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ONTARIO isEiili^ir^ PIUNCIPLKS iiy- mi; SOLUTION OF EQUATIONS OF T'lK HIGHER DEGREES, )VITH APPLICATlOxNS. BY GEORGE I'AXTON YOUNG, lOIlONTo, ( A.NAI'A. [/{(•((./ bf/yrc tlir Cona'U'an Int() 1 111 m-2 m-1 where «/ in rational, and a,, 6,, etc., involve only surds subordinate to I L^ 7. The equation F (x) = has an auxiliary equation of the (»u - 1 )'»» degree. §35, 52. 8. Tf the roots of the auxiliary be A,, 5,, ^,, . . , ^m - i> the m — \ expressions in each of the groups ji i_ III . m ^1 'U-1' sr s "• 2 in ^ m »»».»« 1 m — 2' 2 m — 4' _3 1_ _3 1_ 1 1 . . . . , C-, A "* ^1 ' > 2 m — 1 1 '^2 ' 3 1 ^x ^n-^' ^ -^TO-e'---' ^TO-i'^3 • and so on, are the roots of a rational equation of the (w - l)*"" degree. 7H '~ 1 The terms _i i_ TO - I ' ^2 ^TO-2« 1 TO J[_ TO ' TO — 1 TO +1 ' — ^ — 1 degree. i^39, 44, 55. 9. Wider generalization. §45, 57. 10. When the equation F(x) = is of the first class, the auxiliary equation of the (m — l)'** degree is irreducible. §35. Also the roots of the auxiliary are rational functions of the primitive m*'* root of unity. §36. And, in the particular case when the equation F(x) = is the reducing Gaussian equation of the m^^ degi'ee to the equation - , „ , m — 1 .'<;'* —1=0, each of the — - — expressions. _i l_ m ^ m has the rational value n. §41. Numerical verification. §42. *or instant rdinate to on of th« the m — 1 !)*'> degree. th I degree. bhe auxiliary Jso the roots ^th root of tion F{x) = the equation 11. Rolution of the Gaussian. §43. 12. Analysis of solvable irrediicihlo eqimtiun» of the fifth dogree. The auxiliary biqijadratic oither in irreducihio, or has mi irroduoihle sub-auxiliary of the second dogree, or has all its root« rational. The [three cases considered separutely. Deduction of Abel's expression [for the roots of a solvable quintic. §.58-74. PRINCIPLES. §1. It will be understood that the surds a{q>earing in the present {paper have prinn, numbers for the denominators of tlieir indices, mless where the contrary is expressly stated. Thus, 2^^ may be 1^ 1 regarded as A* , a surd with the index J, h being 2:*^. It will be mderstood also that no surd appears in the denominator of a fraction. ror instance, instead of we should write 1 - V~3 1 + V^ ~ 2 ^hen a surd is spoken of as occurring in an algebraical expression, tt may be present in mere than one of its powers, and need not be iresent in the first. §2. In such an expression asV2 + (l-f- V2),v/2is suSordi- \nate to the principal surd (1-1- v/ -) > t^^© latter being the only prin- oipal surd in the expression. §3. A surd that has no other surd subr rdinate to it may be said to 1 be of the first rank ; and the surd h " , where h involves a surd of the |j|« — l)**^ rank, but none of a higher nmk, may be said to be of the t'''' rank. In estimating the rank of a surd, the denominators of the ulices of the surds concerned are always supposed to be prime Jbumbers. Thus, 3^ is a surd of the second rank. %. 1 i §4. An algebraical expression in which J "* is a principal (see §2) '.tf 1 ijpurd may be arranged according to the powers of J "* lower than the #»th, thus, If -^ \g\ -t- Ai J "* + ai J 2 3 \-'i wi 1 1 -f- eiJ "* + h (1) 42. Isvhere Qi, ky, ai, etc., are clear of Jj §5. If an nlgobrical exprestiion r\, arranged as in (1), be zero, wliile the coefficienttt (ji, ki, etc., are not all zero, an equation 1 wJj = li (2) must aubHist ; where <« is an in^^ root of unity ; and /i is an expression involving only such surds exclusive of J *" as occur in r\ . For, let the first of the coefticients hi, ei, etc., proceeding in the order of the descending powers of J *" , that is not zero, be ui, the coefficient of m J "* . Then we may put mri 1 m «1 {/( J,"*)!™ mJj"* + etc. - 0. Because J is a root of each of the equations /(x) = and «* - Ji = 0,/(x) and a;"* — Ji have a common measure. Let their 11. 0. M., involving only such surds a« occur in/(.T) and oj"* — Ji, be

p^ Now c is a whole number less than m but not zero ; and, by §1, m is prime. Therefore there are whole numbers n and h such that en m 1 1 Therefore, if (wj tt»2 ■•)" = <"> *"d 'i '^i (" 1)*" •■ Pc' *" "^i - ^^• §6. Let Ti be an algebraical expression in which no root of unity 1 having a rational value occurs in the surd form 1 "» . Also let tlere be in ri no surd J not a root of unity, such that [where ^i J. exec Ptho same i [said to ha r> ^ 2 lore, a ro [2 + V J 2 - v/3) §8. Let 1 I'here J. §9. The I generic syj ^» »'2. »"3, ei flirds in vol iird is uni account. nay be c(] (2) expression- . For, let ider of the joefficient of a;) = and eaHure. Let in / (.r) and of «»" - J I, !, etc., are dis- ^l ' '1 . (3) [where «i is an PxprnsHion involving no MunU of ho high n niiik an J f^xcept Huch 08 either are rootn of unity, or occur in ri Iwing at 1 hhe same time distinct from J, The expression ri nmy then be [said to have heen atmpli/ieU or to be in a $imple state. §7. Home illustrations of the definition in i^6 may bo given. The jot 8* cannot occur in a simplitiud expression »i ; for its value is ho, lo being a third root of unity ; but the ecjuation 8* = 2/3)4 - (2 - v'3)(2 + V3)». m — l m m—l m §8. Let piJi + i'>Ji + .. + p^ = 0; (4) 1 irhere J is a surd occurring in a simplified expression ri ; and pi. 2, etc., involve no surds of so high a rank as J , except such as either #re roots of unity, or occur in ri being at the same time distinct ■^ lh}m J m The coefficients pi, p2, etc, must be zero separately. 1 id, by §1, m is ch that n )cn = p^. 1 w Jj = ll. ) root of unity Also let tlere 'yor, by §5, if they were not, we should have w J = ?i, w being an il*h i-oot of unity, and ^i involving only surds in (4) distinct from '4 * m ,■4. ; an equation of the inadmissible type (3). §9. The expression 7'i being in a simple state, we may use /? as a ^neric symbol to include the various particular expressions, say l|» ^2< ^3» 6tc., obtained by assigning all their possible values to the •iirds involved in ri, with the restriction that, where the base of a 0i'd is unity, the rational value of the surd is not to be taken into •Bcount. These particular expressions, not necessarily all unequal, May be called tlie particular cognate forma of R. For instance, if f* = li, 7? has two particular cognate forms, the rational value of the ^ .1 third root of uiiity not iMjing counted. If ri - (1 -f V 2)*, K hii» Hix particular cognate forniH all unequal. Should r\ -^ (2 -»- V '^)^ f- (2 — V ^) (2 + V 3)i, R haH six particular cognate lorniH, but only three unequal, each of the unequal foruiH occurring twice. §10. ruopuHiTioN I. An algebraical exprowiion r\ can always be brought to H Hiuiple wtate. i_ For ri may bu cleared of all surda such an 1 "* having a rational 1 value. Suppose that r\ then involves a surd J. , not a root of unity, by means of which an equation such as (3) can be formed. Substitute for Jj in n its value e\ as thus given. The result will be to elinji- 1 nate J from ri without introducing into the expression any nevj 1 HI surd as high in rank as J , and at the same time not a root of unity. By continuing to make all the eliminations of this kind that are possible, we at last reach a point where no equation of the type (3) can any longer be formed. Then because, by the course that has 1 been pursued, no roots of the form 1 *" having a rational value have been left in ri, r\ is in a simple state. §11. It is known that, if N be any whole number, the equation whose roots are the primitive N^^ roots of unity is rational and irreducible. §12. Let N be the continued product of the distinct prime numbeis «, a, b, etc. Let w\ be a primitive n*** root of unity, 0\ a primitive fl.'i» root of unity, and so on. Let w represent any one iodiflerently of the primitive «*•> roots of unity, any one indiflferently of the primitive a*^ roots of unity, and so on. Lety(<«i, 0u etc.,) be a rational function of wj, ^,, etc. Then a corollary from §11 is, that if f{to\, 01, etc.) = 0,/(w, Of etc.) = 0. For h being a primitive iV**" root of unity, and t representing any one indifferently of the primitive jV'th roots of unity, we may put /(wi, Oi, etc.) = axh + dy etc.) = ait + + etc. = 0, etc. and/(w, where the coefficients ai, a2, etc., are rational. Should these coeffi- cients be all zero,/(a*, 0, etc.) = 0. Should they not be all zero, let ar be the first that is not zero. Then we may put * •!!? Therefor the Hnni« whoNo rt of i\w mtioiiHl {wf , ■ imtiun i^ (r) 0, wh \^ (x) mv\ V (x) have a connnon nn'MHure. Kut by Jjll, ^ (x) in irn'iliiciblB. Thrrffoifl it in a nicuHiirf of y» (r) ; nml tho motH of tlw npiiition }^ Ix) - are roots of tlie tM|uation y (x) - 0. Therefore, /{w,0,iHc.) = Hr )y (0( - 0. ^1.'). Another corollary is, that if /{"'\> ^\> OtC') '■= h\vii + hjutx ~ -f . . 