (een LER (7 — t'*)) &e. ; 3. Now, if we denote that which the numerator of the’ expression for ¢ becomes by the substitution of ¢/, t/”, t/’, &e. for t, by Q’, Q”, Q/”’, &c., we then obtain, by means of the results in 2, O/ | feta ah at SW) WH) (1)... UC) ap Ee ROEM a FO Oy OA 9 SS 4 0) (0) i Hy. aan (/—t) Hh O/” Pe (TV) 71%) 71)... (0 ay &e. 297 From the form of these values it is evident, that when ’=t’, the denominators in the values of y/ and y”, are both = 0; whence, by § CX LIV, it may be concluded, that these values cannot be determined singly by a rational expression, but depend on an equation of the second degree. In like manner, when we put ¢t/ = t” = t’’’, the denominators in the values of CE EAL vanish, and consequently in this case these values must depend on a single equation of the third degree; and in a similar way it holds, when more values of ¢ are equal to one another. 4. In the first place we assume, that the equation IT _ has no more than two equal-roots ¢/, t/’. First let them be unequal, and let them differ by an infinitely small _magnitude A, so that ¢” =t/ + h. Further, let, for shortness’ sake, or) @ 0)... Ho) = te) (Eat ere evest (7 —t) = _ we then have ‘= ibaa fe = iy 2 a (t’ pee i4) Il’ — ATI’ ip teeta GSE SN So OQ” A Mh Sen (/ — ¢) TI” ~ Art’ and .*. 1 Q” Q/ a Aaa sa 5. If we omit the infinitely small magnitude in this last equation, we then can put II’ = II’, and we con- sequently have 2 a 298 Ly, ene YN j a But according to Taylor’s Theorem ae ears LA i rd | sang Nap If .*. we divide this expression by /, and then put h=o, Q” — + &e. we obtain 6. We now assume, that the equation II has three equal roots, which are ¢/, ¢/, tv”. As before, ‘consider again these roots at first as differing, by an infinitely small magnitude, and put t/ = t+ h, t= ‘+k; further, put (f—@”)¢ —t) (¢ —t”)...... ¢o—i@) =T71; =) 0 (at) ( — ) (ea par t’”) (ite ge t”) eK jaa tY’) anil (ey oN t‘*)) a TI” Then we have (3) Q/ 1 Q/ / — Ss 8 ¥y (t/ ake t’’) (t’ An ee) iT hk ; yl! as QQ’ ne 1 Q” (C7 Aa: t’) (ee aye i) ii; hich h (h—k) ; ate. Q/” 1 Q/” Wie 2 a ES oe een IY tee Tite 1 1), kk) If we add these three results, we obtain 1 O/ 1 Q/ j OQ” oy + 0 gE OR aaa ty EP ee a ee FN hle gird: peek A en eee ae or, when we omit the infinitely small magnitude in this 299 last equation, and put II’ = 11;= I, Tay 4. jf aa ar OE EMS ee BRE) ats ke 7. But by Taylors Theorem VO k PO! ii OF a he Berd at die. 119 Heder es dQ k | &Q! | AO ke a 1 * we Tat We ie.3 If we substitute this sum in the expression for y/+y +y’’, and omit what ought to be left out, we then obtain + &e. Oo” = QO/ + ete fi PO! 1 PO) h+k G71 2 ee abe a 2 oS Now, if we put h and k=o, we get i! PO! 1 cae Sah 1. 2I1, y ty ty"= +e) = ! ase Sark) 8. In like manner, if four roots ¢/, t/’, t/”, t/”, of the equation IT are equal to one another, when in the beginning we assume these roots as differing by an infinitely small magnitude, and t?=l/+h, t¢”=t+k, ¢”’=t +1, but after completing the calculation, we put h, k and l=o, we then find the following result : BOS See ie ena 162,20 when we put (ef — 0”) @ - 0%”) @— 0)... — #) = Th 9. Hence we may perceive the law. Thus, if v roots 300 M/s, UY, U..004t are equal to one another, we have ad’ Q/ 1 J / ‘// (v) Be 4 Dey Or PENS hal tim oe ae aT ony tae Y de OT 8 £8 ad en when we put (0) Yt)... 0) = I. 10. The expression II’ contains the roots t’*', ¢’*?, weeeeel™, Now, since it may happen that we know no other root of the equation II, except ¢’, it remains to be shown, how we can determine this expression directly from the above equation. 11. By the assumed nature of the equation II, when we put (t—t?*?) (t—t°*) (t—tor) vecvee(E—t™) — TI we have (t—?t’/) Tl = t” + Pi™" + QU” + &e. If we differentiate this equation v times successively with reference tot, and then substitute ¢’ for t, we then obtain (foregoing §) d’ (t’” + Pt’ + QU". + &e.) dt’” or, when we actually differentiate it once J Py 2 AG Peet tied By ran 123 2S a’ (nt!"—" + (a — 1) Pi’ + (7 — 2) QU™ 4 &e. at’ 12. If we substitute the value of II’, which we derive from hence, in 9 we then obtain 301 Pei 20) <6 109 ARO Uae Ml odie CR MD ES y +y => *ey +y a a(t’?! + (7 —1) PU + Xe.) the differentials taken with reference to (’. 13. We have .*. found the sum of the values corres- ponding to the equal values of y. But in like manner also, we may find the sum of their squares, cubes, and so on. ‘To effect this, we only require in the equations in 3, § CXLII, for the function y to substitute its square y?, its cube y°, &c. Since by these means the magnitudes 2), 2,, 2,,+-++--2x—1, only undergo any change, nothing remains to be done, but to change the expression QO = 2, + Pope + Vong F veeeee + U2, accordingly, and moreover to retain the formula just found for y/ +3’ ty +......+y. Having obtained these sums, we | may likewise always find the equation, which has the values y/, yy y//s......y as roots, and this equation must necessarily be solved, if we wish to find the above values. Remark. From what has been’ here said, we see the reason why it was said in 10, § CXXXVI, that it would be sufficient to solve the equation for the coeffi- cient p, in order to find the other coefficients q, 7, Xc., immediately, and without the solution of any other equation. For since p, g, 7, &c. are all homogeneous functions of av’, «, x”, &e. from the known numerical value of one of them, we may represent the numerical values of all the others by mere rational expressions ; because the cases in which the denominators of these expressions vanish, belong to the exceptions, and can only occur in particular 302 equations, and not in general ones, of which we treated in the above-mentioned §. SECTION CXLVII. For the sake of the use which we might, perhaps, make of this, I shall now arrange the results found in the foregoing § together, and for the greater generality, instead of the function y itself, I shall assume any power of it y*. If we denote the symmetrical functions expressed by * the coefficients of the given equation y" + ylts ne fs MS, + (y'™) ly + ty” ce ial He ma) ee + (>) Cy" >" pRa fe as ty //x a UR llle aS iE Ape of: (79 (yo) t/7Ny!* 4- Yin—lyl + Halle + a + Gyr (y™) in the order in which they succeed each other, by 2, z,, Bi Vereee Z,-1, and put, for shortness’ sake, al/®—' + (a —1) Pt" 4+ (7 —2) QU 4+ &e. = DY 2-1 + D ed Fear + Dig av. te dias «e's eeu -|- (Ke, = Q/ (in which P=t+P, QV =t? 4+ Pl'+Q, Ra=t2+ Pt? + Qi’+H, &c.); we have for a simple root of the trans- formed equation for ¢, O/ y= 973 for a double root 2dQ/ //% _ y* + Yy dp’ ’ for a three-fold root 3d2Q/ y”* + yy” tft — Bap’ : for a four-fold root ad 1 ae +} ae + oe + ae ta ar : and, in general, for an v-fold root 4 vd’Q/ y* + y/* + Jbl + eee + Cy x _— Teen 4 all the differentials taken in reference to ¢’. By means of these formule, we may find the sums of powers of all those values of y, which belong to the complex root ¢/. Having now found these sums of powers, we may also, by § IX, find the equation on which - they depend. I shall now elucidate what has been just advanced by an example. Exampte. I shall assume, that in order to solve the equation I. «*— 3r° — 3x2 + lly —6=0 we have transformed it into another II. @ — of + 21¢ + 9 —54l? + 32 =0 when we put ¢ = a‘ + 2”. I shall further assume, that we are able to find a root of this last equation, and that we now wished from it to determine the value of the func- tion y = 2/2”, Here we have the following corresponding values of ¢ ° and ¥ a! tall, al tal, a! teal”, ol peal, al eal pal! pall ala, alll, ala! allyl allel, ly!” and these give, when « = 1, aoc! all peal al” allel! al al gl gl? = [12] 304 gym (al ta”) ala” + (0 eae) aa + (ao tal") ala’? + &e. om {1 2] 2 (al bal! tala! (a! bal Pala + (ala! Px!” + &e. = [13] + 2[2°] | 2m (2! a)8a a! + (a! par!)3a/a!/ + (a! a”) 3a! x!” + &e. = [14] + 3 [23] 2 a eae BED ah ad f + &c. = [15] +4[24] + 6 [32] gpa (a! ta)8a! a! + (a! a! Pala!” (a! +2!" )Px/2/” + &e, = [16] + 5[25] + 10 [34] If we take the numerical expressions from the annexed Tables, and then put for 4, B, C, D, their values 3, — 38, —11, — 6, we then find zx = — 3, 2, = 24, 23 = 90, 2; = 390, s, = 1542, z, = 6174. If we sub- stitute these values in the expression for Q/, we then obtain, since here P’/=t/—9, Q/ =? — Ot’ + 21, R = 2 —9t2 +21 +9, S = t4*— ot? 4 2102 4 9 — 54, TY = t? — 9t4 + 21t8 + Ot? — 54t, after the usual reduction : Q/ = 2, + P’s,+ Q’/z, + R’s, + 8/2, + Tx = — 3t% + 51t4— 189 t? 4 57¢? + 3008 Also dp = 60° — 4504 + 840% +4 270% — 1080 consequently — 3t° 4 51t/* — 18918 + 57t? + 3000 6t® — 4504 + 841% 4 27%? — 108 One root of the equation II, ist=1. If we substitute this root for ¢’ in the value of y’ here found, we get y =x’ = —6. Of the accuracy of this result we can convince ourselves by solving the two equations w + a” = 1, a/x” = — 6; for by these means we obtain / 305 3 and — 2 for 2’ and x, and these are actually two roots of the equation I. Another root of the equation IT, ist= 2; and this root substituted for ¢’/ in the value of y’, gives y/ = 1. But from a/ + 2/ = 2, and x/x/=1, we find a= “ = 13 whence it follows, that x = 1 is also a root of the equation II, and that a double one. But t= 41s also a root of the equation II. If we / . J . O substitute this root for ¢’ in the value of 2’ , we find y=—, Oo which denotes that 7/ can be determined from ¢/ in no other way than by an equation of the second degree. If, however, we differentiate Q/ and ¢’, we then find dO/=(—15t4 + 20417 — 5676? 4. 114¢/ + 300) dt’ P= (3200/4 — 1800? + 2520? + 54t/ — 108) dt’ ¥Y + yf = oe = —15t/4+ Sar + 567t? +114t/ + 300 " 30t/4— 18007 + 2520? +4 54t/ — 108° If in this we put ¢/ = 4, we then get y + y” sate: 6. In order to determine y/ and y” singly, we must now find the value of y? + y/”. With this view, we put k=2; we then have Sy 1/2z//2 4 &e, = [27] £,=(0/ +2”) vx! + &e.=[23] B= (a + 2” )Pa!?2/? 4+ &e. = [24] +2 [37] = (a +9" Px?2!? + &e.=[25]+3 [34] == (a +0072? + &e. = [26] +4 [35] + 6 [4°] $= (2 + 2")r?r’? + &e. =[27] + 5 [36] + 10 [45]. 2R | 306 If we take the numerical expressions from the annexed Tables, and then put for A, B, C, D, thew values 3, — 3, —11, —6, we then find s,= 63, z, = 102, 2, = 336, z, = 1188, z, = 4668, z, = 18492. If we substitute these values in the expression for Q/, and at the same time for P’, Q’, R’, S’, T’, put the above expressions, after the requisite reduction, we find O/=2; + Pe, + Q’z, + Re, + S82, + T’2, = 63 t°—465t4 + 74149 + 873 t?—1452t/ —1056 dQ/ = (315t/4 — 1860 #9 + 2223 t? + 1746 t/ — 1452) dt’ the values of ©’ and d®’ remain the same as before. We obtain .°. 24/29 315t/4*— 1860 t? +- 229317241746 —1452 PY =" “Sout 1800+ 252074 54t/— 108 If in this we put t/ = 4, we obtain yl? + y/? = 18. We have now .:. the two equations y + 7 — 6, Vise +y” —_— 18, whence it follows, that the two values y’, ¥/”, depend on the quadratic equation Yi Gy 9 = 9 which contains the double root 3; and.*. y/=y/=3, That the result is correct, appears immediately, when we solve the two equations wv +a” = 4, a/a/=3; for these give 1 and 3 for the values of 2 and x”, which are actually two roots of the equation I. Besides, because here y’ depends on an equation of the second degree, wemay infer from hence, that t = 4 must be a double root of the equation IT ; which is also correct. 307 If we put ¢ = — 1, which is also a double root of the equation II, we find, when in the above expression for 4/, Oo ; vay we put — 1 for v/, y/ =—, as required. But if in Oo the two expressions found for y/ + y” and y® + 9, we put — 1 for ¢’, we then obtain ty =—4 y+ y?=8 and consequently the values of y’, y”’, depend on the equation yt 4y+4=0 which has the double root y= —2. We have .°. yf =y" = —2. But when we solve the two equations av +a" = —1, ve’ = — 2, we then obiain for x’ and x the values 1 and — 2, which are actually two roots of the equation I. Besides that for y, as well as for t = 4, and for t=—1, we found such quadratic equations as have double roots, is merely accidental, and this will only be the case, when the equal values of y also correspond to the equal values of ¢. SECTION CXLVIII. Pros. Let ¢ and y be any two functions of the roots of a given equation: required to find a general method by which, from the known value of one, to find the value of the other, however the functions are constituted. Solution 1. In order to solve the problem in its most general form, we shall assume, that both functions contain all the roots of the given equation. This supposition 1s 308 always allowable; for if one function does not contain all the rootsat the same time, we then can, as was already observed in § XLIX, add those that are wanting with the coefficient 0. Thus, if we had the function 2/272”, and the given equation were of the fifth degree, it would only be necessary, instead of these, to put 2/02’ + 0 . ee ee 2. The method in § CXLII for determining the numerical values of y from the numerical values of ¢ assumed as known, in the case in which both these func- tions are symmetrical, may also be applied, when they are not so, by merely making the alterations which are requisite on this account. It was said in the above place, that, when ¢’, t, t/”, ...... t™ denote the unequal values of forms of t, and 7, /, y/, ...... t™ the unequal values of forms of y, the function ty! + t/y + UY! $A eevee + (Ey is symmetrical, because the function ¢*y can have no more unequal values of forms than those of which the former function is com- posed. ‘This is correct, when the functions ¢t, y, are no longer symmetrical, because they do not in this case change, or remain unchanged, at the same time. 3. But the function ¢y/ + ty” + .... + EP y™, in every imaginable state of the functions ¢, y, are as- suredly always symmetrical, when ?’, t”, t/”’,...... {™ and Ys ly coves Y denote not only the unequal values of forms, but generally all the possible values, which arise from the transposition of the roots a’, a, x’, &c. whether equal or unequal. That in certain cases, and in 309 certain forms of the functions ¢, y, we often get a much less number of these values, is nothing to the purpose ; because here we only are treating of the general method applicable to every case. ‘ 4. The method in § CLXIII for determining the numerical value of a function y from a single known numerical value of t, may in like manner be extended to functions which are not homogencous, provided by ¢’, t/’, Ei rin vowed ALIN Oy Sy Shree oan y™, we merely denote all the possible values of forms of t and y, which arise from the transposition of the roots a’, v’’, 2’, &e., and the transformed equation II be composed of all the values of forms of ¢, and not, as has always been the case hitherto, only of the unequal ones. This equation, however, will be found by the following method: I shall assume, that amongst all the 7 values of forms there are x unequal ones, and that the equation for these last ¢* + pt*~' + qt"? + rt’ + &c. = 0 1s already found. Further, if we put =v, then vis necessarily a whole number, because by § LV, Corollary, « is always a submultiple of z, and all the w values of forms, taken » and » together, will then be equal. The equation II, which, as is now required, is composed of all the values of marmane et fil!" 5 55 <3 «™, is consequently no other than (? + pt) + gt? + rit +. &e.)” = o and it may .*. be obtained by solving this equation. If the values of forms 7’, t”, ¢”, ......¢ be all different 310 from one another, then v = 1, and the equation IT is the equation ¢” + pt*' + qt" + &. = 0 itself. 5. With respect to the equation on which the nume- rical value of the function y depends, two cases must be distinguished ; viz. 1st, the ease in which. the given equation is the most general one of its degree, and con- sequently whose coefficients are in no way combined ; gndly, the case in which the coefficients are determinate numbers, or else have some relation to one another. 6. In the first case, the equation ITI can only contain roots which are all unequal, when the values of forms ?’, UP EEO t™ are all different from one another; and if this be the case, as we have seen in the foregoing §, the numerical value of y may be expressed rationally by the numerical value of ¢t. But if the above values of forms of t, consequently also the roots of the equation II, are equal, taken v and vy together, then each of these roots 1s v-fold, and consequently the numerical value of y (when all the particular relations between the functions ¢ and y are first laid aside), necessarily depends on an equation of the vth degree, which may always be found (§ CXLVI); and this equation gives the » values of y, which at the same time correspond to this root. 7. In the second case, on the other hand, it may happen, that this or that root ¢/ of the equation IT, besides the » — 1 equal values, which arise from the 311 identity of the values of forms, has also other equal ones, which have thei bases in the particular property of the given equation itself, and consequently in a case of this kind the numerical value of ¥, which corresponds to the root ¢’, must necessarily be given by an equation of a higher degree than the vith. 8. Hitherto we have not noticed, in the general inquiries respecting the dependence of the numerical values of the functions ¢ and y, the particular nature of these functions; it is now time to consider this. We have already seen, in the preceding §, that, when the above functions are sym- metrical, the function t/*y/ + t/%y// + 2.06. + EPYy becomes symmetrical, when for ¢/, t’’, i’, ...++ t™ we merely take the unequal values of forms ; by which means not only the calculation is essentially shortened, but likewise in the case, in which the transformed equation for t has equal roots, the numerical values of y, which correspond to these equal roots, are expressed by lower equations than we should have obtained if we had intro- duced all the values of forms of é. But a similar abbre- viation may generally be used, when the functions ¢ and y are such, that when the nature of one of them is expressed by the equation A= AW = AX =...=A® = AM =... = A® between the p types A’, A”, AM”, ..----AO, AMT s..000e A®), the nature of the other is determined by the equa- tion 0 EY: fom, LL EW, My merely between the k types A’, A”, A/”,.,. A”. For 312 if we try to find all the unequal types, which a function contains, whose nature is expressed by the type-equation A= AY = A” =...... = A™, and then find all the values of forms of the function ¢ and y corresponding to these unequal types; consequently, when ¢’, t”, ¢”, see ™ and y’, oy, y!, ..s..y™ denote these values, the function ty’ + ty!" + ..... + Ey will neces- sarily be symmetrical, because there is no value of forms of t’y, which is not included amongst those of which this ageregate is composed, 9. As for the formation of the transformed equation for tin the assumed property of the functions t, y, we must distinguish the two cases, where ¢ or y is that func- tion, whose nature is determined by the equation 4’ = AV =A = ...... =A. If the first be supposed, then the values of forms 7’, t’”, t/”,...... t™, are all different from one another, and the equation II, which is com- posed of these values of forms, is actually, as in the case of homogeneous functions, only the result of the unequal values of forms. But if the second supposition be taken, then amongst the values of forms ¢/, ¢’”, t/”,...¢°~ there are several equal ones; and when we put the numbers of the unequal ones amongst these, consequently the number of the unequal values of forms which that function has whose nature is determined by the type-equation 4/= A’ = Al” =,..= A®, equal to u, the number: 7 is -a multiple of the number pn. If .:. we put 7 = py, and assume that ¢* + pt’! + qt"? + &. =o is the equa- tion, which is merely composed of the unequal values of forms of ¢, then the equation II,: which is composed of 313 the values of forms /, t”, t//”,...... t, is no other than the developement of the equation (i + pit*—' + git + &.)” = o. 10. Further, since to each root ¢ of the equation II there are v corresponding values of y, consequently the nu- merical value of y depends necessarily on an equation of thevth degree. If the functions ¢ and y be homogeneous, then y=1, and consequently this value depends only on an equation of the first degree, as required. But all this only obtains so long as the given equations are general ones; for in particular equations it might certainly happen, as was already observed, that the equation for y were of a higher degree. 11. Besides the relations given in 8, between the functions t, y, there are numberless others, in which the calculation may, in like manner, be simplified. Such a simplification as this is always practicable, when in all the values of forms of ¢, which arise from all the possible transpositions of the roots a’, 2”, a/”’, &., such as ¢’, t’’, ee t may be omitted, which are all either different, or the periods of the different values occur more than once, and at the same time are such, that the function ty! My! UP Yl! Be + EM Py is symme- trical. 12. Although there are cases where the calculation may be simplified, when, instead of all the values of forms of the function ¢, we only use those which possess ae load 314 the properties just mentioned, yet for the determination of the value of y from the value of t, there is no further disadvantage arising from it (with the exception of ‘the calculations being extremely prolix). It may indeed be objected, that in this case the equation for y rises to a higher degree than is necessary, and that it may happen, that we cannot solve an equation of this kind, notwith- standing, perhaps, that in the calculation properly arranged, we arrive at a solvible equation. But since in this case amongst the roots of the former equation, there must be more than one which are equal, and in the sequel it will be shown, that an equation of this kind may always be reduced to another, which only contains the unequal roots, consequently in the present case the lowest rational equa- tion for y, .*. this objection is removed of itself. SECTION CXLIX. Pros. Let ¢ and y be two functions of the roots of a general equation of any degree: required to give the : degree of the equation, by which the numerical value of y 1s determined from the known numerical value of ¢. Solution. In the function ¢ perform all the transpo- sitions of the roots 2’, a/’, 2/”, &c. for which its value of form remains unchanged ; in the function y perform the same transpositions as in ¢. Let » be the number of the unequal values of forms of y, which we obtain “by these means ; then the equation between ‘y and ¢, with reference to y, is of the yth degree. For since the equal o15 values of forms of ¢ have only a single numerical value, the » unequal values of forms, on the other hand, » different numerical values, consequently v numerical values of y belong to a single numerical value of ¢ ; the former can be determined from the last in no other way than by an equation of the vth degree. Examrtx I. With respect to the general equation of the fourth degree, let t= f: (2) (a) (a/”) (/%), Y= 6: Q’) @”) 0”) (v’’), and let the nature of these functions be expressed by the type-equations SFE) @) GO") AY) =fi OO) (#) 0”) w= FP @) 2) OY) 0 =H YA) @) ee aE Cans!) G0) ahs a) (ts). Cx’) (a!) =! C1) @)@%) ON” Now, if we try to find the equal values of forms of ¢ (§ LV), and then perform the same transpositions in y, we then obtain the following corresponding values of t and y: 2 (2") (2!) (e) (a/") 2@) @) @™)@”) : (a) (a) (a). C2’) : G4) ir) Gry) (ih) 5) (alY): (24) (0) > (a!”) (a!) (x!) (2!) : (a) (2'") (x) (2) : (2/”) (a!) (2) (2’) EGC) Gan : (2’) (a’’) (ia) Gale aly) @”) @) : (x #y (v’) (73) (xt) 2 (x) (a!") (e) (we) : feat xo) (2’) (v’’) : (es) Ca) (a) (x’) CAC! OE ZINC) SSSA SS, bs = — Sn — ie — in a > Se > a Of the eight values of y, which we have here found, the four first, as well as the four last, on account of the sup- 316 posed nature of this function, are equal to one another ; consequently two values of y belong to a single value of t. Therefore the equation, which gives y in terms of t, is of the second degree. To the numberless functions of the assumed nature, the following ones belong, viz.: t= a/a!/ + a//x’ " y — a’ al! — ala"! a ox” a8 on or y — a’ aL al’ = a’ + a!’ + 0a/” +0a/%. Consequently, if the nume- rical value of 2/2’ + a’x’” be known, we may find from it both the numerical value of 2/2’, and that of 2’ + 2”, by the solution of an equation of the second degree, which agrees with § XLI, where we merely had to solve an equation of the second degree. Examese II. For any general equation, let t= alata A al”, y= a! — a" = a! — a! + oa!” + 2X”). Now, in order to find the degree of the lowest rational equation, by which y may be determined from t¢, pro- ceed as follows : Equal values of forms | Corresponding values of forms of ¢ of y alata! 4. ylV a — a + 0 (x + al’) Talal! 4 gl? x — al 4 0 (a + 2!”) allalal! 4 gl¥ vw! — a! +0 (x + a”) alallal a gf a! — al 4 oa! + a!) wala! 4. lV gl! — a! + 0 (a + a/”) aMayllyl 4. gl¥ gl a! 4 a (x 4 ay Consequently six different values of y belong to a single valueof't, viz.202/ —2", al, ol ere ae el”, | 317 ES tel tty/tl 2 eS é a —a’; and .*. y can only be determined from ¢ by an equation of the sixth degree, when its coeflicients are rational functions of t. Besides, since the values of forms of y, taken two and two, are equal, this equation .*. only contains even powers of y. SECTION CL. Pros. Let t, y, be any two functions of the roots av’a!’x’ &e. of a general equation: required an operation to find the lowest equation, by which the numerical value of y may be determined from the numerical value of t, under the condition that the unequal values of forms of t only are made use of. Solution. Find, as in the foregoing §, the equal values of forms of t, and the corresponding values of forms of y, and from these last take away the unequal ones; let them be of! of//5 of" evsecesns y™: then the required equation will be of the vth degree, and it has the above values for roots. Let y + py + qy v—2 + ry + &e. —0do be this equation; then p = y/ + y/ +9” + &.9q= yy! + of yl! + yf yl! + &e. 7 = aly! yl! + &e. Se. Consequently, since the functions p, q, 7, &c. with refer- ence toy’ yy yy w.s006 y are symmetrical, then in those transpositions of the roots x’, 2”, et” Se." tor which the function ¢ continues unchanged, they in like manner undergo no change. If .*. we denote the une- qual values of forms of t, by t/, U/, t/, veseceee. &, then 318 wr py pls UMM, cagtesese («™)p™, are all the pos- sible unequal values of forms of tp, and in the same way Peay sit! Gg GUN 7a 20 Bt (t'™)*¢™ are all the possible values of forms of ¢‘g, and so on. But if this be symmetrical with reference to 2’, 1”, a/”, &C. .....000 then also are the functions tp! + tp” + t/p// + ... RP dC sill iy he ae sea ee sees got cama Sa Fire LE by Arch Pg -|- (t™)g™ necessarily symmetrical with reference to x’, 2”, a///, &c., and consequently the unequal values of forms Ui Zabel oes leans t™ are sufficient for the determination of p, 9,7, &c. Therefore the operation given in§ CXLVI may be applied immediately, and, without any alteration to the coefficients p, g, r, &c. Thus, if we wish to deter- mine p, we find, in the first place, the transformed equa- tion for the function ¢ according to the third chapter ; it is : 7 + Pe + QI? + Ri™ + &e. = 0. Having found this, we immediately get | a He Pl OS hs cote 23 rt +ae—l . Pi 4+7—2 . QI? + &e. in., which ).Po= ¢ +P, OC 2 1 Pio Qeke., and the symbol of the form z, denotes the numerical value of the symmetrical function t/“p/ + ¢/‘p// + tp” +... apie + (t™)‘p™. For the coefficients g, r, &c. the same equation and the same expression obtain, with this exception, that by z, we must understand the numerical values of the functions ¢/‘q! - t/q!’ 4+ U/"q// 4 cose arti (EP) Ig, 2 rl Mt et coedenet (Er, &e. 319 SECTION CLI. So long as 2’, a’, a’, &c, may be considered as the roots of a general equation 2” + 42""'+ Ba" * + &e. =o, consequently of one whose coefficients 4, B, C, &c. are undetermined, we shall always find rational functions of t for the coefficients p, g, r, &c. But if these roots relate to a particular equation, then it may happen, according to the nature of the function ¢, that the common denominator wi™—'! + w—1. Pi™ + 7-2. Qt??