4- /<„ a 0, where hi, Aj, ftc., urn clear of wi, ihe cot'tHcients Aj, h-^, etc., are all iiqual to one anolluM*. For, by {^12, because / (*»\, 0\, etc.) = 0, /(w, 0\, etc.) = 0. Therefore m j /(w, //,, etc.) | = 0. In to I /*(<«, ^'i, etc.) I give t» HUCccHHively its rt — 1 diffuront vuIuch. Then, in addition, 7iA, = /(, + /*,+ . . + An. Similarly, nA, - A, + A, + . . + A„ . • . A, = A,. In like manner all the terms Ai, h-i, etc., are ecpial to one another. §14. Proposition II If the simplified expression ri, one of the [•articular cognate forms of K, be a root of the rational equation F (x) = 0, all the particular cognate forms of R are roots of thiit equation. For, let r2 be a particular cognate form of R. By §12, the law to be CHtabliMhed holds when there are no surds in )'i that are not rootn of unity. It will be kept in view that, according to §1, when root« of unity are spoken of, such roots are meant a» 1 "• , m being a prime number. Assume the law to have been found good for all expressions that do not involve more than n - \ distinct surds thit are nrt roots of unity ; then, making the hypothesis that r\ involves not more than n distinct surds that are not roots of unity, the law can be shown 1 still to hold ; in which case it must hold universally. For, let J ' not a root of unity, be a surd of the highest rank (see .^.3) in i'\ Then F {r\) may be taken to be the expression (1), and F (r-z) to be the expression formed from (1) by selecting particular values of the Burds involved under the restriction si^ecified in §9. In passing from _i JL ri to 7*2, let J , m, etc., become respectively J , a^, etc. Then m — 1 m — 2 m {F{ri)\ = Ai J, + ei Jj + etc. = 0, m— 1 and m | F (n) } = Aj Jj "* + *2 ^j + ®^- 8 i By §8, because n is in a simple state, and F{ri) = 0, the coefficients /<■, fli, etc., are zero separately. But hi is clear of the surd J . It tlierrfitiv docs i;ot involve more than n — 1 distinct surds that are not roots of unity. Therefore, on the assumption on which we are proceedinij, liecnuse hi = 0, Aa = 0. In like manner, cj = 0, and 80 on. Therefore F (rj) s= 0. §15. Cor. Let the simplified expression ri be the root of an equation /' (x) = whose coefficients involve certain surds -1 _! « , »i , etc., that have the same determinate values in ri as in F (x). Then, if /•2 be a particular cognate form of R in which the surds z ,11 , etc., retain the determinate values belonging to them in ri, ra is a root of the equation F (x) = 0. For, F (ri) = 0. Thci-efoje, by the Proposition, F (B) su 0. Let E, restricted by the 1 I n 8 condition that the surds z , u. , etc., retain the determinate values belonjriiis to them in n, be R'. Then F (R') = 0. A particular case of this ivS /'(r-j) = 0. The corollary established simply means that 1 i_ ft S the surds z , u , etc., may be taken to be rational for the purpose in hand. §16. The simplified expression ri being one of the particular cognate forms of R, let ri, Va, etc. (5) be the entire series of the particular cognate forms of R, not necessarily unequal to one another. Then, if the equation whose roots are the terms in (5) be JT - 0, JTis rational. In like manner, if those jmrticular cognate forms of R, not necessarily unequal, that are obtained when certain surds z , n , etc., retain the determin- ate values belonging to them in r\, be ri, »v, etc. (6) and if the equation whose roots nre the terms in (6) be X' = 0, X' \ 1 involves only surds found in the series ;; ,u. , etc. This is sub- stantially proved by Legendre in his Theorie des Nombres, §487, third edition. §17. Pfl the )[^eneri iiv the r( ^iiiequal \n the series As in §i Je the ten Deiug X = It has a rai Ithe equnti ,^tiite, all \ '1(pliile at tl bf the equ >eing irrec Inequal te Jecausft X (x) is a t.s roots tl '(x) = mce F (x is F(x)r tetX = ) )rms of R -fognate for ■f: I M §18. Coi ibrms of in ate v&i ) occurs] (6) an ivolve or irreduc raroken in| ilpive the tllrdR occij ■•Urds thai X" =■ 0.1 thiit equal tterma in -fQiiid rooi 9 9 coeificients r . It Js that are |ich we are = 0, and §17. Proposition HI. The unecuml pmticular cognate forms of /f, the (i^eneric expression under which the simpliHed expresaion rj falls, iro the roots of a rational irroducible equation ; and each of the niequal particular cognate forms occurs the same number of time« in the series of the cognate forms. As in §16, let the entire senea of the jtarticular cognate forms of R )e the terms in (5), the equation that has these terms for its root.s Deing A' = 0. By §1^. -^ is rational. Should X not be irreducible, It has a rational irreducible factor, say F{x), such that r\ is a root of ^he equation F (x) = 0. By Pioj). II., because r^ is in a simple ^tate, all the terms in (5) are roots of the equation Fix) = 0, 'i|(phile at the same time, because F (.c) is a factor of X, all the roots pf the equation are terms in (a). And the e(|uation F (x) = 0, Ibeing irreducible, has no ecjual roots. Theiefore its roots are the %nequal terms in (5). Should F (x) not be identical with X, put I X ^ {F{x)\ jy(x)l. Jccause A' and F (x) are rational, f (x) is rational. Then, since (x) is a measure of X, and the equation F (x) — has for its ts roots the miequal roots of the equation A' = 0, the equations ~'(,r) = and

s, §487, third 10 X* = (X") {X"'). Then, by the line of reasoning followed in the J Pro|K)8ition, X*" has a nieasuio identical with X". And so on. Ultimately X' = {X^. §19. Cor. 2. If rj, one of the particular cognate forma of /?, be 1 zero, all the particular cognate forms of R are zero. For, l)y the | proposition, the particular cognate forms of R are the roots of a; rational irreducible equation F (x) = 0. And r^, one of the roots of j that equation, is zero, but the only rational irreducible equation that, has zero for a root is a = 0. Therefore F (x) = x = 0. In fact, in the case supj)osed, the simplified expression ri is zero, and Ji has no particular cognate forms distinct from ri. §20. Proposition IV. Let iV be the continued product of the, distinct prime numbera n, a, etc. Let wi be a primitive n^^ root ofi unity, Oi a primitive «**• root of unity, and so on. Then if the equation ' F(x) = x-d 4- 6,a:d-i + ij^^-a ^ gtc^ _ q be one in which the coefficients 6i, 62, etc., are rational functions of «>i, Oi, etc., and if all the primitive n*^ roots of unity, which, whoii substituted for \. For, assuming that there is a term «>2 in (7) additional to wi, wf may take «>2 to be the first term in (9) after u»\ that occurs in (7) M and it may be considered to be <«i , which may be otherwise writttjf. (-,) = Ur {iui ). x' f ('"1 )> ^ [negative, [these terr fin (7), sai If (w,) = |lHH;au8e si because t lit is not J Iwhich, w [Therefore (less than \z=- (h I which, b( 'in (9) aft [ unalterec j(8), vvhi fin (7). [the eye ■expressii (0). T ' Because F{x) = where j. lit I is cl Therefl Then, if F (x) be written ^ (wi), we have by hypothesif Here, the 11 kwed in the I And so on. [ma of if, be For, by the |e roots of a the roots of bquation that In fact, in Ind R has no Iroduct of the re n^^ root of if the equation lial functions of r, which, when made up of a (181 to the first, but prime) the cycle (9 form of F {x) if v.d— 1 + (10 Up (oil) = ^ (ttfi ). Therefore, by §12, changing tu\ into ui\ , tp (wx )sBr I ^* , ^' Uf (w\ ). Therefore (p (wi) = y» {io\ ). And thus ultiuiately f (wj) = if (<"! )> 01' V ('"i) = *P (<«i ). ~ being any whole number fMwitive or ; . \\ [negative. Hut m\ includes all the terms in (8). Therefore each of [these terms is a term in (7). Suppose if possible that there is a tern) I II) (7), say tu\ , which does not occur in (8). Then, just as we deduced ^9' (wj) =

\ int( J»2» (^i. etc., are tional to wi, wf t occurs in (7) bherwise writtei. 5 by hypotheaif i9' 3 »;u -)- Au I because still farther

\) =.