+4 &e. in the expressions for p, q, 7, &c. is equal to 0, and that it even continues equal to o, when it is differentiated more than once. We now assume, that we must differentiate it z—1 times before the denominator ceases to vanish, consequently it follows, from § CLXVI, that the coeffi- cients p, g, 7, &c. depend upon the same number of equations of the th degree pt +a'p*! + b’p*? + cp? + &.=0 qt +algt + bgt? + e/g + &e. pt pagllpt-) 4 b//ph—? 4 oy 4. &e, = o &e. which may always be found by the method there given, and in which the coefficients a’, b’, c/, &c. a’, b/’, c’’, &e. al”, b’”, c/”’, &e. are all rational functions of ¢. Il ° All that has been said in this chapter respecting the function y, may also be applied to the function 2. Thus, if we wish to determine a root, say 2’, from the known value of a function t= f: (v’) (”) (@”)...(2”), nothing further is necessary, than to put y=a’, and to proceed besides in the way already pointed out. We now perceive the reason why we are not able, from 320 the known value of a symmetrical function of the roots of an equation, whatever the nature of this function may be, to determine these roots. or since a function of this kind, in all the transpositions of the roots, always retains the same value, .-. it must necessarily give all the roots at once ; and, however we begin it, we shall consequently always again get an equation, which is not different from the given one. ( 321 ) VIII.—A GENERAL METHOD. FOR THE SOLUTION OF EQUATIONS OF ALL DEGREKS. SECTION CLII. IN § CXLI we have seen that the requisites for the general solution of equations may be reduced to two ; viz. first, to find such functions of the roots, by means of which the equation, into which we have transformed the given one, is adapted to the selution ; and secondly, to determine the roots from the known value of the assumed functions. The second requisite we have handled in the foregoing chapter; the first, together with its application to the general solution of equations, will form the subject of this chapter. In order to render the notation more easy, and the inspection more convenient, I shall henceforth omit in the types the letter x, together with the superfluous brackets, and for the dashes substitute numbers; thus: f: (12345 Reese n) instead of f: (a’) (2) (2) (2’”)...(x™), and me: (342651) instead of f:..(2’), @/"). 2”) (2), @D rar): SECTION CLIII. Rue. When from the period of n types A,, 4,, A,, A,......4,,....4,......4,, which may be derived from the equation 322 f: (123456 ...... n—1n) = fi: (284567 0 nl) we take away any two 4,, 4,, and find all the possible types, which may be derived from the transformation-rule A,,=A,; then we shall get a period, which, in the case where » — yz and y are prime numbers to each other, consists of all the n types 4;, A,, A;,...... A,; on the other hand, when v— and » have a common measure m, this period only consists of — of these types. The m types, which we successively obtain by deduction, succeed each other in the following order : kis, bigks Rio: Elias AMSA so that the dashes ny, v, 2v—p, 3v —2u, 4v—Sy, &e. form an arithmetical progression, with the difference v —j, when from all the terms of this progression, which exceed the number 7,. we omit this number as often as possible. Thus the equation Sf: (12345678) a (23456781) gives the period Pyle coeetbers fi: (12345678) CGY EEL J: (28456781) oe STE RO Dae fi: (84567812) Ag orererrevess Jf: (45678123) Ag ivecucassas. fix) (O01S1204) Peer Are oseeee fs (67812345) pe el Na Mae Sf: (78123456) Aguavipe teens fi: (81234567) If we equate every two of these types, we obtain 323 For the ' equation The period A= A,| Ay 4 Ay, A 4,= A,| A, A,, 4, A,, As, Ay As, Ag A, — A, ae, A, A, me A, A, A, As, Ay, A;, A,, A. A, Aa si iy Aas hs, be A, = A,| A,, A;, A, A,, As, A,, 4;, A, hod, Nid “bss MAL Ag nde dep A: As Alt Awe Aad lacke Hct Me hed Ag Aes Ae Ai, ade, As &e &e. The reason of this is easily found, and depends on the properties of numbers. Corollary I. The rule is also correct, when, instead of the type-equation 4,,=A,, we take the type-equation A,=A,, if in the progression which is taken away, viz. V; My, 2Bu—v, 3u—2v, 4u—3v, 5u—4v, &c., when we come to a negative term or o, we add the number 2 so many times, till it becomes positive. ‘Thus we have For the equation The period 4, = A, A,, A,, A, Ay, A;, A,, A,, A; A, = A, | 4,, 4,, 4, A; Ape Alcda tt APL As, Ac, As,) 4... A, A, a Aalto Aims hal ee e t a dh Ais Hee) A, = A,| 4, Ai, As, As A, = 4, Ay, A,,'4,, As, Ay, As, Ag, Ay z &e. &e. 324 Corollary 11. If .. m be a prime number, we then always obtain the same period again, whichever two of the types A;, A,, A, ......... A, we put equal to one- another. SECTION CLIV. Transpositions of the kind, which the equation 4,, = A,, or 4, = A,,, gives in the preceding §, are called re- curring transpositions, and the periods which are obtained from them, recurring periods. The characteristic feature of transpositions of this kind consists in this, that each root is removed one place forwards or backwards, and in the first case the last takes the place of the first; but in the second case, the first takes the place of the last, so that there is a kind of circular motion ; as if, for instance, a number of persons stand in a circle, having their backs to each other, and all walking at the same time either backwards or forwards. The transpositions are called recurring ones, when only some of the roots move in the manner just mentioned, but the remaining ones retain their places. ‘Thus the equation f: (12345678) = f: (34512678) only gives recurring transpositions of the first five roots. The Jaw of the preceding § is also true for this, when we take for » merely the number of the roots to be transposed, and the remaining ones are considered, as though they did not exist. SECTION CLV. Ruve. If the equation 2+ Av’ + Ba" + &e. =o, by the introduction of afunction t=f: (12345...n), o® 325 be transformed into an equation of two terms ’ — A =10; then the roots of this last equation t,t”, t’”, ....+ t are always the numerical values of such values of forms of the function t, as, taken together, form a period. Proof. The roots of the equation t’— A =o may always, as may be seen from the fifth chapter, be expressed by t’, at’, a7t/, a°t’, ...-.. a’—' t/, when a de- notes a primitive root of the equation 1”—1=o. There- fore t/ = at’, = at”, 1” = at’, .....08 = aera t= at. Now, let 4, 4, As, Ay, ....-- A, denote the values of forms, which correspond to the roots ¢’, edict aia veeeee £®, then also must 4, = a4,, A, = 2A,, A,=aAsz, Wee _A, = aJA,_,, A; = 24,; and since ‘every such equation Avy = «/,_,, independently of the particular values which we may assign to the roots and EEN &e. must be true, it will also remain true, when in the two parts of this equation we transpose the above roots in any way, provided it be the same, because this is precisely the same as when the values of these roots are changed. It .*. we assume, that in A,_,, we have so transposed the roots 2’, v”’, x’, &c. that it becomes A,, and that by the same transposition A, is transformed into any other value of form 4,; then we have also A, = aA, Butif also A,,, = 24,, consequently 4,,, = A,. Uence it follows that 4,,, is generated by the same transposition from 4,, as A,is from A,43 .*- Ae is produced from A,,as A, is from A,, as A, from A,; and so on; lastly, as A, is from A,_,, and A, from A,. Since .°. all the values of forms 4,, 4,, 4s, Ay veveerees Ay are deduced from one another by the same transposition-rule, and. - Sa 326 from the last ,, the first 4, is again obtained; .:. it follows, that these » values of forms constitute a period. SECTION CLVI. Pros. Amongst all the possible functions of the roots a’, x’, of the general equation of the second degree xv — Av + B= o0, find that one which is fit for its solution; under the supposition that we know not how to solve any other equations, than those of the first degree, and those of the form 2 — K = 0. Solution 1. Let t =f: (12) be that function, which is fit for the solution of the given equation. Since it has two values, viz. f: (12), f: (21), then the equation for t, taken generally, is of the second degree. If we only wished it to be of the first degree, then must f: (12) = Ff: (21); but then f: (12) would be symmetrical; and the roots x’, a/’, could be determined from the known value of ¢ only by the solution of the given equation itself (§ CXLIX). There now remains nothing further than to assume that the two values of forms f: (12), f: (21), are the roots of an equation of the form ?? — K = 0, because it was assumed that we know not how to solve an equation of the second degree of any other form but this. That they may be so, appears from hence, that they form a period (preceding §). 2. Since K = — t/t’, when @’, t/’ denote the two roots of the equation t? — K =o; then also K = — J: (42) x f: (21); and since this product remains 327 the same when we substitute x’ for a’, .:. K is a sym- metrical function of these roots. Consequently this magnitude may be expressed rationally by the coefficients of the given equation. 3. Since f: (12), f: (21) are the roots of the equa- tion t? — K =o, therefore f: (12) = — f: (21) and this is the only condition which we have to fulfil. Having once found the numerical value of t = f: (12), then also the roots 2’, x”, may be determined without the solution of any other equation, because the values of forms fi: (12), f: (21) are different (§ CXLIV). 4. This condition, however, is evidently sufficient, when we put f: (12) = @: (12) —@: (21), where it is allowable, for @: (12) to assume every arbitrary function of «’, 2 which is not symmetrical. For from f: (12) = p: (12) — @: (21) we obtain by the substitution of x’ for x”, f: (21) =: (21)—¢: (12): consequently f: (12) = —f: (21), as was required. 5. Hence it follows, that all functions of the form p: (12) — g: (21) are fit for the solution of the given equation. SECTION CLVII. Pros. Solve actually the general equation of the second degree 2? — Av + B= o. Solution 1. We have seen in the foregoing §, that all functions of the form t = : (12) —@: (21) are fit for 328 the solution. Amongst the infinite number of functions which we can assume for @: (12), the root a is the most simple. Put .:. p: (12)=2", then é=@: (12) —@: (21) =21/—a’’. But the equation ?@—K =o gives K=?= (a! — a")? = o/2 + 7/2? — 2a/a” = [2] —2[1°] = A°—4B; the transformed equation consequently 1s ? — (A? —4B)=0 and this gives t= +./(A’—4B). We.:. have the two equations ML aptee BAL a! — a! = + / (A? — 4B) and hence Paes ak Gam) /= co INAS a as was required. ¢ SECTION CLVIII. ° Pros. Find the functions which are fit for the solu- tion of the general equation of the third degree a? — Ar? + Br —C=0 under the supposition, that we know not how to solve any other equations than those of the first and second degrees, and those of the form #? — K =o. Solution 1. Let t = f+ (123) represent all those func- tions which are fit for the solution of the given equation. Since the roots a’, 2”, a/”, admit of six transpositions, consequently the function ¢ contains six values; and these are Ff: ARB Fe A2SL ery ole) J (218), oF 82) loon me 329 Consequently, taken generally, the equation for ¢ is of the sixth degree. 2. The six values of forms in 1 are arranged in recurring periods. Thus, in the first horizontal row, we have the recurring period f: (123), and in the second Ff: (213). If we assume, that the three values of forms of the first period are the roots of the equation of two terms, viz. (®?— K =o, then XK is the product of these three xoots,tand «y= fi: C123) Ff : C23 LyX fi. (S12 But this product, as may be easily seen, is such, that in all the recurring transpositions of the three roots x’, Teepe a’, x’, it undergoes no change: for if we perform these transpositions, we then obtain the period Fe (128)! xf 9(231) 2 fs (812) BPEL) x of 28(312) x fe (123) Poa x fs (1s3)" xt fee (23) and these three values of forms of X are evidently not different from one another. The six values of forms of K, which arise from all the transpositions of the roots a’, a’, x’, are .*, equal, taken three and three, and consequently this function has no more than two different values ; and these are fi: (123) x f: (231) x f: (312) fi: (218) x f: (182) x f= (821) of which one can be derived from the other merely by putting the roots x’, «” for each other. 3. Since .:. K has no more than two different values, consequently this function depends on an equation of no higher degree than the second, and the roots of this equa- 20 330 tion are these two values, But these values admit of 2 more simple form; for since A = ??, and t = f: (123), then also K = ( fi (123) ee and since the second value, as we have already seen in 2, 1s obtained from the first, merely by putting the roots a’, 2” for each other, con- sequently (f: (213) )° is the second value of K. 4. Let K?—pK+q=0 be the equation, on which the function K depends, .° p the sum, and g the product, of the two values of K. Consequently p = (f: (128))* + (f+ (218))° q= (f2 (128) xf 18) J and these functions p, q are such, that in all the trans- positions of the roots 2’, a”, a/”, they suffer no change. Since p and qg are symmetrical functions of the roots a’, a’, a’, they may always .-. be expressed rationally by the coefficients of the given equation. 5. We have consequently reduced the transformed equation for t, which originally was of the sixth degree, to two equations e— K=0 Kk? —-pK +q=0 and we are always able to determine the coefficients p, q, from 4, B, C, when the function f: (123) is known. Having once determined the coefficients p, gq, we then obtain, by the solution of the second equation, the two values of K, and if these be successively substituted in the first equation, by its solution we obtain the six values of t.