(i is the first term S in (9) after wi which, when substituted for wi in y ('«i), leaves v (ci) ^ unaltered, is impossible. Hence, no term in (7) lies outside the cycle ;> (8), while it has also been shown that all the terms in (8) are terms ■^f in (7). Therefore the terms in (7) are identical with those constituting |the cycle (3) We have now to determine the form of /'(«). The •expressions, Ci, C^, etc., taken together, aie the sum of the terms in ;(9). Therefore Ci -f Cj -j- . . . . + C,„ = - 1. (II) if Because (9) contains all the primitive n*"* roots of unity, we may put l\x) = x-'i - |p + (p + p^) wi + (/; + 7^2) "»! + etc. |x'*-i+etc.j(12) where p, pi, etc., are clear of wi. But F (x) remains unaltered when <«! is changed into toi . Therefore otn + etc. } x'^ — ^ + etc. F{x) = x^ — {p -^ {p-^ pi) oji Therefore, equating the coefllcients of a:^~i in (12) and (13), (13) {P — Pi) + + (Pm + i — Pi) wi -I- etc. = 0. Here, by §13, the coefficients of the diflferent powers of wj have all the same value. And one of them, p — jh, is zero Therefore 12 Hi 9! ! N i! rvj ilill /8" ^ p^ _^ ^ = pi. That is to sav, the coefficient of wi or ojj is the same as that of «*i. In like manner the coefficients of all the terms in (8) are the same. Therefore one group of the terms that together make up the coefficient of x^ — ^ in (12) is properly represented by — (p + p\)C^. In the same wny another group is properly represented by — (/> + PiW^, and so on. Hence F{x) = xd- ^p + {p-^pi)Ci + {p + p2) C.> + etc. Ja^-i + etc. And by (11) this is equivalent to (10). The form of F{x) has been deduced on the assumption that the series (7) contains more than one term ; but, should the series (7) consist of a single term, the result obtained would still hold good, only in that case each of the expressions Ci, C2, etc., would be a primitive >i*'' root of unity. §21. A simplified expression will not cease to be in a simple state, if we suppose that any surd that can be eliminated from it, without the introduction of any new surd, has been eliminated. §22. Proposition V. In the simplified expression ri. one of the particular cognate forms of B, modified according to §21, let the 1^ 1^23. Co R ; an( come in ruis of ] Ifierent n lese tern , and rj lue is o> taken ii. pt the surds d the sim] St fa + 2 ^hen Ta + cpression lerefore lould, by ipossibl( inner a) '.r J24. CV ms in ( surd J " of the highest rank be not a root (see §1) of unity. Then, 1 in if the particular cognate forms of R obtained by changing J in ri successively into the different m*** roots of the determinate base Ji, be ri,rz, ,rm, (H) these terms are all unequal. For the terms in (14) are all the particular cognate forms of R obtained when we allow all the surds in rj except J to T-etain the determinate values belonging to them in ri. Therefore, by Oor. 1, Prop. III., each of the unequal terms in (14) has its value repeated ijear of t the same number of times, in that series. Let u be the number of the unequal terms in (14), and let each occur c times. Then uc ^ m. Suppose if possible that u = I. This means that all the terms in (14) are equal. Therefore, ri being the expression (1), mn = ri + r2 -f- .... + etc. = gi. 1 Therefote the surd J can be eliminated from r^ without the intro duction of any new surd ; which, by §21, is impossible. Therefore u is not unity. But, by §1, m is a prime number. And m = uc. Therefore c = 1 and m = w. This means that all the terms in (14) are unequal. l|9ots of I lie respec ll^eoted, 13 u)i is the same | le terms in (8) gether make up f—{p+lH)Ci, I c. ja^-i + etc. F (x) has been more than one arm, the result the expressions U'23. Cor. 1. Let Va ■■- i I'e any one of the particular cojjimte forms 1 m I m R ; und let J . ^a -^ i , etc., l>e respectively what J , //i , etc., come in pasHJug from ri to I'a + i . Also let the'm (mrticular cognate irms of B, obtained by changing J in rj „ i sticcessively into tbe pfferent m"' roots of Ju , i , be • • > ^a f »/i • ( » ^ ) l_ m a simple state, rom it, without \ ri. one of the to §21, let the E unity. Then, 1 ngmg J in n inate base ^i, be (14) late forms of B to retain the fore, by Cor. 1, I value repeated the number of Then uc = m. ill the terms in thout the intro le. Therefore « And m = uc. he terms in (14) Tlit'se terms are all unequal. For, because J ' is a principal surd in 1 1^ , and ra is what ri becomes when J is changed into a surd whose m fliluo is c'x J , tui being a primitive »»*•* root of unity, the view may Ibe taken that r^ involves no surds additional to those found in i'\ , «|i(. pt the [iriniitive 7/i*^ root of unity <«! . Therefoie vi — r-i involves no surds distinct from primitive in^^ roots of unity that are not found in the simplified expressicm rj . Therefore ry — rj is in a simple state. 1 m J m JLet /'a + 2 be what Va + i becomes by changing J into tuj J 'Ihcn Ta + I — Ta + 2 is a particuhir cognate form of the generic fXpressioi) under which the simplified expression ri — r^ fulls, therefore Va + i — t'a + 2 cannot he zero ; for, if it were, ri — r^ would, by Cov. 2, Prop. IIi',, be zero ; which, by the proposition, is ijJDpossible. Hence, the first two terms in (15) are unecpial. In like aianner all the terms in (15) are unequal. §24. Cor. 2. Let Xi = be the equation whose roots are the iWms in (14). When Xi is modified according to §21, it is, by §16, 1 nt can he sliown to bo unequal to tlm cirresjiondin!; coolhcieiit \Xi in the same way in whicli the terms in (1.5) were proved to be [unequal. 26. Cor. 4. Any two of the terms in (IG), as Xi and A'l . being cted, the equations X\ = and A'l =: have no root in common. ^•, siippose, if possible, that these equations ha^'e a root in common. i^ing the forms of Xi and -Yi in (17) and (18), since ri is u root of llil equation A'l = 0, m c m-1 r"+ {bz.^ + etc.) r™ -f etc. = 0. (19) C JiM the surds in this equation excei)t s, occur in ri . It is in\poasible + etc. — u. .^. ^ j^ 1 1 , . Uilt z can occur m ri : tor, z, occurs in ri ; and z„ = ftiz • i expression ri . Tf^ 2 ' ' ' i ' ' 2 ' i l^^boing a primitive c'** root of unity ; but this equation, if both z and s,, occurred in ri , would be of the inadmissible typo (3). |"*-^+etc.=0. % 4- -' Since ,~ does not occur in ri , it is a principal (see §2) surd in (19). ■yS^f niay, therefore, keeping iu view that ri is the expression (1) in the equation •wfcich J is a principal surd, arrange (19) thus, , = 0. (18) m — 1 c — 1 c e_—2 c 15) are roots of U are all unequal. lar cognate forms ' V (-^1 ) = "^l (P'^-2 + ^'2«2 + «?tC.) «i — 2 e — 1 c — 2 711 , t V + ^1 ( rd z^ , jikequal pt ie series .