- 33] 6. Since all the values of ¢ are different from one another, we may always determine (which is known from the foregoing chapter) the values of the roots 2’, x” yall’, immediately from the values of the function ¢ already found, and that without the solution of any other equation, however constituted this function may be. 7. It now only remains to determine the function t = f: (123) in such a way, that the three values of forms f: (123), f: (231), f: (812), may be the roots of an equation of the form t®? —K =o. If this be the case, then these three values must have such a rela- tion toeach other, that PoC lao) = ape 29a) pe Cole): In order to perform this, we assume any other arbitrary function @: (123), and put ff: (123) = Ag: (123) + Bo: (231) + Cog: (812) in which 4, B, C, denote coefficients hitherto unknown. From this equation, by a proportional transposition of the roots, we obtain f: (231) = Ag: (231) + Bg: (312) + Co: (123) fi: (812) = Ag: (812) + Bo: (123) + Co: (231) and when we substitute these values in the foregoing pro- portional equation, we get Ap: (1238) + Bo: (231) + Cp: (312) =a (Ag: (231) + Bg: (312) + Co: (123) ) =a? (Ag: (312) + Bg: (123) + Co: (231)) If we equate the coefficients of these values of forms, we obtain, for the determination of 4, B, C, the following equations : 332 A=zaC=a?B B=aA=2?C C= aB = 2°A. Since a?=1, the first gives B=ad, C=a* J, and these values verify also the second and third. ‘The coefficient A remains undetermined, and we may .*. put it equal to 1. Consequently f: (128) = @: (123) + ag: (281) + 22g: (812). 8. We can, as was observed already, for ¢: (123) assume every arbitrary function; yet, for another reason, those which undergo no change in the recurring trans- positions of all the three roots, cannot be used. For in the case where @: (123) is a function of this kind, we have @: (123) = @: (231) = 9: (812), and. con- sequently f: (123) = (1+a+a’) @: (123) =0, because 1+a-+a?=o0. This restriction .*. might with good reason be omitted, since it is a necessary consequence. SECTION CLIX. Pros. Required to solve actually the general equation of the third degree a? — 4a* + Br —C=0. Solution 1. In the foregoing § we saw, that all func- tions of the form ¢ : (123) +a: (231) + a%¢: (812) are fit for the solution of equations of the third degree. Consequently there are numberless ways in which these equations may be solved. The most simple supposition 333 is@: (123) =a’; then p: (231) = a”, g: (312) fis (2S ah ib oak ote oF aff 2. Hence we obtain (f: (123))> = [3] + 6[19] + 3 a(alla!? $alal!!? ala!) 4 3a(al xl? + a! a/!? + a/y///2) and when in this we substitute 2’ for x’, we get (f: (213))> = [3] + 6[1°] + Baa(ala!2 allel 4 el! x!2) + 8 2 (aa? lly? 4 ie or when, for shortness’ sake, we put [3] + 6[13] = x a/2 + a! gll/2 tal 4/? = Q, OG? al gl A al ll?) i (f: (123)?) = P + 32Q + 32°R (f: (213))? = P + 3aR + 3aQ. 3. Hence, by 4 of the foregoing §, we further obtain p= (f: (128))°+(f: (213))' = 2P 4+ 3(a + a?) (Q+4+ R) or, since a +a? = —1, andQ+H= [12], p = 2[3] + 12[1°}] — 3[12] and when for the numerical expressions we put their values taken from the annexed Tables, p = 243 — 9AB + 270. 4, Further, by the foregoing § g= (fs: (123))8.x (Ff: (@13))? = (P + 3aQ 4- 3a°R) (P+ 3aR + 32?Q) =P(P+3(a+a*)(Q+R)) +9(a+2)QR+9(C+ R2) or, since 2 + a = —1, Q4+R= [12], QR=[37]+ 3[27] + [124], Q? + A? = [24] + 2[123] 334 q = ([3] + 6[1°] ((3] + 6[1°] — 3[12]) + 9([24] 4+ 2[123] — [3?] — 3[2%] — [1°4]) or when for the numerical expressions we put their values, qg = 4 —9 4B + 27 4B — 7B = (4 — 3B). 5. Consequently, the two equations in 5 of the fore- going § are e—K=0 Kk? — (248 — 9AB + 27C) K 4+ (42 — 3B =o. 6. Let K’, K” be the two values of K, and U, t’’, the corresponding values of ¢, then 3 Cif F128) re Per vai \/K! 3 Coys (213) —_ al -- ar’ +. azyl/ or ASK". We have .:. for the determination of a’, 2”, «/”, the three equations bet ELE tial? ol ee 3 le ae ry eI ee \/K! 3 v4 aa! + ate = \/K". If we multiply the third equation by a, and then add it to the two first, after dividing by 3, we then get, since l+a+te’=0o, , At VK 4 e\/ KR" 3 xv If we multiply the second by a, and then add it to the other two, we then get 3390 peek a) Kl +. N/R! 3 Hi Lastly, if we multiply the two last equations by «, and then add them to the first, we obtain 3 3 fe A+ a(N/ K’ “2 N/K) i ae a oy 7. But since each of the two irrational magnitudes 3 3 / Ve \/ K’’, has three values, for instance, the first 3 3 3 the values a\/ Ay a\/ K’, a\/ K’, and the second the 3 3 3 values aN/ Ky‘, a\/ Aik a\/ kK”, we must first deter- mine which are to be taken. I assert, in the first place, . 3 3 that the two roots \/. K’, \/K! » must always be com- bined with one and the same power of a. For let 3 in which the exponent v may either be one, two, or three. If in this equation we put the roots 2’ and 2” for one another, we then obtain 3 el! raw “aia! = a /. ‘Kl! because in this case K’ is transformed into A”, .°. also 3 3 \/ K’ into / K’. Hence it follows, that in the two last of the three equations, consequently also in the results 3 3 derived from them, the roots \/ Eis; \/ K’’, must always be combined with the same power of a. It only remains now to determine the exponent v. 336 8. With this view, if we put in the values of a’, a’, a’, 2 3 3 3 found in 6, a°\/ K’, @\/ K’forN/ BK’, N/K, respec- tively, then 3 3 8 3 3 hg es: A+ at /K! of aK! i, ped Arat(N/ Ke e \/ K) eae GI Set aE HDT RI YL Now, since these three roots must also be found, when for « the other primitive root «2 is substituted, then also in Aber (N/ K+ NTR”) ; muyst be a root. But since this one is not to be found amongst the three here given, consequently no other assumption is allowable, except this one, that a’t! = 2 9 Ree. Consequently aK’, a : kK”, must be substituted for \/ Kee / Kk”. Tf we actually do this, we find the three following equations : (4+ VK + VK: 8 (A + aN/Ki 4 @\/K"): 3 (4 4 at\/ Ki 4 A/ KRY): 8 in which it is only necessary for K’ and K” to substitute the two roots of the second equation in 5. 9. Tf, for the sake of greater simplicity, we put A = o, then this equation is transformed into K? — 27CK — 27B° = o 337 and the two values of A are 27/30 + WGC + 7 B)). If we substitute these values in the three roots in 3, we obtain VECEVGC+EB)]+ V/aC— VGC+3B) a\ {aC + V(2C24+3,B)] +a°\/[3C— VEC? + 3B] aV/[§C+ VQC+EB)]+a\/[3C— VAC+ EB] which agrees with Cardan’s formula. SECTION CLX. Prox. Find the functions which are fit for the solu- tion of the general equation of the fourth degree, viz. at — 4x? + Bx?— Cr+ D=0 under the supposition that we only know how to s equations of lower degrees, and those of theY i AP = 6, Solution 1, Arrange the twenty-four values of i . ff: (1284) in recurring transpositions, under and oppasi each other; (the symbolical function and the brackets. — VOG, are omitted for shortness’ sake) Ss 2341;)8412/4128 83142114283 |,4231 1243);2431/438312 134213421/)/4213 $82414,;2413|)4132 214314114382 )432T. Thus, in the first vertical column, we first put f: (1234) with its recurring transpositions of the three first roots, yi 338 this gives the recurring period f': (1234), fs (2314), f: (3124). Then we in like manner put /: (2134) with its recurring. transpositions of the three first roots, and we obtain the’ period f: (2134), f: (1324), f: (3214). . From the value of form f: (1234) by a recurring transposition of all the four roots, we further derive the values of forms f: (2341), f: (3412), f: (4123), and place them near f: (1234) in a hori- zontal row; we do the same with the remaining five values of forms in the first vertical column, so that in each horizontal row there is a recurring period. 2. Since the four values of forms in the first horizon- tal row form a period, they may .*. be the roots of an equation of the form ~~, té#—_ K =0 (§ 155). Now, since — X is the product of all the four roots, we have —K=f : (1234) xf: (2341) x fi: (8412) xf: (4128) and this product is such, that in all the transpositions of the roots we can obtain no more than the following six different values : fs (1234) xf: (2341) xf: (8412) x fi: (4123) SF: (2314) xf: (8142) xf: (1423) x fi: (4231) SF : (8124) x fi: (1243) x fi: (2431) xf: (4312) SF: (2134) xf: (1342) x fi: (3421) xf: (4213) f : (1824) xf: (8241) xf: (2413) xf: (4132) J : (8214) xf: (2143) xf : (1432) xf: (4321) ° which arise merely from the transposition of the three roots 2’, 2”, a/”, 339 3. From the equation t* — K = 0, we obtain K = p= GF: (1234) )4 Consequently also Ch (1234) )4 must be such a function, that in the recurring transpo- sitions of all the four roots it remains the same, and con- sequently has no more different values than those which arise from the transposition of the roots 2’, 7, x”. Therefore the six values of K can also be expressed thus : Ce: (1234) )4, (f: (2314) )4, GF: (3124) (f: (2134))4, (f: (1324))4, (Ff: (8214))* 4. Since the function X has still six different values, consequently it necessarily depends on an equation of the sixth degree. If thisequation be solvible, then it must admit of being reduced to such equations, whose solution is assumed to be known. [I shall .-. assume, that the three functions (/: (1234) )4, (f: (2314) ),* ( f: (8124) )*, which arise from the recurring transpo- sition of the three roots 2’, x’, x’, are the roots of an equation of the third degree I. kK? —pK? + qk —r=o0 consequently the coefficients p, q, 7, are no longer ra- tional, because otherwise K can have no more than three values. They must .-. depend on certain equations, which we shall now seek. 5. Since (f: (1234))*, (f: (2314))4, (fF: (3124))4s are the roots of the equation I, then p=(f: (1234) )4+ (f: (2314) )*+ (Ff: (8124) )* - 340 q=(f: (1234))*x (f: (2314) )* +(f: (1234))*x (f: (3124) )4 +(f: (2314))*x (f: (3124))4 r= (f: (1234))*x (f: (2314))*x (f: (3124) )* The functions p, g, 7, are evidently such, that in the recurring transpositions of the three roots 2’, 2/’, 2” they undergo no change. But in the recurring transpositions of all the four roots x’, «, x”, x’”, they in like manner suffer no change, because the functions ( Fe (1284) )4, (f: (2314))4, (f: (8124))4, remain the same after this operation (3). 6. Consequently the functions p, g, r, can have no more than two different values, viz. those which arise merely from the transposition of the roots a’, a’. If.-. we put, for shortness’ sake, p = f’: (1234), then p has no more than the two values /’: (1234), f7: (2134). Let these two values be the roots of the following equation of the second degree : p— pp +7 =o then P= f8 (1284)) xf 2 (213A) q’ =f": (1234) x f’: (2134). The functions p’, q’, are .*. such, that when 2’ is substi- tuted for x’, they remain unchanged. But since in the recurring transpositions of the three roots 2’, a”, 2’, as also in the recurring transpositions of all the four roots, they also suffer no change; they .-., are necessarily sym- metrical, and consequently admit of being expressed rationally by the coefficients of the given equation. 341 What has been here said of p, may, in like manner, be said of g and r. Consequently these coefficients also depend on equations of the second degree with rational coefficients. ~ 7. The function t =f: (1234) .°. depends on the equation of two terms of the fourth degree, viz. t+— K=0 and the coefficient K depends again on the equation of the third degree K? — pK? + qk —r=0 whose coefficients p, g, 1, are represented by three equa- tions of the second degree Pit! Bish Ge Bie GP Fi girh hse r—pr+g=o whose coefficients p’, 9/5 Pis Yir Px qi, are all rational functions of the coefficients 4, B, C, D. 8. It only remains now to determine the function f: (1234) in such a way, that the values of forms f: (1234), f: (2341), f: (3412), fe (4123) may be the roots of an equation of the form “#—K=o. If this be the case, then they must have such a relation to one another, that fs (1284) =af: (2341) = af: (3412) =a°f: (4123). Now, in order to perform this, we put in a way similar to that in 7, § CLVIII, fs (1234) = Ap: (1234) + Bo : (2341) +Co: (3412) + Do: (4123) and derive from hence the values of f : (2341), f: (3412), 342 f: (4123). If we substitute now these values in the foregoing proportional equation, we then get AQ: (1234) + Bo: (2341) + Cp: (3412) + De: (4123) =« (40: (2341) + Bo: (3412) + Co: (4123) + De: (1234)) . = a2 (49: (3412) + Bo: (4123) + Ce: (1234) + Dg: (2341)) = a3 (do: (4123) + Bo: (1234) + Cp: (2341) + Dg: (3412)) and when we put the coefficients of these values of forms equal to one another A=aD=2C=a'B B=aA=e’D= 22°C C=aB=e?A = 2D D=aC = eB = 23AJ. The first equation gives Baa A, C= 02d, D=a'A; and these values verify also the second, third, and fourth equations. We have consequently fe 1284) %— ps (1234)+agp: (2341) +27p : (3412) + a5 + (4123). SECTION CLXI. Pros. Solve actually the general equation of the fourth degree at — Ax? + Br? — Cr+ D=o0 under the same conditions as those of the foregoing problem. Solution 1. We have seen in the foregoing §, that all functions of the form @: (1234) + ag: (2341) 4 ah: (3412) + ap: (4128) are adapted to the solu- tion. If, for the sake of greater perspicuity, we put 343 gp: (1234)=.2", then fs (1234) =a! + wal + aa!” + ar!” or, when we briefly substitute for « one of the primitive roots of the equation 2* — 1 = 0, say + /—1, fi: (1234) = af — al + (eX — 2") J 1. | Hence we obtain SF: (2814) = 2! — a! + (0% — 2") / —'1 fe (8124) = a! — a2" + (2X! — 2) /—1. 2. By 5 of the foregoing §, we.*. have p= ( ao! aol! + ( pl oall) fr A see ( al! aa! 4. (al -gl?) of -¥ )s a (als! + (x/-0/") /-1 \a q= (2/0! + (a!/22/") KI )4 x (altlat (a//-2/") RP} }4 bi ( a/-a!"/ + (a/!=a/¥) f1 )4 x (al/-a!! + (a/22'") Wet yA aS (2-2! + (a!/-2!") NASA )4 x (0!/-2" + (a/-2!7) A y4 r= (a-v-- aa") ot Vee (a!/-a! + (a//!-4/7) ES )4 x (a!/-0!’ + (a/-2'”) a | Nis The functions p, qg, 7, are evidently such, that each of them can only have a single value which is different, viz. that which arises from the one here given, when we substitute 2” for 2’, and, vice versa, a’ for 2’. Conse- quently each of these functions depends on an equation of the second degree, which may be found by the method already well known from the foregoing §. As the subject contains no difficulty, and the calculation is rather diffuse, I shall dwell no longer upon it. 3. In the preceding chapter we have seen, that in two homogeneous functions the value of one may always be determined from the value of the other by a rational 344 expression, so long as we have to do with general equa- tions. Consequently also the values of g, r may be found immediately from the known value of p. Now, since p has two values, consequently the magnitudes p, 4, r may be determined in two different ways. Every such determination gives an equation K? — pK? + qK —r=o0 and we .