| nipditied : putting equations A'l = 0, ATi =: 0, etc., has m unequal particular cognate forms of li for its roots, and since, by Cor. 4, no two of these equa- tions have a root in common, the mc roots of the equation X2 = are unequal particular cognate forms of B. m w* I'OOtS and 6*2 to I •t'tlie cloJ Whicii is tf the roots subject to| 0% in Xe I (21) ) cannot Ic (o,,J^'^ ) = 0. But in like equation, and is impossible. J. m \2S. PRoi'ttHiriox VI. L«'t tli«' Niir)|i]ifio lii<{hfst rank in r\ , .V in a inultiplf of m Hut it' ri involve y Hunls (liiit an) roots of unity, one of tlu-ni bcin;,' tin- |iiiuiitive root of unity, X in a iiiuhi|ilc of a nifUHurc of // — 1. Fii-st. let J , not a root of unity, l)«' a hui»I of tlu! Iiiglicst rank /•j . Takiuj^ the cxpre.ssion (1) to h lt;t A' I lie fi urincil us in and Ift it be inodilitd acoording to §21. It is ch-ar of the Should it involve a surd that is not a root of unity, let be t'onned aH in §27. Setting out from rj we arrived by one . tep in ^ (J'" ail' al Xi , an cxpreHsion clear of J , and such that the roots of the eauation ^Vi = are uneijual i)articular cognate forms of Jt A = ?i , CO being _]_ (J )exclu8ive _1_ c ^1 «nd Htej) brought us to X> , an expression clear of the additional Ulrd z , and such that the mc roots of tlio equation X2 = are ttequal particular cognate forms of li. Thus we can go on till, in e aeries Xi , X,> , etc., we reach a term X^g i"to which iu> surds Then /, enter that are not roots of unity, the rnc . . . . I roots of the ecpiation jr« = being unequal jtarticular cognate forms of A'. Should A'l. niotlified according 'o §21, not be rational, its form, by Prop. IV., jMltting (/ for j/ic .... ^, is cept the primi L . This makes 3). Hence the on. 16 terras in (16). , in the same r. 2, each of the .rticular cognate of these equa- quation X2 = X,=a;d-(;nCi+ .... +y>„,(7,„)a/i-i + ( , attd (?2 to bect)me C3 , and so on, C,„ becoming Ci . As was explained »fe the close of §20, the cycle (8) may be reduced to a single term, which is then identical with Ci . It will also not bo forgotten that the roots of unity such as the m'** here spoken of are, according to !:;i, subject to the condition that the numbers such as n are prime. When Oi in Xf is changed successively into C'l , Co , etc., let ^V, become V V V y (m - 1) (22) 18 If ^V^ 4 1 lie the cuntiiiiu'd product of tlir- tonnH in (22), the dm rootH «if the e(|iiiition Xe + i => oin ha Hhown to he niie(juiil |iHi'ticul>ir «• tgtinte forniH of Ji. For, no two terms in (22) hh X, an«l Xt arc iih'nticiil ; hecanHe, if tliey wore, Xg wonM rcnjiiiii unalttred by the B ' . B • '•litin^f of (111 into «i>i ; which, hy Prop. IV., liecauHe w, is not a Uy,u\ ill tin- cycle (8), is injpoHHihh?. It folhtws that no two of the ('(juations Xf = 0, Xc = 0, etc., have h root in common. For, if the eqnatit n.s Xf = 0, and X, = hail a root in common, since Xg and X, are not iilentical. X^ would have a lower measure involving only surds fouml ill Xt , liccanse the surds in Xe nre the same with those in Xt . Li-t I.' (.r) he thi.s lower measure of X, , and let rj be a root of the equ i- tion 9' (x) = 0. Then, by Cor. Prop. II., all the d roots of the epuvtion ^V, = are roots of the equation

f lave a root in unequal | ar- 1 according tn uld Xe + 1 'i«>t Going on in f snoh that the qual i)articuliir with the eqna- I., the equation late forms of /?. are all unequal Ji, Xt must be x). But F (x), Xt is identical the N^^ degree, m. This is the 1 it rank J, not fcp writ's of int«'gor.H r», r, etc., of which X i«* tlit« ronh'nui'd product, retluot'd to its lirMl U-rni. If j'l iiivtilvo only siinln tliaf are ro<)t« of y, n — 1 is a inultiplti of X ; for X tm m . . . . g ; thiTi-fore, use X i« prime, it is etpial to m ; hut »»« = « — 1 ; thorfforo - 1 s 8X. \\IZ SOLVABI-K rimEntCIULE KqI ATION OK THK ;/j'^ pKOnKE, JH PlUMR. §30. The i»rii,cipleH that have Itfi-u fst^iMishcd may l»o illustrated %y an examination of the solvahle irrediicihlf rational erpnition of the •§|'*' degree F (ji) = 0, vi heiti;,' prinie. Two nisfs may In* tlistinguished, Ibougli it will he found that the rt)ot« can in the twt) casos he hroiight (Ipder a cotnmoii form ; the one i'iVM' hciti;,' that in wliicli tin; simplified Ipot ri is, and the other that in which it is not, a rational function of )t8 of unity, that is, according to §1, of roots of unity having the (nominators of their inilices pritnf innnl)i'ix The equatitm /'(x) =5 iy he said to he in the former ciise of thf. first class, and in the latUM- the second class. i^ The Equation F(x) = or the Fiiist Class. ■I ': §31. In this case, hy Cor. Prop. VI., Vi being modified according to 1^1, if one of the roots involved in rj be the primitive n^^ root of onity ti»i , n — 1 is a multiple of m. Also the expression written JCt in Prop. VI. is reducetl to x — ri , so that n = piCi + P2C2 -f Thc m roots of the equation F (x) hkve n = pi Ci + P2C2 + . r-i = PmCi + piCi -f- . • • • • + PmCm • being ri , r-i , etc , we must • ■ + rmCm , • • + pm-lCm> (23) rm= P2 Gi + pzC^ + + IhCf^ Por, by Prop. II., because ri is a root of the equation F (x) =s 0, all the expressions on the right of the equations (23) are roots of that '«quation. And no two of these expressions are equal to one another. For, take the first two. If these were equal, we should have (Pw - pi)Ci-\- (pi- p2 ) C2 + etc. = . Therefore, by §13, each of the terms pm — pi , J>\ — Vi > ^*'^'» '^ ^^''^o. This makes Pl , P2 , etc., all equal to one another. Therefore n = — ;5i ; so I involve a surd that the jjrimitive «"• rt roots of unity, re of n — 1. ,y, X = m', for. 90 P'J. lift r\ b« oiitt of the imiticiiliii- co^nutn foriiiH of the g(!ticn<- rX|nHiim ri fnlLs. Tiiuii, liocii\im-, hy J'i'i)|i. 11., ull thu |)articiilai' cdgiiate foriuH of H uro ruut.-s of till' I'ljuiitinii /'(./■) = 0, r\ Ih p(jiial tu one of tlio m t<;riiiH ri , /"j , (ttc , Htiy tu rt . 1 will now hIiow tiiiit tint chaiigfu of thu Hurd^ iiivolv«;(l that caijsi! ry to hocoiiiu /'i , whohu valuo w r, , ciiumo rj to nrt'ivt! the valiir >', + i , iintl ra to ivc'«!iv«! the value r, ). -j , and ho on TluM may appeal' ubvioiiH on thu face of the eiiuationH ('J3) ; but, to prevent luiHundei'Htautling, the utepu of the ileditution are given. Any rlianyes nuide in r\ uniHt tiauHibrni f i into f, , oiits of the ;u terms C'l , C'i , etc. In pa8.sing fiom r^ to /'i , while Ci becomea C» , lot r liocoMie /o , and /^i become ;>i , and /;•.» become /j^ , and ho on. The change that cauhiiH C\ to become C, trauHforms C'a into C, + i , and Ca into Cj + 2 , »nd ho on. Therefore, it being underntood that /'m + 1 . t/m+ I > tJtc., are the saujo as pi , C\ , etc., rcHpectively, r\ = p\Ci -I- P'iC, + 1 + etc., and ra = ;;„,C,4- piCn. 1 + etc. ; which may be otlierwiso written n = ;'m + 2 - « C'l 4- «,ft + 3 _ I Ca + etc., ) , , , ^ (24) »*2 = i>m + 1 - # Ci + ;>m + 2 - » (^2 + etc. Therefore, form (24) and (23), ' t (h{PmJr'i-» —Pm +2..») + Ci {pm +3-« — />m + 3 - «) + etc.= 0. I I Therefore, by §1 3, ;>,» + 2 - * = /% + 2 - a , i^m + 3 - » = /^m + 3 - ., etc. Hence the second of the equations (24) becomes ri = Pm + \-zCi + Pm + 2-tC2-\- etC. = T, + y. Thus r-z is tranHformed into ^^ + 1 . In like manner rg receives the value r^ + 2 > 'md .so on. §33. By Cor. Prop. Vl., the primitive n*** root of unity being one of those involved in r\ , n — 1 is a multiple of /u. In like manner, if the primitive «^ root of unity be involved in ri , a — 1 is a multiple of viy and so on. Therefore, if ti be the primitive m^^ root of unity, t\ is distinct from all the roots involved in rx . fe — '•, ' Is reached An I HO oil fi . fi , ei by ^\2, HI m*'' rootR mi-c — (ri §35. P Ihe tornii ire the r ' For, j^ognato rticulil th root fa in ri , contain! numbeil indifferl J'he coj lean \)ri 31 tliH genoric FhIIm. Thuii, li uro roots iiiiH ri , r-i , lot' the Hurds ciiuMc r> t(» I , unci HO on. \{'i:\) ; hut, to 1 given. Any tho m tenus §34. From tliis it foIlowH fhnt, if tin* rirc'o of loot-* r| , r^ Iw iirnuigiil, tK'ginnin:; with r, , in tin- order r, , r, an ''c — (''I + »'2 + otc.)= mr, — (ri + rj -f> etc.). Therefore re=r, §3.'). PuoposirroN V^H. Putting 1 m 1 ^•i + ^^-j + /, ni + ''m > 1 — ■> 4 + t a(m - t) 1 ^•m w -1 -1 -2 1 m > (26) e terms, •^\ 1 -^i i ^3 > • ■ ■ "»! — (27) the roots of a rational irrediicible equation of the (m — l)*** degree (a-) = f), wliich may he said to he nuxiliari/ to the equation (x) = 0. For, let J he the generic expression of which Ji i.s a particular gnato form ; and let J' denote any one indifferently of the m — 1 rticular cognate forms of J in (27). Because, by §.33, the primitive f(i^^ root of unity does not enter into ri , ro , etc., no changes made ri , r.2 , f:tc., affect -\- etc.) = I'l + t" r-y + etc., s being a whole number. This may be written rm + 1 - * + '"m + J - * *^i*-'-, i'Je unequal, ri ^ 1 ; which, because , etc., are clear of , ete., are not rational. We may take the primitive «"' root of unity (Di to be present in these coellicienl.s. But wi occurs iu Vi , r-j , etc., iind theref«)re also in Jj , only in the expressions Ci , C2 , etc. Theiefore .Ij := (/j Ci -|- .... -f , etc., cannot all be equal ; for this would make Ji = — di ; which, by §21, is impossible. Hence 711 unequal lues of siveiv S/lOW I thc.it' M lerefore eause d\ Dce tlu'i 'roj). It lowei ierefoie (27) is ;§37. Co hicli is ^ fneric e »r /j HucJ , ^2 , e| t^e Prop! I lllictiun 4% satistiJ §38. n Pm =- {' Multipl P,= ( Hence 1%, illK •23 ilie express than »i — 1 lal purticulai taking the ling ti and t" ^tc, etc. (28; .11 the 111 terniD conies lUiHottli*' j»t'nfiic oxpnissiou J aici h Ci + + '/. c, Tto Hiiow tlmt tlie.se exprcMsion.s uro all iineinial, take tlw liiNt two. If the.«ie were eijiial, we .should have ('/. , etc., are iiot all njual to oiu' aiiotlier, is impossilile. Since then J lia.s at lea.sl in unecjiial partieular eo^fMate t'oviiis, Ji i.s. hf I'rop. ffl., the root nf a rational irredneilile e(|iiati<)U of a ilf;,'ree not lower than tin; ;/i"' ; which, I'V I'rop. VII., is ini|»ossil)le. Therefor*' Xj , {■■, , etc., are rational. Hence each of the e.\pie.s.sions iik (27) is a rational function of , ■i"» root of unity : in 7'i , r-i . etc., IS Ci , C2 , etc. etc., are clear ot il ; for this wouKl lence m unequal §37. Cur. Any (^xpre.ssion of the typo, ^"i + Aj ^1 -f A-.; ('\ + etc . 1(fhich is such that all tlie une(jual ]).irticular connate forms of the Sneric expression under which it falls are obtained by snl)stitutinu r /i successively the diflen-nt primitive //i"' roots of unity, while i|l , ^2 , etc., remain unalteied, is a rational function of ^1 . F'or, in ti^e Proposition, Ji or k\ + ^''2 'i + *-'*''• ^^'"^ shown to Ite a rational iftnction of t\ , the conclusion being based on the circumstance that 4I1 satisfies the condition specified. §38. Proposition' IX. If j/ be the sum of the roots of the equation f(x) = , 1 2 3 A2 = -^ ((/ + J,'" + "I -J, " + h -J,'" + III I 1 1 III + .iJ/" +/mJ, ) (29; ^^»JFor, ^ bfiug one of the whole nunjOers, 1,2, .... , in — 1. put W=- ('•! + '1 '■•■i + ''^''^3 + ftc.) (n + ^1 ro + ti /•, + etc.)-'. (.30) JJultiply the first of its factors by C{~ and the second l)y f\ . Tht;n Pt = (r-i + t' n + ti'n + etc.) (/•, + /i Vi + ti (-4 + etc.)-'. (31) jiflence /i, does not alter its value when we change r\ into rj , r-i into f^ , and so on In like manner it does not alter its value wIkmi we rh;iii^'o j'l iiitn /•„ , rj into ?•„ + , , and so on. Tlierefoiv, V)y §33, /j, is n(jl cli!Ui!.'(il liy i«ny altciatious that may hv. niaiio in )'i , ro , etc., wliild /i icinains nnaltcroil. Consoqnontly, if pz lit? a iiarticular C()<,Miatt' f'diin of /^ all the nnrcjnal jiartionlar oof^nato forms of /* niv ohtiiiiH'fl l)y substituting for ti sncccssivoly in p^ the different prinii tiv(? »/t*^ roots of unity, wliile ri , 7'2 , etc., remain unaltered. There fore, l)y Cor., Prop. VIII., p^ is a rational function of ^i . Wlien c = 2, let }>! = (ii ; when s = 3, let p^ = bi , and so on. Then, from 1 in _2^ I III in (20) and (30), J '" = (^i J /" , J.'" = bi J,'" and so on. But, from (27), since g is the sum of the roots of the equation F (x) = , m . =v to + J, + j; +••■■ + ->,:- . )■ By putting wi J '" for J '" , bi J '" for Jg'" and so on, this T>ecomes (29). Because r/j , bi , etc., are rational functions of ^j , while Ji , tli> root of a i-ational irreducible equation of the (m — !)»•> degree, is als. a rational function of (\ , the coefficients rti , bx . etc., involve no suri that is not subordinate to J §39. Pkoposition X. If the prime number m be odd, tlit expressions III \ m .X^\ 1 III o » , J 1 m 1 1/1 -.1 )(i + 2 » (3-J (ffl jvth — ;5 — I degree. By ^32. when I'l , is charged into r, , j'2 becomes r, + 1 , r^ beconii^ I'z+o , and so on. Hence the terms ri r-i , ro r^ , .... r„,rt , form cycUs the sum of the terms in which may be denoted by the synil'' Si . In like manner the sum of the terms in the cycle r^ r^ , ro rj . . . . , r,„ r-i , may be written 2^. And so on. In harmony wi': 2 2 • '■ this notation, the sum of the m terms r\ , r-i , etc., may be written Si Now Tx can only be changed into one of the terms r^ , r-i , etc. ; ai. we have seen that, when it becomes r, , ra becomes r^ + i, and so o: Sucl: changes leave the cycle r\ r^ , ri r^ , etc., as a whole unalterr ierefo Itiiiiial IS a ra $Ht 2.] tli'^i-efoT 1 . "> Hfnce, s <82) are flviOy in foots of J i540. number obtain t one of if n* roots ndt'i'itif/ ■docinii a^ 1 Let H (8), v.l, sjmibo ainoe s i| QB r(>ns()| pairs of <>ycle coj 25 u >-i , r2 , etc.. ii |);irticuliir ornis of /* are ilVcront |)rinii- ilttred. There- of ^1 . Wlien )ii. Then, from oil. But, from W = . m, this >ieconief <, , while Ji , til- )ih degree, is ills. . involve no siir! m be odd, tin derofovf, \>y Proji. III., S? i« the mot of a simple equation, or has a tioMfil value. In likt- nianimr oach of the expressions (33) «k.s a rational value. From (20), by actual multiplioation, lut "S.