*. obtain generally six values of K. If we put K=/f': (1284), then the six values of forms, which correspond to these numerical values, are those which arise from the transposition of the first three roots (§, CLA, 3); i227 1234); (7: 314)S 77 = (3124), SJ’: (2184), f/: (1324), f2: (3214), of which the three first correspond to the three roots of one equation for K, and the three last to the three roots of the other. 4. Let K’, K”, K”’, be the three roots, which cor- respond to the values of forms f’: (1234), f’: (2314), f’: (3124). If we substitute these values of K in the equation ¢*— K =o, we then obtain for ¢ three values 4 4 4 N/K’, N/K", N/K, and to these .-. the values of forms f: (1234), f: (2314), jf: (3124) correspond. Now, since f: (1234) = a’ + ax” + a2’ 4 afx’’, we then have, including the equation 2/+2” + a¢// + a/" = A, the four following equations : ar! Wx ng TRUE onl Clete lll vw par! + ay 4 oy” — N/K! ze! + axl! + aia! + abs!Y ox N/ Kv a cea! te 8 4. a8ylY = A/ KM, 345 5. If we multiply the three last by a, and then add them to the first, we obtain, after dividing by four = A and the remaining roots 2’, 2, 2’, are all of the form ad + b\/ K! + oN/ K” + d\/ K", in which a, 8, c, d, denote certain functions of a, which are different ' for each of these roots. 6. It may now be shown in a similar way with that in 7 § CLIX, that the roots \/ K’, \/ Kl, A/ Kl", must be combined with the same power of a. For if we put 4 a! ba! baby! + obx!¥ = a\/ K! then also must ae! + cn! + 2a! + aba!” = a? \/ RV and vl + gal + oe! + 08x! = a \/ RW because A’ in the first transposition is transformed into K”’, and in the second into K’’, but a remains the same. Now I assert, that necessarily v = 3, since other- wise a2 would not vanish from the value of x”, and conse- quently amongst the roots v2’, 1”, x/’, there must be another of the form of the root x”, because for « we could also have substituted the other primitive root of the equa- tion z¢—1=0. Weare certain .’., that At RAN RUAN RM Teh i pian in HD is a root of the given equation ; and we could also have 2y YH 346 found the other roots, if we had given ourselves the trouble to solve the four equations in 4. SECTION CLXII. Pros. Find functions, which are fit for the solution of the general equation of the fifth degree xv — Aa* + Bo? — Cr? + Dx — E=0 under the supposition, that we know not how to solve any other equation but those of lower degrees, and those of the form ¢? — K = 0. Solution 1. Arrange the 120 values of forms of the function ¢ = f': (12345) in recurring periods, as follows : (symbolical functions and brackets are omitted) 12 3\4 5 23114 5 31 2|4 5 21345 13245 32145 23415 Dinky 4 Bad 12436 23451 OL 452 LS? A 1593 13452 32451 214.58 8A 1°52 14253 24351 34512 Pa523 24531 34521 245138 14532 41528 42531 43512 45 123\|5123 4 4523 1 di, 553 Je 4521 3 45132 45321 15234 25314 35124 5231 4 53124 5213 4 51324 53214 52341 53142 51243 438216132154 21543115432|/54321 Thus in the first vertical column we find the 24 transpo- sitions of the four first roots arranged under one another, 347 in the same way as they werein],§ CLX. ‘The four following columns contain the recurring transpositions of all the five roots, and in such a way, that in each hori- zontal row there is a period. 2. According to this arrangement, the 120 values of forms of the function f: (12345) may .°. be generated in the following way. From the two values of forms, f: (12345), f: (21345), which form a period of recur- ring transpositions of the two first roots, we derive, by recurring transpositions of the three first roots, the six values of forms, f: (12345), f: (23145), fi: (31245), fi: (21345), f: (13245), f: (32145). From these, by recurring transpositions of the first four roots, we get the 24 values of forms which are contained in the first vertical column in 1; and lastly, from these again, by recurring transpositions of all the five roots, we derive all the 120 values of forms. 3. Let the five values of forms f': (12345), f: (23451), fi: (34512), f: (45123), f: (51234), be the roots of the equation of two terms o—K=0 then K is their product, consequently K =f: (12345) x fi: (23451) x fi: (34512) x fi: (45123) x fi: (51234). If, for the sake of brevity, we put K = f/ : (12345), then f: (12345) is a function such, that in all the recurring transpositions of the five roots, it remains the same, be- eause in each such transposition, one of the five factors, of 348 which it is composed, merely changes place with another. Consequently all the values of A include 24 times five equal values, and.*. this function can contain no more than 24 unequal values, and they are those which corres- pond to the 24 transpositions in the first vertical column in 1, and consequently those which arise exclusively from the transposition of thie four roots, a’, v/’, 2”, x’”. 4. The equation for t, which, taken generally, is of the 120th degree, is consequently already reduced, by the introduction of the function , to an equation of the 24th degree. Each root of this last equation gives five values of t, viz. \/K, a\/K, a\/K, or /K, \/K, and .*. all the 24 roots together give all the 120 values of t. 5. Since K=t and t=f: (12345), then also K =f’: (12345) = f: (12345)%. The 24 roots of the equation for K are .-. no other than the results of the transpositions of the four first roots in ( vi (12345) )°. We must now endeavour to reduce this equation. 6. With this view I shall assume, that the four values of forms f’; (12345), f’: (23415), f’: (84125), J: (41235), which together constitute a period of re- curring transpositions of the four first roots, are the roots of an equation of the fourth degree K —pK?+ qK?—rK +s=0; then the coefficients p, g, 7, s, are symmetrical functions of these four values of forms, and, consequently, in each re- curring transposition of the roots a’, 2’, a’ a”, they remain 349 the same, because in each such transposition, one of these values merely changes place with another. Therefore they can contain no more unequal values than those which arise from the transposition of the roots a’, x”, 2/3 conse- quently six values. Therefore the coefficients p, q, 7, s, depend on equations of the sixth degree only. 7. Since p, q, 7, s, are homogeneous functions, be- cause they all change only when the roots 2’, a’, x”, are transposed, we are .*. always enabled, from the known value of one of these coefficients, say p, to find directly the corresponding values of the remaining ones q: 7, 8 (§ CXLITI). It is consequently quite sufficient to solve the equation for p. Moreover, the six corres- ponding values of p, 9g, Tr, s, give six such equations as those in 6; and since each of these equations gives four values of K, consequently all the six equations together give the 24 values of KX. | 8. Ifwe put p= f”: (12345), then f”: (12345), J” : (23145), f” : (31245), f’: (21345), f”: (13245), ff’ : (32145), are the six unequal values of forms of p, which form two periods of recurring transpositions of the three roots 2’, 2’, 2/’”. I shall now assume that the three values of forms of the first period f/” : (12345), Sf’: (23145), f” : (31245), are the roots of an equa- tion of the third degree in Bn hls iret .-. p,q’, 1’, are symmetrical functions of these three values, and consequently in each recurring transposition of the three roots 2’, 2”, a”, they remain the same. They .°. can have no more than two different values, viz. those which arise merely from the substitution of a/ for a’. Besides, if p’ be found, then also g’, 7’ may be found immediately, because these three functions are homo- geneous. 9. Let p’ =f”: (12845), then f’”: (12345), f”: (21345) are the only two unequal values of forms of this function. If .-. we assume, that they are the roots of the equation pe hig £ g// erie then p’’, q are symmetrical functions of the roots ie al’, a’, a”, x”, and may consequently be expressed rationally by the coefficients 4, B, C, D, E, of the given equation. 10. We have now .*. reduced the equation for t of the 120th degree to the following equations : I #& —K=0 Il. K*— pK? + qk? —rK +s=0 il. p —ppt+qp—r=0 IV. p? — p/p + q’ = 0. Having found the equation IV, we obtain from it two values of p’. If we substitute one of these values of p’ in the equation III, and for q’, 7’, the corresponding values, we then obtain, by the solution of this equation, three values of p. Lastly, if we substitute one of these values in the equation II, and for g, r, s, their cotres- ponding values, we then obtain four values of AK, and from one of these values that of ¢. 35] 11. We now wish to inquire, how the function ¢ must be constituted, in order that the five values of forms SF: (12345), f: (28451), ff: (84512), fi: (45123), ff: (51284), may be the roots of an equation of the form t?—K =o; for this was the supposition with which we set out. If this condition be fulfilled, we then must have: SF: (12345) = af: (23451) = a?f: ($4512) =e oN 45 E25 | mk a LeSal, Of this kind, however, are all the functions of the form o: (12345) + ap: (23451) + ap: (34512) + a°h: (45123) + atp: (51234). Consequently all functions of this form are fit for the solution of an equation of the fifth degree, under the supposition that we are able, from the known value of this function, to determine the roots 2’, a’, a/”, 2’, «”, 12. But I affirm, that this last supposition is always correct, whatever function we may assume for @: (12345). For since t, in every recurring transpo- sition of all the roots, changes its value, .*. it can have at most only twenty-four equal values, viz. those which are in the first vertical column in ¢. Now, since amongst these values there is not a single one which has x” in the first place, consequently, these can only at the same time give the roots a’, wv”, a”, 4”, but the root x” would always admit of being determined rationally from ¢. Therefore, equations of the fifth degree may be solved in an infinite number of ways, and we shall see in the sequel, that this is generally the case with all equations. 352 13. Since f’: (12345) = ( f: (12345) )°, and p = fs (12845) +f": (238415) +f" = (34125) +f": (41235), then also, since we have put p = ff”: (12345), fl: (12845) = (fs (12845) )> + (fs (28415) )® + (fi: (34125) )° + (fi: (41235) )5 Further, since p’ = f”: (12345) +f”: (23145) +f”: (31245) = f’”: (12345), then also, when the requisite transpositions of the roots are made, f+ (12345) = (fi: (12345) )P+ (fs (23415) )> + (fi: (34125) )o+ (fs (41285) )* + (f: (23145) )+ (fs (31425) )* + (f: (14235) )5+ (fs (42315) )? + (fi: (31245) )'+ (fs (12485) )> + (fs (24815) )5+ (fs (48125) )> If in this we substitute 2 for 2’, we obtain Sf! + (@1845) = (fs (21845) )>+ (fs (13425) )> + (fs (34215) )®+ (fs (42135) )> + (fs (18245) )5+ (fis (82415) )® + (fi: (24135) 5+ (fs (41825) )> + (fs (32145) )54 (fs (21435) )> + (fs (14325) 54 (fz (13425) )% Hence we see, that the two functions f’”: (12345) and jf’ : (21345), taken together, give all the possible values of (f: (12345) )>, which arise from the transposition of the four first roots, consequently all the unequal values of this function. ‘Therefore, since p’ = f’”: (12345) +f” : (21845), p” is a'symmetrical function of x’, 1”, av’, x”, x”, which is obtained immediately from the function -( fet (12345) is when we take the sum of all 3098 the values of this function, which arise from the trans- position of the four roots 2’, x”, a/”, x”. 14. Further, because q/ =f” : (12345) x f’” : (21345), we then immediately obtain the function q’’, by taking the product of the above two values for f’” : (12345) and 7! : (21345). Besides, it is evident, that both in p’ and q’’ the root a must vanish, because otherwise t would have more than 120 values. SECTION CLXIII. Pros. Solve actually the general equation of the fifth degree : a — det + Br? — Cr? + Dr — E=0. Solution 1. Since for @: (12345) in the foregoing §, we can assume any arbitrary function, in order .°. to simplify the calculation, I shall assume for it the root 2, and put @:; (12345) =a’. Then ¢: (23451) = x”, @: (84512)=2", p: (45123)=a, @: (51234)=2"5 hence t =f: (12345) = a + aa!” + aa!” + aba!” + ate” and .°. (f: (12845) )> = (a! + aa” + ata!” + aba!” 4 altel)? to which expression we can also give, as in § CXL, the form (cer! aPa! 4 aby 4- aha’ + Lama 2. If we solve this expression according to the powers 2 2 354 of a, it has, for the reasons given in 12, § CK XXIX, the following form : Kl 4 Eg 4 Elia? 4. EW gS 4. EP aA and then &/ = [5] + 120 [1°] + aly! nV pg! BqMM IV A gl yl /Bq ll A lB VV A209 glyMMBq% Ae gla M3qh¥ 4 glally/V3 4. gllly/V8zV ? ala? a! pl! V3 _ al al 2 V2 allyl Rap! VB A. oplD yl IDA all x V2 VE 4 4.30.2 gly gh gly! oplV2 4. yl Bpl/2q/V 4. yllI2q1V V2 tg l2ylV2yh 4 gl Byll/2zV EM — 5 ied -. gl bg lM eg MMlAg lV 4 Co ete oe) -- 10 (aia 4 y/2yM//3 4 g/3yV2 1 gll27/V3 | thon Hh) gl34 M1 gh + a! y/Bx” we xl lg V3 A. ap! al//3 yh 2x! 27" Ais gl all2y//2 + gl tal Vah2 4 el al2 x! V2: a gl" lV 2y V2 + G0 et ee eee piped, +20 ( +30 ( gl — 5 (gl ae rae a al ln Ma) +. 10 CORE ce eases red AW Peete a) 4.30 af 2 al a /F2 oe gel 2g M2 MV be gpl gf MAg V2 4 yll2g/ Il pve Hf ap!/2y/V249 / i 60 5 ata ieee) + lal af V24V 4. all plll2yl¥ a 350 EIV ial 5 (PFA? pe lal 4 al AY lf M4 iglll VA) + 10 rates 0 + 2/3772 pe gl l2g M13 4 gla lV3 alley * +20 CHE ante a ae) ae al3y/V xy + 30 (aPal Bz. tt eR A gl Bai Mi Ee) ie 60 ei a lA gI Va Y Be al gl 2g lll ap V2 ; + al all al ayl¥ ae al a!al Vx? ) &” = 5 (alba? al al4 pe aly! 4 A alg /"4 +a/%r") 13/1/12 2 JV +10 (x L + a’ ao Ba iS ll i cers eo RL A JV ry" y/ al gl IV A gh ByMM 4 ¥ 4 yl gl! y¥3 + gly l¥3yV cy ( a!2y/ ay 24 al yl? V2 + 24 M2 y/V ) + al al V2 a"? + al l2y/V27V rr 60 (aehinte + a!a!27!/" 7’ + ikke) a a! ll yl!" yf V2 a allyl yl" yh? +20 ( 8. Now, if in the expression &/ + &/a 4+ Ea? 4 £/"a3 + &’ a‘, we perform all the possible transpositions of the roots x’, a”, «’”, a/”, and denote the sum of all the results thus obtained by Cp Mee Mok + C8 + L% a! we then find 2/= 24 [5]+8 . 20 [173] +8. 30 [127] +24. 120 [1°] 2/=6.5[14]+6. 10 [23]+4. 20 [1°3]+4. 30[127] +6. 60 [1°2] and for Z/’, 2”, Z”, the same values as for 2/’. 'There- fore 356 Ch + Cae Catt CaP op Coe = + (a+ a+ a + at) = 2 — 0"; because a + e& +a? + at =[1] —1= —-1. 4. But by 13 of the preceding § the coefficient p’’ 1s the sum of all the values of forms of (3 (12345) )*, which arise from the transposition of the four first roots ; we have .