-, , ^3 , etc., are n'spectively identical with il,,,, S,„_i , etc. ri»;refore 1 .0^" j;'_^ = si + (S^) (/i + fV'} + (S')('i + /r')+etc.(34) ^ence, siuc(^ tlin terms in (3.3) are all rational, and since the terms in 1 III 4B2) are respectively what J ' J "' becomes by chanduK 'i aucces- ' ' * \ III — 1 J ^^ ■ m — 1 ■lively into the terms ^i . t\ , etc., the terms in (32) are the lecree. 1 m + o » (3-2 yth I degree. r, + 1 , ra becomes . . . . r„,ri , form •ted by the syinl>i cycle ri r^ , r^ ^j In harmony wi*: may be written Si 5 rx , r-i , etc. ; hi, s r, + 1 , and so o; a whole unalten -^ I deg I ^40. For the solution of the equation x^ — 1 = 0, « being a prime liumVier such tliat m is a prime mensuro of n — 1 , it is necessary to ■f tlu^ 7/;''' degree to the equation I . ;r'' — 1 = . ■••■Ji- §41. PtioPOSiTiox XI. When the equation F {x) =0 is the re- icing Gaussian (see S^iO) of the ?m"> degree to the equation aC* — 1 = 0, rach of the — -. expres.sions in (32) is equal to n. Let the sum ot ilic^ ]>riinitive 7i**^ roots of unity forming the cycle ■JP), which sum has in jprocoding .sections l)een indicnted by the #inbol (\ , bo tlie root ri of the equatior! /'' (,»•) = 0. This implies, l^ce s is the numbor of the teiTiis in (S), that //?.res.sion for n . The cycle (9) being that whioL contains all the pi-imitive n"» roots of unity, let u.s, adhering to the notation of previous sections, suppose that, when on is changed iutc wi, Ci or ri becomes C-z or ;-.j , Co or r-i becomes C3 or 1-3 , and so on On the same grounds on which every term in (8) occurs the saui- number of times in the value of rt , each terrain tlie cycle of terii)> whose sum is C^ occurs the same number of times ; and .so on Therefore ^ be selc Of those o ftimi {'M) rf = « -f hi Ci + ho C2+ .... + h,„C\a r-i C, Tm = s + hi Ci + h.i C-i -\- + hi C,„ . Therefore, keeping in view (11), -1 ^ ins — (hi + ^2 + . . . . + /*/„) But n'^ — s is the number of the terms in the value of rj which ait primitive n^^ roots of unity. A nd this must be equal to s {hi+ .... + h,„). Therefore hi + 7/2 + .... + h,t, = s — ] . • . 2'J = ms -|- 1 — s = n — s Again, because r\ is made up of pairs of reciprocal roots, and becaust therefore unity does not occur among the »'-' terms of which ri Vii- the sum, n 1-2 = h Oi + k-2 C2 + .... + k„t C,n , r-z ra = k,„ Ci + A:i C'2 + + ^«i_ 1 0,,, , Ixk like mfi It^ve the V 'pi. Tv !ceding ; tlie third d w|iich giv( 3JI» next Equations latiou WJlJch giv I'm >'i = ^z Ci + k-i C, ■\- .... + ^1 C„i ; where h\ , k^ , etc., are whole numbers whose sum is s. Therefor- -2 = — s. In like maimer oneh of the terms in (3.3) except the fiiv is equal to — « . Therefore (34) becomes -^1 ^,n _!=(«- •"') - « {h + tl-{- etc.) 1 ■m m 11. inity occurs • term iu (M). ill the cyel'' in tbo value It is to siiy loccurs exactly lig that whicL Vlhering to tin IS changed iutc j-3 , and so on bcurs the aauie cycle of terii)> au( so oil Gtn > 1 m • ft xiH reaisiin nnvv ou the aKsuiii|»tiou that the cycle (8) is not made of jtairs of recijtrucal roots. It contains in that case no reciprocal its. IW the same reityoning as aliovt; we get - i = — s . As re- rls the tiMiuH in (33) after the tirHt, one of the terms C\ , Tj , etc., C't , mu.st be such that the «"' roots of \inity of which it is the 11 aie reri[)rocal-' of those of which C'l is the Hum. In jMi.ssing f rom :to Cf , we change /'i into r, . In fact, C'l being ri , C, is r, is being Kept in view, we get, by the same reasoning as above = n — 8. But, if any of the expressions C'l , C'-z , etc., except Cg be selected, say C'a , none of the roots in (8) are reciprocals of any ol those of which C'a is the sum. Therefore - o = — s . Therefore. from (34) ■Vv 1 1 . m m , K ' ~^ J, J , = — » -f (?( — s) r 1 1(1 — 1 ^ '1 't2 + • • • • + /*».) of 7'i which are al to 1 — s = n — s, 'oots, and becaust 1 of which I'l v'ii- + 'l ) z— 1 = ?i. -« |(^l + 'i+ 1^-1 like manner every one of the expressions in (34) can be shown to ^e the value n. ^42. Two numerical illustrations of the law established iu the bceding section may be given. The reducing Gaussian equation of tlie third degree to the equation x^^ — 1 = is .t^ — x^ — 6x — 7=0 ; w|lich gives n = i (- 1 4- jj + jj ), 2Ji = 19 (7 -I- 3 ^ 3), ': 2J2 == 19 (7 - 3 v/ 3), J, 4 = 19. next example is taken from Lagrange's Theory of Algebraical |uations, Note XIV., §30. The Gaussian of the tifth degree to the equation x^^ — 1 ^ is jc^ + x* — ix^ — Sx^ -\- 3x -\- 1 = 0; wnich gives ri =^ * ,i ' ' (- 1 + -Ji"+ J2+ Ja^ J4); L is s. Therefore 3) excei)t the fii-st c.) = n. 4 Ji = 11 (— 89 -- 25 ^/ 5 -I- 5p — 45^), 4 J2 = 11 (— 89 + 2.') ^/ 5 — iop — ;V;), 4 J4 = 11 (— 89 - 25 y/ 5 — 5p + 45,?), 4 Ja = 11 (- 89 -f 25 x/ 5 -f- iop + 5q), ;> = v/ (— 5 ~ 2 v/ 5), -7 - v/ (- 5 + 2 v/ 5), pq = — ^ b .-. Ji Ji = 111' . t'y 38 '7 §43. Pkoposition XT I. To solve tho Gaussian. The path wo Iiave heen foHowing loads directly, Rssuming tho )ir. raitive w**' root of unity ^i to he known, to the solution of the redurjn; Gaussian «''|natiori of tho //i'** dogreo to the equation x-" — 1 =( For, as in i^-il, the roots of the Gaussian are C\ , C^ , etc. Tberofon g, the sum of tiie roots, is — 1. Therefore -L _1_ -1- By Prop. VIII., Ji , J2 , etc., are rational functions of t: . Therefp: ^1 If //( — , 2 »n — 1 K\ + k2 h -\- kz (i + + «,„ ), n is known. §44. Propositiox XIII. The law established in Prop. X f-il under the following more general law. The m — 1 expressions i each of the groups Bssisely cklmged sue ($jf) whose )^i|up are tl 0, §45. Cor under a yel e root I iDcmg i\ 1 m - 1 ' 1 m 1 m ./,-3 ' 1 3 ui 1 m 'm-2' 1 j'" 1 nr ■'m-6 ' 1 1" 1 nj — 1 _2 /?i — 1 3 >n 'm—1 1 "' \ 1 m the valuu ,) > (3: il(/'2, i_ This is a I equation J :(1 and so on, are the roots of a rational equation of the (m — 1)*^ defi;r ^^ff^^" 29 In* HI — 1 terniH in the first of tlif y §38, ;i,„ _ , is a rational |»n — 1 2(m — -l) km tj ; flUaction of ti ; and, by Prop. VIII., Jj i.s a rational function of c 111 , 111 , 7(1 J J J 1 2 3 m — 1 (38> ' m — 1 J But, by §13, tl I to one anotlie _ - etc., are knowi if!>l^e root of a rational equation of the {m — 1)"^ degree, if Therefore, fioi a + -'ft + 3c + . . . . + {ni - 1 ) s = Wm , in Prop. X fal ^^^i'lg '^ whole number. For, by (30) in connection with (26), 1 expressions i ..