°. also pl al Ma Med 4. U8 4 Cat =m OY. Now, if we substitute for 2’, 2/’, their values already found, we then get p’ = 24[5] — 30 [14] — 60 [23] + 80 [173] + 120 [127] — 360[1°2] + 2880 [1°] and. when for the numerical expressions we substitute their values taken from the annexed Tables, pl! = 24 A> — 150 AB + 150 AB? + 250 42C — 250 BC — 1250 AD + 6250 E. 5. By a method not much different from this, we may also find the coefficient g’’. Having, however, found p”’ and q’’, then the solution of the equation IV in 10 of the foregoing §, gives the value of p’. Having obtained p’, then we may also find the coefficients q’, 7’ of the equation III; and the solution of this equation gives the value of p. From the known value of p we may now again find the coefficients g, 7, s, of the equation I. But the calculations by this method would be extremely troublesome, and almost impracticable. I shall, in the third part of this collection, show how it may be short- ened essentially, and at the same time give the complete 307 solution of equations of the fifth, sixth, and seventh de- grees. 6. Let K’, K”, KR’, K’", be the four roots of the equation II, consequently Mae eeamoy ae, ys (care ee fi’: (34125) = K”, f’: (41285) = K’’. If we substitute K’, K’”, K’’, K’”, for K in the equa- tion J; we then get for ¢ four values \/ Kl, / Ke / KM, / K’", and the values of forms f: (12345), ff: (23415), f: (84125), ff: (41235) correspond to these values ; we .*. have eps’ (TAS 46 ye \/ Ki! 5 of 2 (23415), = \/ Kit if: (34125) = J BR", f: (41235) = A/ KM. If we substitute here for f: (12345), its value a’ + awl -+ ata!’ + abr!” + atx’, we then get, including the equation a + 2” + x” + 2” +24" = A, the five following equations : diet OA ee a) 2 CE Aol dl ig gh A 5 ao + an! £ oa!” + ot’? + ats” = \/ K! 5 sag”! + el” + oe + ate” = CWA kv 5 v4 aa” + a! + ar! 4+ ata” = \/ Kk” 5 ear! ate ada! 4 ate! = \/ tLe 7. Ifwe multiply the four last equations by a, and then add them tothe first, since 1 + « + a + a? + a*=‘[1]=0, we get immediately 358 5 5 5 At a Rt Kt KY VR”) yee 5 | and the remaining roots a’, a’, a”, a”, are all of the 5 5 5 5 form aA + tN/ K! + \/ K” + N/K” + A/ K” in which a, }, c, d, e, denote certain functions of a. Hence now we may conclude, in asimilar way as in equa- tions of the third and fourth degrees, that AT KE A KY N/R 4 N/R 0 TR TIOy oa, ae Bik Ll GER: 0 Eis ed is a root of the given equation. 8. Having .:. solved the equation II, we immediately have a root of the given equation, and the remaining roots may be determined from the five equations in 6, by elimination, if, after having performed the calculation, we merely substitute a\/ K’, at\/K", ab\/ RM, b\/K” for \/K', A KY, N/ KY, 7K”. SECTION CLXIV. Pros. Find functions, which are fit for the solution of the general equation of the sixth degree 2 — Ax’ + Bort — Cx? 4+ De? — Ex + F=0 under the supposition, that we know not how to solve any other equations than those of lower degrees, and those of the form t® — K = 0. Solution 1. With the view of arranging the 1.2.3. 4.5.6 = 720 values of forms of f: (123456) in a re- 359 curring order, if we put the 120 values of forms in J, § CLXIT, in a vertical column under each other, and add to each the root 2” in the last place, we then have 120 values of forms, all of which end with a’. From each of these, if we derive, by a recurring transposition of all the six roots, five others, we then get 120 periods, each consisting of six values of forms, consequently the 720 values of forms of f: (123456). 2. I shall now assume, that the six values of forms of the first period f: (123456), f: (234561), f: (345612), J: (456123), f: (561234), f: (612345), are the roots of the equation rae by a Vr then — K is the product of all these roots, and hence —K=f: (123456) xfs: (234561) xf: (345612) xf: (456123) xfs (561234) xf: (612345). This product, however, evidently undergoes no change in each recurring transposition of all the six roots 2’, 2”, a’, a/”, «”, 2%’, because in each such transposition, one factor merely changes place with another; consequently the 720 values of forms of K, taken six and six together, are equal. Therefore K can have no more than 120 different values, and these 120 values are no other than those which have 2”” in the last place, and which consequently arise merely from the transposition of the five remaining roots. 3. Since K = ¢®, and t =f: (123456), then also K= (/f: (123456) )°. Consequently the function ( f: (123456) )® can have no more different values than 360 those which exclusively arise from the transposition of the rootsa’, 2”, a’, x”, 2”. For shortness’ sake, I shall denote them by /’: (123456), and assume, that the five values of forms f’ : (123456), f’ : (234516), f”: (345126), fi’: (451236), f”: (512346), which arise from the re- curring transpositions of the five first roots a’, a”, «/”, v”, x”, are the roots of the following equation of the fifth degree : Ko — pK* + qK° — rK? 4+ sK —u = 0, then p, g, 7, s, u, are symmetrical functions of the above values of forms, and consequently undergo no change in each recurring transposition of the first five roots. But since they also remain the same in the recurring transpo- sitions of all the six roots, they consequently can contain no more different values, than those which arise exclu- sively from the transpositions of the roots a’, a”, a’, x”. Therefore each of these functions depends only on an equation of the 24th degree. Since they are homoge- neous, it will be sufficient to have determined one of these functions. 4. If for the sake of brevity, we put p=f” : (123456) then f” + (123456) can only undergo a change when the four first roots are transposed. I shall now assume, that the four values of forms f” : (123456), f”” : (234156), ff’: (841256), f’: (412356), are the roots of the following equation of the fourth degree: ” p’ — pp + 7p? — rp + = 0; then the coefficients p’, 9’, 7’, s/, are symmetrical func- tions of these values of forms, and consequently in each 361 recurring transposition of the four first roots they remain unchanged ; and since they also remain the same in the recurring transpositions of the first five and of all the six roots, consequently, amongst the 720 values of forms, there are no more than six which are different, viz. those which arise exclusively from the transposition of the three first roots a’, x’, a/”. 5. I put p’ =f”: (123456), and assume that the three values of forms /’” ; (123456), f’ : (231456), J: (312456), are the roots of the following equation of the third degree : ps — p/p’? + q'p! —1r’/ = 0; then p”, q’’, 7’, are symmetrical functions of these values of forms, and consequently in the recurring transpositions of the three first roots suffer no change; and since they also remain unchanged in the recurring transpositions of the four and five first, likewise of all the six roots, .°. each of these functions can have no more than two differ- ent values, viz. those which arise from the transposition of the two first roots. 6. If.-. we put p’ =f” : (123456), and assume that the two values of forms f’” ; (123456), f” ; (213456), are the roots of the equation pl? — pp! + qf = 0, then the functions p’”, ¢/” undergo no change in the transposition of the two first, three first, four first, five first, and all the six roots, consequently they are symme- trical, and .*. may be expressed rationally by means of the coefficients 4, B, C, D, E, F, of the given equation. 3A 362 %. The equation for t, which originally was of the 720th degree, has consequently been reduced by these successive operations to the following equations : I. #& —K=0 Il. K®— pKi + qK?—rk?+skK—u=0 Ill. pt —pp+qp—rpt+s =0 IV. p?— pp? + q/p' — 1’ = 0 V. pe—plp! + q/ = 0 which are so constituted, that the coefficients of each of them depend on the solution of all the following equa- tions. The equation V gives two values of p’’, .*. the equation IV six values of p’, consequently the equation III 24 values of p, and .*. the equation II 120 values of K, consequently the equation I 720 values of ¢. 8. Therefore, if the function ¢ be such, that the values of forms f: (123456), f: (234561), f: (345612), J: (456123), f: (561234), f: (612345), are the roots of the equation °—A’=0, consequently it is always fit for the solution of an equation of the 6th degree. But nothing more will be required to effect this, than that J: (123456) = af: (234561) = af: (345612) = af: (456123) = a*f: (561234) = af: (612345) when @ denotes a primitive root of the equation /—1=0, But all functions of the form p: (123456)+ ap: (234561) + a?h: (345612) + a>: (456123) +a4h: (561234) + a5 p: (612345) are of this nature. Consequently all functions of this 363 kind are fit for the solution of equations of the 6th degree. Besides, we need not be apprehensive of not being able to determine the roots of the given equation from these functions ; for, since the equal values of forms of f: (123456), if indeed it should have any, can only be found amongst those which have 2” in the last place, consequently w’ cannot be amongst the roots, which correspond to the equal values of t=f: (123456), and .*., by the foregoing chapter, this root at least must be determined from the known value of ¢ by a rational expression. SECTION CLXV. Pros. To solve actually the general equation of the 6th degree in the foregoing §. Solution 1. If, to facilitate the operation, we put gp: (123456) = 2’, then Ff: (1238456) = a’ + ar! + aby!” +4 by!” + aba” + a. We have .°. fs (123456) = (f: (123456) )° = (a! + aa 4 altel + aba!” + abe” + abr”) = a— (ar! + aa 4 aba! + atal” 4 aby” + abx’’)® = (an! + aba! + aa!!! + ate!” + abe” + abx”)® This transformation was performed merely in order to make the dashes over x agree with the powers of @, by which means the solution of the polynomial is rendered more easy. 2. By 2 of the foregoing §, f’ : (123456) is the sum 364 of all the results which are obtained from. the recurring transpositions of the five first roots in’: (123456), con- sequently also the sum of all the results, which arise from the transposition of the five first roots in ( fs (123456) )°% Further, by 3 of the foregoing §, //”: (123456) is the sum of all the recurring transpositions of the four first roots inf”: (123456), consequently also the sum of all the recurring transpositions of the four and five first roots in (Ff (123456) )° By 4, /’” : (123456) is the sum of all the recurring transpositions of the three first roots in J’: (123456), consequently also the sum of all the recurring transpositions of the three first, four first, and five first roots in (Fi: (123456) )° Now, since by 5, pl =f” : (123456) +f" : (213456), .*. also p’” is the sum of all the recurring transpositions of the two first, three first, four first, and five first roots in Gf: (123456) yf consequently the sum of all the values, which are obtained from this function by the transposition of the five first. roots. 3. In order .*. to find the coefficient p’”, we must first solve the power (a2! + ox! + adr’ + ode!” + abt” + abe) ; this solution, since a&&=1, a’=a, a®’=a?, &. assumes the following form Ca Mog pe Mak t CVod 4. Lat 4 Zo in which 2’, 2”, 2’, Z/", 2”, Z"’, are functions of 2, a’, a, a!” x”, a’, without a If we then perform the 120 transpositions of the roots a’, a”, a/’, a/”, x”, then the sum of all the results thus obtained, gives the coeffi- eient p’”’. 365 4. Further, if we multiply the two functions /”” : (123456), f’”: (213456) together, we then obtain also the coefficient q/’. The equation V of the foregoing §, will then give the coefficient p’”’ ; and having found this, we may directly find q, 1’ by the foregoing chapter. Now, the equation IV gives the coefficient p’, .*. also q’, 7’, s’, and lastly the equation III gives the coefficient p, and at the same time also the remaining coefficients of the equa- tion IT. 5. Let K’, K”, K’”, K’”, K”, denote the five roots 6 6 6 of the equation II; then \/ RK’, N/K", N/K", 6 6 / Aes \/ KY’, are the five values of t, and the values of forms f: (123456), f+ (234516), f: (345126), fis (451236), f: (512346) correspond to these values ; we .*. have the six equations My pe allie gO ee oll ee par =o 8s A a’ 4- ax!’ 4+ ay!” + ay!" -. air’ |. afr — N/K! al! tat!’ + eal” + ar” + ade! + a" = N/K all! + axl 4 ody” + ae! 4 ate” + aa” = N/K vw” at” + oa! + abe 4 of! 4 abt” = N/K" wv tal +t” + ade + ata!’ + aba’! = \/ RK’. 6. Hence now, when we multiply the five last equa- tions by a, and add them to the first, we immediately obtain oy At aN KIN RON RUN] RUAN KY Serene oT 366, and when for the same reasons as in the case of lower 6 6 6 6 equations, we put aN] RK’, aN SR", a K&N h ny a\/K", for \/K!, \/R", N/R, NI] RN RY so that @ vanish, we get a root of the given equation 6 6 6 ¢ : ieee A4NSEIAN] KUN RUAN RARE ; eye MCLEE WM eee Tee CRE CORR ON Paey MN Ps and we obtain the remaining ones from the above six equations by elimination. SECTION CLXVI. Pros. Solve the general equation of the nth degree Re Ag oe Bypet 7 Sel 0. Solution 1. Let t= fz (12345 .:..., n) be a function of such a nature, that the 2 values of forms, which arise from the recurring transposition of all the n roots, are the roots of the equation i —K =—0. Since then K is the product of all these values of forms, it must consequently remain the same in all these trans- positions. ‘Therefore its values of forms, 1.2.3. 4 acess n, taken n and m together, are equal. Hence it follows, that K has no more than 1.2.3.4.,....n—1 different values, and that these values are those which arise exclusively from the transposition of the n — 1 first roots, 2 2. Since .*. K still depends on an equation of the 1.2.3.4......2 — 1th degree, we must consequently 367 endeavour to reduce this equation. ‘To effect this, I put K = fis (12845 biped n’, and assume that the x — 1 values of forms of this function, which arise from a reeur- ring transposition of the m —- 1 first roots, are the roots of the following equation : K*> = pK? + qk" — rK** + &e. = 0. Then the coefficients p, g, 7, &c. are symmetrical functions of these values of forms, and consequently in the recurring transpositions of the m — 1 first roots remain the same. But since they also undergo no change in the recurring transpositions of all the z roots, because by that means fis CUB84 son008 n) suffers no change, consequently only those of their values are different, which arise from the transposition of the x — 2 first roots. Therefore each of these coefficients has only 1.2.3. 