^ 2 i i 4- — /';> J, , J = pz "J. ) '»i'd so on. Therefore (38) ha.-* 'l ') _i_ in tho value a-l- 26-f 3«+ .... +((/. - 1)/ b c {P2pi ) Ji in be H' , ur (/My'y . . . .) Ji 2 '^ This is a rational function (jf ^i , and therefore the root of a rational equation of the (/a — 1)H' degree. (m — l)*** degr 30 u The Equatio.v F{x) = 0? the Second Class. §■16, We now siippoHe tlmt the simplified root rj of tlie rationu irreducil)le equation F{x) = of the »i''' degree, ni prime, involve- •when modified according to §21, a principal surd not a root of u)iitv It must not be forgotten that, when we thus speak of roots of unity we mean, according to §1, roots which have prime numbers for tL nt in m — 2 m—\ , hi J, (3D Prop. 1 UM, by 5$2; iOnne order m is rs. \ •of«l, let and 80 eliminate J from rj , introducing in its room the new surd 1 s with — for the coefficient of its firat power. We may then put 1 / A ^ ^ ' 3 m w — 2 m 1 — 1 m where g , a\ , et<;., are clear of A 1 1 sively into A^ , >;j'*' root of unity. ^rop. \l., tlif tortus in (11) ino tin- mots of flic <>(]iiHtion I of the rationa ^^^B) = "• 'I'ltkiiig r„ , any uiie of tlic pHrticiiliir cognate forniR of prime, involves a root of uiiifv • jgt, j '" an , etc., bo n'Hpectivply what J,"' , oi , etc., become in pass- roots of unity ^ n '1 « lumbers for tL —„ )n8 can be estal U^ {rom ?•] to r„ ; and when J^^ is changed successively into the been consideio.l ^gj^p^nt ;//'»' roots of tlie determinate base J„ , let r^ become I F(x) = hit roots are tlie m m-\ on— I) (42) By Prop. II., the terms in (42) are roots of the equation F (x) = ; ~m And, by ^2'^, they are all unequal. Tlieiefore they are identical, in 1 • (3? 4Qnie order, with the teims in {\\). Also, the .sum of the terms in .ble. Also, sul^ <*^) ^« y- Therefore ,7 is rational. in (37), the lav ||8. Proposition XI V^. In r\ , as expressed in (40), J is the ary in §45. ff -}- J^ -I- «i Jj + etc (43) om the new surd ^6 may then put , _j By Prop. II., ri is equal to a term in (41), say to r„ . And, by §48, -Axj7^);(40>^"^'"~^^"'^"" TO mr„ _2_ changed succes- " ./ r « — 1 j i n- 1. i ^ t Therefore, etc. (44) (41) ^ Jj (1 - ^.-l) 4- J^ («'i - «i «„ _i) + etc. = 0. (45) .^E TliiH (■(fiiation invoIvoH iin surds rxoopt, tliosc found in thn siinitlificii (■x|ireHHinii ri , tojifthcr with tlii' |iiiinitiv«! //t"' root of unity. Then lore the <;x|irt's.sion on tlu* h-ft of (ITt) is in ii sinijilr stiitf. Th«frefi»ri \n- for hor \o 1 m l>y §8, the cootlicients of the dillerent |»o\vei"s of J are separatflv r I wxo. Therefore /n-1 = 1. ^i = «i , f>\ = 'n » ""«' «" o"- I^ut) »• 1 wiiH shown in Prop. V., s lioing a principal surd not a root of unity (see !; till immai (^ in the siniplitied expression a\ , oi ciinnot l)e ecpnil to ^x unltJsa z can be eliminated from a\ witliout the introduction of any new surd In like manner b\ cannot bo equal to 6i unless z can be elimiuatiMi from b\ . And so on. Therefore, becau.so «! = ai , and hi = /*i . • coeffio At the coe and so on, z. admits of being eliminated from ?•_ without the intro duction of any new surd, whicli, by §21, is impossible. Next, Iti tion ti »! be a root (see §1) of unity, which may be otherwise written Oi Let the different jtrimitive c*^'' roots of unity be Oi , O2 , etc. ; and, when Oi is changed 8ucces8iv*>ly into ^i , O2 , etc., let vi become sue ffbing on) mv., J m cessively ri , ri , etc. Sujipose it possible that the c — 1 teruh *jl«e|>t J c Ti , ri , etc., are all equal. Since z is a principal surd in ri , we c- 1 .fi-2 may put ri = h(\ + /c^'j +....+/; where h, k, etc., art clear of Oi . Therefore (c — ]) n = d ~ {h + k + etc.) Thus /i wlMre (/, d z. may be eliminated from ri without the introduction of any new surd ; which by §21 is impossible. Since then the terns ri , ?'i , etc., are not all equal, let ri and ri be unequal. Then I'l is equal to a term in (41) distinct from I'l , say to /•„, Expressing ini'i and 7nr„ as ii. (43) and (44), we deduce (45) ; which, as above, is impossible. §49. Proposition XV. Taking ri,r„, J , etc, as in §47, an equation m n (4G) where d , {8f since J^ n:t II tho Hiinjilifid f unity. Tlicn ate. Tlujiefiin I are Heparatil so on. But, II Dt a root of unit to ax uule8S z 1' any new Huitl in be uliniiiuit( ; , and bi = h\ ithout the intro jible. Next, It! formed , wliero t is an wi**» root »)f unity, and c m u whole kber loHH than m but not zero, and p involves only MurdH HulKirdi- (Hce §3) to J or J I m By §-17, one of the ti'iniH in (12) i.s cfjual to ri . P'or our argument ilil immaterial which be selected. I^et r, = ri . Therefore » — 1 1 m (A„ J,^ -f. .„ J„ + . . . . 4. j^ ) - (A, J, 4- «i J, +....+ J, ) = 0. (47) m Tib coefficients of the different powers of J here are not all zero, tli the coefficient of the first power is unity. Therefore by §5, an j tion < J == li Hiibsista, t being an tn^^ root of unity, and ^i in- 1 m ing only surds exclusive of J that occur in (47). By Prop. XfV., J is a surd of a higher rank (see §3) than any surd in (47) 1 e c — 1 teriib «jM»pt J Therefore we may p\it vise written Oi , O2 , etc. ; ami t r\ become sue surd in ri , \vf e h, k, etc., an t + etc.) Thu. tion of any new ni s 7'i , ri , etc., s equal to a term and mr„ as ii^ npossible. , as in §47, w (46 1 J m 2 TO in — I m 1 m h = d-{- di Jj -I- , < iliil tiiuiili wfth mill Mries 1, 2 m Thcr in a curtiii |M. I'F foycts uf J 1 w different powers of J can be expressed in terms of the sui Is i; 1 m n ' n volved in d and of the m'** root of unity. Substitute for J , J^ etc., in (47), their values thus obtained. Then (47) becomes *n seven nd^ roots . By §47 it is inn XT., the tn- l m differ eiit e-(Ai^i +....+ V) = 0; (4 where Q involves no surds, distinct from the primitive m*** lo 1 of unity, that are not lower in rank than J \ which, becaii m the coeflScient of the first power of J in (49) is not zero, is, by ; impcBsible. Hence there must be one, while at the same there can only on© of the ♦n — 1 terms, tfi , d%^ etc., distinct from zero. L By Cor. order, wi .1ft tlic •♦•nil that is not irrn. Tlion t* — i^ a 0. TfnTi-for*' /^ is not zero. Therefore (/ =i 0. Tlior-'fi>ro, putting p for dc , J . An: = /» J, . JO. Cor. J5y thc) proiMjsition, values of tho diflferent jwwenj of etc. =ai d , diJ^ -f etc.), me way in wliii 48) are separate! ti ), etc., must ' factors, ro(biefs, multiples of the prime number m left out, of the terms in tlie B8 1, 2 , in — 1, by the whole number c which is le.ss than Hl"^ Therefore the series c, «, s, etc., is tho serieH 1,2, . . . . , 7u — 1, i||A certain order. **& >1. Proposition XVI. If r,, be one of the particular cognate vs of li, tho expressionB ^■'- In t^s , f" «" ^ ' .