4...... nm — 2 differ- ent values, and consequently each of them depends on an equation of the 1.2.3.4 ...... 2—2th degree only. 3. Since it is quite sufficient to have found p, because Dy Yo 7 &e. are homogeneous functions, I shall put p = fs: (12345 ......2), and assume, that the n — 2 values of forms of this function, which arise from a recurring transposition of the x — 2 first roots, are the roots of the equation pits pp 3! fp died pr? + &. = 0; then p’, q’, 7’, &c. are symmetrical functions of these values, and they .*. remain the same in the recurring trans- positions of the x — 2, n — 1 first roots, and also of all the n roots. Therefore these coefficients have only 1.2.3.4...... n—3 different values, viz. those which 368 arisé from the transposition of the n — 3 first roots, and they consequently depend only on equations of the 1. 2. aa Ey n — 3th degree. 4. In a similar way we successively form the equa- tions om — pl! pl +9! p~ —y!/ ytd +&.= O //n—4 plot yfyf 4 gpl — pln" 4 Be, = O pl — pl Mp8 4 gl pf — ay! Mp8 4. Be, == 0 &e. viz. the first from the recurring period of the x —3 first roots of the function p’ =f’: (1234....... n); the second from the recurring period of the n — 4 first roots of the function p” = f/": (1234.........)3 the third from the recurring period of the n — 5 first roots of the funchion 9! i= ff" 's (12840... n); and so on. We continue this operation till we arrive at an equation of the second degree ( bra des OTe + Gat = Ue (n—3) ? then p gq", -aresuch functions of a’; 7’; a’; 42.22. ... &, as remain the same in the recurring transposi- tions of all the m roots, also in the recurring transpositions of then — 1, n— 2, n — 8 first roots, and so on, and likewise of the two first roots. ‘They .:. undergo no change in all the transpositions of the roots, and conse- quently they are symmetrical. Therefore they may be expressed rationally by the coefficients of the given equa- tion. 5. We consequently have a series of equations 369 “—-K=0 K’"' — pk’ + gk"? — rkh*™™ + &e. = 0 p us pp n—3 +. gq ots* AS fpr? -}. &e. = O ae 0 1% — pp BE of n> ab. cor -|- &e. ; (p?-)2 es meri + cone —0O which are so constituted, that the first coefficient of each depends on all the following ones. Now if the first coefficients p, p’, p”, p’’, &c. are found, then also the remaining ones q, 7, &e. q’, 1”, &e. q”; yr’, &e. &e., may be found by the foregoing chapter. G6. It only remains now to assume for ¢ = f: (12345... vseeeem) a function, such, that its values of forms, which arise from the recurring transposition of all the x roots a, a, x’, &c. may be the roots of an equation of the form t?’—K=0. With this view, we assume any other function z = @: (12345 ......%) at pleasure. Let a!, 2’, 2, 2, ...... 2” denote the values of forms of z, which arise from the recurring transposition of all the 1 roots, and « a primitive root of the equation 1” — 1 = 0: assert, that then tae! tag! + oz! + aFal” + ocesge ct arg is always a function of the required property. For since in this function, in each recurring transposition of the roots 2’, 2, v//’, &c., the values of forms 2’, 27, 2'", 2!¥, ...... 8” are in like manner transposed in a recurring order (for by these means 2’ changes place with 2/,2/ with 2’, 2” with 2”, and soon, lastly 2 with 2’), there- 3B 370 fore the function t, in the recurring transpositions of thie roots a’, a’, a/’, &c. has the following values : th =a! tas” tare!’ 4 a327+ ..... A a a tal! + asl + call 4 bY A occ uee ae! al + aa!” + ahh aa! A wees ee vee Fae! tM eV ta ta2sY 48h 4 oo QP Ng//1 &e. and we immediately see, that t” = a®'t/, t’” =a" “t’, i” = at’, andso on. Therefore the functions ¢’, t”, t/’, &c. have exactly those relations which they ought to have, in order that they may be the roots of an equation of the form «” — K = 0. 4%. That the value of the roots 2’, 2”, a/”, &c., may . always be determined from the known value of the function 2/ + az” + ate!” 4 a2” + ......00 + at ig” let the function z be what it may, appears from this, that the root a never can correspond to the equal values of this function, if it should have any, because these equal values must necessarily be amongst those which arise from the transposition of the n—1 roots 2’, a”, xv/’, x”, wate eee x’-, Consequently this one root at least may always be determined from the known value of ¢, without solving any equation. But then the remaining ones may also be found, when the solution of equations below the nth degree is pre-supposed. 8. Therefore all equations may be found in num- berless ways. If, in order to make the calculation more simple, we put z = a, we then have t= + ar”! + allyl 4987/7 4... nce hee 37] Hence we immediately obtain K=r= (a! + aa! 4 aie 4 Bi a Ma 2 9. Now, in order to solve actually the given equation, we must first of all endeavour to determine the coeffi- cients p"-», g’—) of the last reduced equation in 4. Since p is the sum of the n—1 values of forms of K, which arise from the recurring transposition of the n—1 first roots, and p/ again the sum of all the values of forms of p, which arise from the recurring transposition of the n—2 first roots ; consequently, also, p’ is the sum of the n—I1.n—2 values of forms of K, which arise from the recurring transpositions of the » —1 and of the n— 2 first roots. Further, since p is the sum of the »— 3 values of forms of p’, which arise from the recurring transposition of the n—3 first roots, .°. also p’ is the sum of the n—1.n—2.n—8 values of forms of K, which arise from the recurring transpositions of the n— 1, n—2, and n— 3 first roots. If we proceed further in this way, we then find that the coefficient p—) is the sum of all the n—1.n—2.n—38....3.2 values of forms of Ky, which arise from the recurring transpositions of the n— 1, n—2, %—3 roots, and so on, lastly, of the two first roots, or, which is the same, that p’- is the sum of all the 1.2.3.4.,.%—1 values of forms of K, which arise from all the transpositions of the n—1 first roots. 10. In order, therefore, to find the coefficient p°—, we must in the first place solve the expression for K in 8 (a’ + aa! + ata! -+- asylV + eceeren + Gear) 372 according to the powers of a. This solution, since a’ =1, act'=a, a? = at, &c. will then have the following form : < E+ SMa Eat st Ea? +... Se ied in which &, &”, &/”,......& are certain functions of 2’, al, ul y.0004.0. Now, if in these we transpose the roots a!, a, a!, ......2°—. in all possible ways, while «™ retains its place, and then add the results together, we obtain an expression of the form Ce Mat OV 4 0! B 4 ELM a which, since it 1s the value of p—, is necessarily sym- // all »)n) 3 ‘J > eeeeer metrical with reference to the roots 2’, a and .*. may be expressed rationally by the coefficients A, B, C &c. of the given equation. 11. In order to find the coefiicient g°~ of the last equation in 5, in the expression &/+&/a+&a*+... +E" complete the n.n—1.n—2......3 recurring transpositions of the n—1, n—2, n—83, and so on, first roots, exclusive of the two first ; then substitute 2’ for 7”, and again make the same transpositions. Now, if we multiply the sum of the n.n —1.n— 2u........3 first results by the sum of the n.n — 1.n—2......3 last, we then obtain a symmetrical function of a’, 2”, a/”,...0%, which will give the value of g-», expressed by the coeffi- cients 4, B, C, &c. 12. If we write the reduced equations in 5 backwards, 373 we have the following series : (p* — pp?) + i — 0 (gs ee aCe sli ge pty — rie = 0 ( py ~% p> ps 1 gi p> ¥ ‘s rt) (pr) ) + s"—) = 0 KK" — pk" + qk" — rK"* + &. = 0 — K = 0. Having found the coefficients p"-», g”— by 10 and 11, then the solution of the first equation gives the coefficient p’ of the second equation; and by the foregoing chapter, we may hence immediately determine the values of g*» and r’ by rational expressions, because p”—, g’— and r°“—, are homogeneous functions. ‘The coeffi- cients of the second equation are consequently fully determined, and its solution gives the coefficient p°-°) of the third equation; .*. also, as before, the coefficients qo, rv", ss and the solution of this last equation again gives the coeflicient p" of the following equation. If we proceed further in this way, we at last find the coefficients p, q, r, &c. of the equation for K. 13. Let K’, K”, KK’, K’’......K°, -be the n—1 roots of this equation; corresponding to these, as we have assumed in 2, are those values of forms of K = f’: (1234...2), which arise from the recurring transposition of the n—1 first roots. Each of these values of K substituted in the equation t’— K=0, gives n values of t, consequently all together they give n.n—1 values of ¢; and these are: N/K, aN/B) BN/ Kiyo al) K! n m2 n \/K", ON er VAR Na nat est a4 / K! n n n n \/RK", aN /K", a SK"... i aN KM n n n nh \/ Ke», aN / Ke, a KO, a / KO) and corresponding to these are the x.n — 1 values of forms ‘of t=/f: (1234...... n), which arise from the recurring transposition of all the n and n—1 first roots. These numerical values of the function ¢ have such a relation to the values of forms of f: (1234...... n), that if we put f: (1234...... n)=aN/ K’, the remaining n—2 values of forms, which arise from f: (1234......m) by the recurring transpositions of the n — 1 first roots, cor- n n respona to the numerical values a'\/ Ke a'\/ Ks n n a’\/ K’ rein fa K°-?, Now since we have assumed that ¢ =a’ + av’! + oa’! + abr” +......4 a", we have .*. the following equations : ag Di ans Aue aches esi, ©. Ti eames ae ie A a! tar! tate! ... +e 2p (M1) 4 gtr) a’\/ K! x! ag! thie! ah) gy geo a'\/ K"’ eel Veen +... tae tal) = a\/ K” nN CO) + cal 4 ota! te oon el PaO 4 alg) = a\/ say 14. If we multiply the »—1 last equations by 2, and 375 then add them to the first, we obtain, after dividing by n, v+1 n n n n ae — A + eae (ASR! - \/KU + WAG 7s \/Ke-) n n and the remaining roots are all of the form ad + BN/ KR! + A/ KY 4 N/R 4 tN) Rom in which a, b, c, dy ...... 1 denote certain functions of «, But since these roots must always remain the same, Whatever primitive root we substitute for & .*. amongst the remaining roots 2/ hea aA x°’—) there must at least be another, which has the form of the root 2; and since this is not possible, @ must quite vanish from the value of 2 already found, and... @’t! = an = 1 > conse- quently y= 2—41. Therefore rat We ARI 4 TK! wu NJ Kem) is a root of the given equation, and the remaining ones may be determined from the equations in 13, when we substitute in them » — 1 for v. Remark. The solution which I have here given, has only this one fault, that we do not by its means obtain all the roots at once, but only one, and that the remaining ones must afterwards be sought by a very troublesome elimination. “I shall .-. give another solution, which has not this fault, and in other respects also is perhaps preferable to the former one. For the sake of perspicuity I shall begin with an equation of the fifth degree. 3706 SECTION CLXVIi. Pros. Solve the general equation of the fifth degree a? — Ax* + Ba? — Cr? + Dr —- E=0 so as to obtain all the roots at once. Solution 1. As in the preceding §, let 8 — K = 0 be the equation for the recurring period of all the roots of the function t= a! + ar! + ay!” + aba!” + aby” then Kat = (a! 4 ar” + ata!” + aba!” + afa”)” or also K = (aa! + 222” + aa!” + atx!” + x) and K can have no other unequal values of forms, as we have already seen, but those which arise from the 24 transpositions of the four first roots. 2. Thus far all is the same, as in the foregoing solu- tion. But further, instead of forming the equation K* — pK? + qk? — rk +s =0 as heretofore, from the recurring period of the four first roots, I shall now assume that it has the four following values of forms for roots : (aa! + ax! 4 aby” + ata!” 4 x")? (ox! + ate + ar! + ads!” 4 2”) (ax! 4 ar! 4 abe’ 4 ofa!” + x”) (aty/ ele asyz!/ + ay!” fi ar/’ =e Dale the three last of which are obtained from the first one, when in it we substitute successively 2°, a, a+ for «. 377 3. In these four values of K, 2/” is always combined with another power of a, and .-. all the 24 unequal values of K may be derived from these, merely by per- muting the roots a’, 2’, v/”. Thus, if the roots a’”, 2”, retain their places; and we merely permute the three first roots, then each of the above four values of forms gives five new ones, and consequently all, together give the 24 values of K. 4. Since the coefficients p, q, 1, s, are symmetrical functions of the four above-mentioned values of Kos. these functions undergo no change, either by the recurring transposition of all the roots, or by the substitution of a for a2, a, at; and consequently they can have no more unequal values than those which arise exclusively from the transposition of the three first roots a’, v//, 2//’. Therefore these functions depend on equations of the sixth degree only. 5. Consequently, if we put p= f: (12345), then J: (12345), f: (23145), fi: (31245), fs (21345), SJ: (13245), f: (32145), are the six values of forms of p. Now if we assume that the three values of forms J: (12345), fz (23145), f: (31245), which arise from the recurring transposition of the three first roots, are given by the equation p* — PP’ + p'p— 1 = 0, then the coefficients p’, 9’, 7’, are such functions of 2’, 2”, a’, x", x", as can only have the single value, which the substitution of 2/ for a” gives. Therefore p/ (and the 3c 378 same obtains of q/ and 7’) has no more unequal values of forms than the two f’: (12345), f’ : (21345). 'There- fore p’ depends on an equation of the second degree only. 6. Let p? — ply +4! = 0 be this equation; then p”, q’’, are symmetrical functions of a’, a’, vw’, x”, a”, and consequently may be ex- pressed rationally by the coefficients 4, B, C, D, E of the given equation. 7. We now have the three following equations : K+ — qK* + gKk*—7rK+s5=0 Pp — p'p? 4 gp —StWwih 23219) pt — p/p +r’ = 0. The last gives the value of p’, from which, by the fore- going chapter, the coefficients q’, 7’ may be determined. The solution of the second equation again will give the coefficient p; and from this again we may’ find the coeffi- cients g, 7, s. Now, by solving the first equation, we obtain four values of K. 8. Let K’, K” K’’, K’’, be these four values, we then have the four following equations (2): (aa! + ofa! 4 adel 4 ata + a”)> = K’: (aa! + ata! + an!” 4 abeY 4 x”) = KM (abr! + aan! + atta! 4 oa” 4+ a? B= KM (abe! 4 oa 4 cba!” 4 a” + al = KM. If, we extract the fifth root. from both parts; of these 379 equations, we then have, including the equation 2/-} 0” + alll + av hy al — A, helps alt le gs wes yi gh) VY at! + ae 4 odyll 4 aly 4 gh = \/K! ata! tate! a agg! Ss oy” +a! = A/K! 08a! 4 ar! + ate 4 aa” 4 oh = \/ KK” ata! + Sr 4 ata! 4 ae 4 ooh \/ K”. 9. If we add these equations together, we then obtain, since 1 ta+teta4a4= fij/=0 iin Rome REE \/'K! 1 \/K! Hi JK ae A/K!, If we multiply the second by a, the third by a’, the fourth by @*, the fifth by «, and then add them to the first, we obtain 5a! = At a/R! 4 o®\/K 4 2X / KM" 4 a/R", If we multiply the second by a*, the third by a, the fourth by 2‘, the fifth by a, and then add them to the first, we obtain ae ee ee If we multiply the second by a?, the third by a‘, the fourth by , the fifth by a’, and then add them to the first, we obtain bal" = At ats /, K! 4 at\ / KU 4 aN / Ki 4. as /, K”. Lastly, if we multiply the second by a, the third by a?, the fourth by a, the fifth by a4, and then add them to the first, we obtain } | Sim A KaN/ Kloof N/K! aX RY po \/ KY, 380 10. If we inspect the values of the roots x’, a/, a’, x/¥, x”, as they have been here found, we shall imme- diately observe, that if in any one of the four last, we substitute successively a?, a, a4, a, for a, we always obtain the other four, ‘Iherefore, all the roots of the given equation may be comprehended in the following expression : ee 1 (A EE a\/K! ~f- PNK 4 oN / Kl! 4. at\/K") when by « we suppose each root of the equation v° — 1. 11. I shall only further remark, that in this solution the root a must certainly vanish in the coefficients p, q, vy, s. For, since the functions K’, K”, K’”, K’”, in 8 are such, that in the substitution of a for a’, a’, at, con- sequently for the remaining roots @, y, 6, of the equa- tion a°>—1=0, they merely change places; .*. the coefficients p, g, 7, Ss, as symmetrical functions of K’, K”, K’”, K’”, must also remain the same in thiss sub- titution, and consequently must be symmetrical func- tions of a, 8, y, 6; .*., by the fifth chapter, are rational, SECTION CLXVIII. Pros. Solve the general equation of the undetermined nth degree x” — Ar + Br* — Cr’ + &. = 0 in such a way, that all the roots may be obtained at once, and yet under the supposition that » is a prime number. 38] Solution 1. As in § CLXVI, let ta xl te ea + a! Lo bt te Ks t= (o! + ar! + otal 4 pat izoy or, which is the same, K Aare ee ae ee ey ee Cait» ame ci The function K, as we have already seen in the above- named §, is then such, that it remains the same in the recurring transpositions of all the roots 2’, a”, 2/””,...a, and consequently can have no more unequal values than those which arise from the transposition of the n—1 first roots, 2. In order to find the 1.2.3...» — 1 values of forms of the function K, which arise~ from the transposition of the »—1 first roots, we first substitute a4, 2°, a?,...q"—1 for a; hence there arise the following n—1 values: (axe taal ail clans sadtce, gD ats eek (arr peated ala abi steal a mE aye (aia! + aha! + cal” + on... a yO) 4 gy (cee cle eran. bcs... ee CE A pnts Since in all these values of forms, because m is a prime number, the same powers of @ occur, and in each the root x°~ is combined with another power of @; it is .°. evident, that we obtain all the values of AK, when in these n—1 values, we permute the roots 2’, 2’, v/”,... ev’ in all possible ways, but let the root x°—” retain its place. Each of these values then gives (in- cluding the one under consideration) 1.2. 3......m—2 382 values, and consequently all together give all the above 1.2.3...n—1 values of K. 8. Now if we assume that the n—1 values in Q are the roots of the equation K’* — pk + qK’~* — rR" + &e. = 0 consequently, by the fifth chapter, the root «# must vanish . in the coefficient sp, g, 7, &c. and they .*. remain the same when a is substituted for ns a, of ... ov. . Hence, however, it necessarily follows, that these coefficients can have no more, unequal values than those which arise from the transposition of the n—2 roots 2’, a’, a/”,...0°—). 4. Since .*. the coefficients p, g, 7, &c. have no more unequal values than those which arise from the trans- position of the »—2 first roots; consequently these are similarly cireumstanced with the coefficients mentioned in § CLXVI. Thus the equation of the 1.2.3... ..m—2th degree, on which the coefficient p depends, may, by the union of those of its values which arise from the recurrig transposition of the n—2 first roots, be reduced to an equation _ pr? — p/p’ + dp" — rp’ &. = 0 whose coefficients p’, 9’, 2”, &c. only depend now on equations of the 1.2. 3....— 3th degree. Further, the equation for p’, by uniting its values of forms, which arise from the recurring transposition of the n —~3 first roots, may be reduced to an equation ovis — pla + Daly erry ri piee° + &e. —_ 0) whose coefficients p’, ¢/,.7/’,-&e. only depend on equa- 383 CA tions of the 1.2.3...n—4th degree, and so on, till we come to an equation of the second degree. 5. Having determined the values of the coefficients Ps 9, 7, &c. by the solution of all these equations suc- cessively, and by means of the foregoing chapter, then the solution of the equation A’-' — pK"? + &. =o, gives n—1 values of K, which I shall denote by K’, A”, K”,......K¢-. The n— 2 values im 2 correspond to these values; we .*. have the following m — 1 equa- tions: (cea! Pt! bese TO 4 2) = K! (a22°’ ie ata! Ba as gr ty) ni ey tie K’ (a lal of a al! + is 4 etn op gn ee Km, 6. By extracting the nth root there arise the following equations : Pte SE nek Sonat AM Sa pe +27™= =A aw! + otal + ebal/ + i. a = SK ofa! A ata! + abr!” + ..... se a” N/R , ax! + oa! 4 aba!” + oeccee + = nL RW a ol gal! oe ic + 2 = N/ Kes If we multiply the second equation by 2’, the third by a’-*, the fourth by a"-*, and so on, and then add them to the first, we get ga! = Aa Ra AK Seas spice) KOT If we multiply the second by a*~*, the third by 2*~*, the 384 fourth by «’~*, and so on, and then add them to the first, we get na!) = Atal) KR af NI) RY ciceee bah) KO, In a similar way, we further find na!’ = A+ aN) Kl 4 at) RY 4. seeeeeba\/ KO nu” = A+ at—'\/) Kl 4 a 8KM 4. ese bad) ROY 7 &e. Lastly, if we add all the x equations together, we get ne” = A + N/K! + N/K! +...+ N/ Ko», It is readily seen, that we can derive the values of a”, a’, x’,......0 from the value of 2’, by substituting AN, ib serts ire te ro a" (=1) successively for a, conse- quently by substituting for @ all the roots of the equation a"—1=o0. We..-. have the following general expression for the roots of the given equation : Bi (4 aN) Bl NJ RY cece batN/ KO, Remark. When » is a compound number, then indeed, for the reasons given in 1, § CX X XVII, this method is not applicable ; but for this case, I shall in the Third Part give a particular method, which has this. advantage besides, that it 1s much shorter. a T. C. HANSARD, Paternoster-Row Press. Ene ee NT a as ERRATA. Tue author has in the original designated the sum of the roots of an equation, the sum of their squares, cubes, and in general the sum of their «th powers by the symbols [1], [2], [3]....[«], and has employed the crotchet for this particular purpose, instead of the paren- thesis. By a mistake in the printing, which was not discovered till it was too late to be corrected, the parenthesis has been used indiscri- minately for this and the usual purposes through the earlier part of the volume. This gives rise to a little confusion, the mere mention of which will be sufficient to prevent any obscurity which might otherwise have arisen. See, in particular, pages 5, 6, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 35, 36, 58, 66, 67, 69, 82, 83, 84, 85, 86, 87, 133, 134, where it will be most material to attend to this distinction. Page 5, line 6, for function, read symmetrical function. 11, — 3, 4, for of the mth order, read taken m and m. — 13, — 15, for letters, read roots. » —18 - - - ditto. — 21, — 6, for =(@—«) read —(B—a). —- 22, — 23, for evolution, read development. — 41, — 6, 7, for odd or even, read even or odd. —— 49, — 16, for number, read number of divisions. ——-, — 20, for numbers of a different kind, read numbers of different kinds of things. —— 57, — 18, The product inthis line should be written thus: (=m)! X (F m/e x (Fm!\e", 3 D 386 Page 59, line 9,10, for in each transformation are changed, read when transformed are unchanged. —- 61, — 1, for the, read this. —— 68, — 14, for 0, read a. — 99, — 13, for By, read B,. —— 106, — 8, 9, for § 4i and 42, read § 40 and 41. ——— 109, — I, 2, for when the roots are transformed and permuted, read by the substitution and permutation of the roots. — ]13, — 1, dele them. — 130, —— 23, for quite as, read as little. —— 132, — 6, for other, read others. —— 133, — 5, The coefficient of the second term should be 2’. ——- 135, —— 12, 13, for transformation, read substitution. ——— 143, —— 23, for merely, read all. —— 144, — 4, for other, read higher. —~—- 149, —— 3, The letters 4’, B’, C’, D’, ought to have been old English capitals. 152, —~ 3, 4, The letters 4’, B’, C’, D’, E’, F’ should have been old English capitals. ———— 209, —- 10, for referred, read in reference. ——~ 226, —— 4, dele try to. 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WIT, —\L +L +HI3—|£ u | | i ; [cet \ r= (éet] e-it — {eel ‘a (-le-lr = to | eti¢ +i¢ —|l = Ee) _—_ | ———_—— “ Par fare CAisie«a : 9 3 ; | 5 TABLE I. = a = Li ee | - | 9 a] So) #1218 [2le 2/%/2/%/8|2|« El = i=? 3) = 1-349 a) = ie deay4o [5] = i d+ d+ af— 5|— dit 5 Pz ss 2] = es [i3} = i al- 344 eS ee ee (is = —1 [2] = I~ 242 [28] eine eal 8 [122] = ih + [173] = M— 24— 1+ 5 Se aa [128y = Ne Sb | f19q] = il— 5 * ; Ca ae | = 6 7 = S Cee ee = ) cal fan fae Q a = = fo 4 =/s{slajzi2le/2lelei. Bote is Sa Se Se” f 7\— 3+ 1l— 6)— 1/4 ¢ — 7j— 44+ J— 8+ 7— 14 74+ 1-7 [24] = ~ 3 2+ 4— 34 4 2 64 6 8l Tel ate la [2] = 1} O- 3+ 34 3I_ al_ 34 3 SS eles a Ss eee, = [4] = ee leo I+ 2+ 4|— 1]-+ 3}— 74 1/- 2- 147 [123] = esi ee Se Sea 2 a 0) 0) = HO 2 et 5 Aaa 7 [i38i == 11 2 a4 6 [223] = 1} «#—al_ iit e+ 3l_ 747 fP2} = I-44 9 (154) = JJ— 3/4 3)—- 14 24 1-7 [12] = 1l—-¢ [1223] = 1l— 3|— 44 6l4 ol_ay (i) = 3 (123) = No #i— 3l4 St 7 4 ee es Se ieee, & (824 = eet ; Pl?) 2 = fe 7 = t= [et] 2 8 {tl = [él] Ss octj9 —\t = [est] < ag Uehara Felice (ioe Meee ESV a 91-6 +p —j*# ft = [ete] €—ili+ig +i¢ —fe —jr = Lever] g —lt +i +18. —le Hie —1r = Lh] SHO iS ie le pe Pe = Le) SO eb Pah Ouch ieU be IP cT =e) (One | SI+l¢ —l6 ~\¢ Hie +H —\* |e +ie —jr = [eést] PotlOI-|9 —lh +e —li tle lp Hin —io —]e = [pet] g+it —ig —|t +le —le tit —lp —lp Hie Hie —lt = [eet] g —ig tig +i¢ —i6 —lo +e Ip te io —le oO oft = Lee) Sris ti cio 8+ is +p le fe tig — ie t[t = Cel 91-6 +iortie —it +Hlor—lp tis —lort+is |r —}® |r —jo —[t = [ret 91—l6 +p +i¢ —it +18 —l¢ +18 +loi—|p —jirtie —l¢ 4+]}r —le —}r = Lear 8 —it +ie Hi —lg +le —}t Hr +le —|a —lp +h —le —|¢ tie tie —]i = [oct] pip —lp lh + —is tip —lo His —lp —ip te ofp tle Hip le fe ot = Lal 8 tls —jg —j8 +14 +t +e lp —]r +18 +6 —ie +1 —le ti9 Tle —|* Ci lin age} g tis —Ip +] +13 —lp +12 —lp —lortls —lo —lo +1e tio “ye iy Hie Je +p lt. = Loa] gti —ig —it +ig —leé +r —lp —!6 tle t+lor—jr +18 Hie —l4t—|ritin —je —l6 +i9 —[n = [AT] gs —ijg +I +ls —ls +io1—ls +lr +lor—is —Irz+ls —ls —lartiret|ze—lg +/2 +o1—jost+is —|t = [sl JSR Ee eae eee ee ewe eee Bieta Se Pectin edna |e ueagh 7 poutine rs 8 Baa tell Gia ik) ty Lak LoL —{[l = [e6et] Gi teas iGo Ll Leeks OSE |a te LIT =a.) Peteb) S7—|SI Foi +19 —le —|lr = [east] 6 TH Fig +t —i¢ He —[t =) [rer] . GHe Tle tie je me te ie = Bt PE Tos—je¢ —i6 +g Hip —le fp —]t = [ecest] TH9 —|I -]o +iottlt —\* ie +i2 —[t = [eet] ge+|ZI—-|g —jo¢ +izi—|Fit+is —jp +H —]o —jr = [rect] 6+it —le —ir He —ig¢ the -le —ip H]e He -(t = [ert] Boag rigs igi ie tigi) lpicig —ie. fe fe dt = Eee] £o—{et+Hlertiz1—|si-|z Hl* ole Hp +le —l* |* |e lr = [eset] foal io 16 Tiel e tle —\2 +19 He —\* jt ie —|t = [yet] Lo—\LI+logt{ii—|6 +iei—l¢ +let—je +/* Jl —j* fo Fit —jo [to = [rect] Le—|II+lo +17 —|6 +lot—|p +14 +le1—|9 —]et+lp —lp +le +11 le —|r = Leet] 6 tit +e +t —le He —|t +16 +p —|o —ip +it —le —le —|@ tle +ie —[c = [yet] e He —le —le tio tie —* i Tle —ie@ He f* le tie —* fj |* fe Jr = [esl gitist—lp —ltit+i* |p +i¢ —\o +1e —]19 —le +/* |e —lg Hr jo —\* |* fe —|t = [reel 6 +16 —\¢ +1c +16 —lp Hie —It tig tle |9 —lo +11 —jo —l* Jo +e —le of +e ir = (ez) 6 +i¢ —\6 —|¢ +le —lrtti¢ —\t1Ho —lr —ln +l*) le —]o —l¢ tie —|* of OE tle fe ott = Lett] sit+lor—~igi—jOl+l* fort, —\e Fle +i +iet—|t +lo9 —|lo —Ig¢ +lo +]! —le f@ Ip +12 —jo —|r = Leer] sttiot—ir —le +l |g Hle —Ist—lottip +himle +Hhyitis +ler—l21—|Pl+le —je ~j4 tir 1% +ih —|t =. [9ei) Seeds GW Tb aera Om OI to ir) tia tite SS |% Hie tei 9. ie 16 E19) iT =. Leet] Ge i616 —i6 Ste +N is Hit ie bp itis let |e ip +]* Palm (pun Phieed Lilet ges tbat el Lane vd ac Gio rie +16 le eerie +i* 16 —i6 tie 6 |e. le +16 tet—le +19 tists Fie tig He Ti 8 Hil) = foe] eens (6 +h aio +6) +81—|9 Ho Fle 16 sip OST il Tis Sie Fle TSI |9 Sie fol tis —|9 Fis ait (13) Caan Fil 6. Flat) +6) HOlTI6 HiFi 19 (st it i6l ier +\é ~\sI+|9 —\6 joe leit it —\4 —pite -ir = ° fet) oD 6 +16 —\6 —\6 +16 —\stt+j6 —|6 —Isttie +)£2—j6 +16 +\stt+|4o—|4o—lost+l6 —|¢ +]43—|st+i6 —jeetier—je +16 +los—|ée+\6 —|r = [6] Hei ele/olele;glelelele lel el el el ele] elelel gs ae ie > > >i rel ol el] pt} ei] > cm) a ‘Se se eX) s to ‘S 8 ol vo bo oy 0 to 6 OS bo » a bo C) o xE 6V — —_—____ —} |] | ee | — |} J 19} = = 10 TABLE ITI. ee J oe | — | —__—_ [1226] [1235] = [174?] i bo ® NO Cs [1525] [1334] = [12224] [1223] [1233] —_—— | | eS | | JF SSS 4) -4-19)— 17} —19]-+15,+18)—i5|4 4|—15}4 6j+19|— 6)— 4/411]— 9/4 4|— 6|—121+30 — 7\— 8}|+31)/—40 foe = lets 2 [3] += 3\+ [lea] = q— i+ 4|— 6)4y7)—15|4 6 [1437] — V+ 3)+ [13223] + 14|—32|—46}+ 100 [1724]. - = x x [1%4}- = + 3}— I}+ [1523] = MN— 3} 7j+12/415)/—60 Os — a) = [1922] [1*2] i Up Ca) ’ 7. fy | a Benya) ae \ oF Fa —* <