••••' <"'-' «« ^„ . <"'-' ''« ^. m n m— 1 m (51) a*| severally equal, in some order, to those in (39), t being one of the l|||§ roots of iniity. %y §47, one of the terms in (42) is equal to >•] . For our argument itiJB immaterial which be chosen. Let rn = ri . By Cor. Prop. 3Bf ., the equations (50) subsist. Substitute in (47) the values of the i_ m = 0; (4 primitive m^^ ro ; which, becai; not zero, is, by ; e same there can ict from zero. L Breiit jiowers of J so obtained. Then e I {l-^pd^ + r-a^«„Jj"' + etc.) - (j/" +«!<" +etc) = 0. (52) r. m I By Cor. Prop. XV., the series Jj" , Jj , etc., is ivlentiral, in sumo dbr, with the series J^ , Jj , etc. Also, by §8, since J^ is a i H' 36 surd occurring in the himpntied expression ri , and since besides J there are in (52) no surds, distinct from the primitive rw*'» root m unity, that are not lower in rank than J^ , if the equation (5: 1 were arranged according to the powers of J. lower than the »n" 1 the coefficients of the different powers of J would be separattl 1 zero. Hence J is equal to that one of the expressions, m 14 O A t- 1 ;> Jj , <- 2 qan Jj , etc. (5? 1 m . 2^ m in which J is a factor. In like manner ai d is equal to that ob 2 of the expressions (53) in which J^ is a factor. And so on. There fore the terms J , (i\ ^. , etc., forming the series (39), are sevei ally equal, in some order, to the terms in (53), which are thost forming the series (51.) §52. Proposition XVII. The equation F (x) = has a rationa nnxUinri/ (Compare Prop. VII.) equation f (x) = 0, whose roots ai' the j/i''* powers of the terms in (39). Let tl>e unequal par+icular cognate forms of the generic expressioi J under which the simplified expression Ji falls be ce ma i, becai todifftiem rotits of a ftttin the s( tiM remain infpduciblc iMi^ihe con terms in (l |53. Th^ of the* exp to the eq •Hiiliaries H. Pr limes J surds .4i |here ci fotttis of J Otmoi bi .for, jus re are | . T J «# equal nin^ *#■ 1 to Jl , Ji , , Jc {b\ By Prop. XVI., there is a value t of the r/i*** root of unity fo' which the expressioi'S ose m m— 1 m — 2 (55 are severally equal, in some order, to those in (39). Therefore J2i 0(jual to one of the terms -J| 1 "1 Jl ,••■., ei Ji , //I Ji (51' loiial ■)v ■^^^ff* lice besides J " 1 tive 771*'' root ^ le equation (5: ^er than the »«•' uld be separattl isiODS, (5? equal to that oii Lnd so on. There 38 (39), are sever , which are thosi = has a rationii 0, whose roots an generic expressio! (5i » root of unity fo' Jl^'$ike manner each of the terms in (54) is equal to a term in (56). MiA, becaust> tlie terms in (54) are uiio<|ual, they are spv»>rally equal toiifftreiit terms in (50). Hy Piop. 111., the tcriii.s in (54 » arc the PD!^ of a rational irreducible (-(luation, .say <,'■{ (.r) = 0. Rejecting i|l|| the series (56) the roots of the equation Vi (x) = 0, certain of i|Ktemaining terms must in the same way be the roots of a ratiotial IWllducible equation V'2 (^') = ^^- And so on. Ultimately, if y (x) huf^'^e continued jiroduct of the expressions ^''i (r), (,''2 (x), etc., the tMnas in (56) are the roots of the rational equation - i^iliaries are all irreducible. 4. Proposition XVIII. In passing from ri to r„ , while J, imes Jn , the expressions a\ , hi , which, by Prop. XIV., involve surds occurring in Ji , must severally receive determinate values, J"^ 6,1, etc. In other words, aj being a i)articular cognate form of ^ there cannot, for the same value of J„ , be two particular cognate fotios of A, as rf„ and «.v , unequal to one another. And so in the of hi , ei , etc. or, ju.st as each of the terms in (42) is equal to a term iu (41), eax'e priuntive m^^ roots of unity r and T'such that the expressions m .j, tJ^ +^^«n'J„ +etc., T J^. +r^ay J^ + etc., an equal to one another. Therefore, if ^JiV = J,. , in which case, by ning suitable values to r and T, J may be taken to be m U to J 1 2 ^,r (^ - 7") + -^r («" ^' - «^v T' ) + et<;. = 0. (57) m — 1 -1 A2^, (55 . Tlierefore J^t 086 if possible that the coefficients of the different powers X 1 cl 4^ in (57) ai-e not all zero. Then, by §5, t J^ = /j ; < being t^ m*^' root of unity ; and h involving otdy surds of lower ranks ; 1 j_ 1 Hence, by Prop. XV. and Cor. Prop. XV , J is a 1 jional function of surds of lower ranks than J and of the .18 primitive »«*•' root of unity ; wliich, Ity the ihfinitioii in ^6, iinpossihle. Since then the coetlicients of the tlitVereiit puw- of J in (57) are separately zero, r = 1\ a,^r^ =«>• T^ , therof. Jj^o. l^HOPOSiTiox XIX. Let the terms in (39) be writt. respectively 111 _i_ in Ills. I • tfl'lll dMUr 1 ' 2 ' 3 m — 1 (5- The symbols J, , 5^ , b^ , etc., are employed instead of J, , J, , J, , etc becanse thi.s latter notation might suggest, what is not necessaii true, that the terms in (5(1) are all of them particular cognate foir: of the generic expression under which Ji falls. Then (compare Pn : XIII.) the m — 1 expressions in each of the groujts 1 JL _L _L J ' (J I .-I'^^a ^n-1^h K m — 1 ' 2 m — 'i' 3 »« — 3 ' 1 2 1 2 1 -,''■ •) (J," <'" _2_ _1_ m . m HI — 2 ' 2 Til (-^1 _L _L 3 1 ' m— 1 2 " I (.);■ iruiinul Hire is t «qtn»l to a nNii of uni oruei il^^nal b( «i- in Pi ■'i ' i/- ' ^n-X ^ .) !,nd so on, are the roots of a rational equation of the (tn — 1)^'» degiv ■»u — 1 Also (compare Prop. X.) the first ~ — terms in the first of tl groups (59) are the roots of a rational equation of the I r — j degree. In the enunciation of the proposition the remark is made that tl series (54) is not necessarily identical with the series Jl I '^2 > 'li . • • • • > '^»i — 1 • The former consists of the unequal particular cognate forms of J ; tl latter consi.sts of the roots of the auxiliary equati.m f (x) = ' These two series are identical only when the auxiliary is irreducili!^ To prove the first part of the prc})osition, take the terms forming tl ;;i — 2 second of the groups (69). Because i^ _„ represents ^] J r«%ecti iuci dlilinct biriiowi ntliional IWitely, roots of iqpplios ■Qoond ]'. iproups ( Kfitb th iiitioii in ^6, (liUl'i-ciit puwe> ly r^ , Uicref(, -^, , -J, . -J, . f ' IS not necessni! lar cognate fuit in (compare Vv s ^. ') _1_ 12" _L I (»i — 1)'1» degiv I the first of t! 39 «i -Ji J, = J. t m fji l»p the generic syniV)ol under \vliicli the sinijdified expression «l flUls. By I'roj). XV II J., when Ji is changed successively into tha « terms in (54), Pi receives successively the deterniinate values :39) be writi. i&f2 Cc \ and therefore e\ Ji receives successively the irininate values of the (^ c is made that il ss ite forms of J ; tl atl-Hi Jj is equal to J , which is the product of two of the terms in (39) occuring 1 Bctivelv at equal distances from opposite extremities of the series. J _l_ mm . d „ in the m in — 2n ^jOthe^ words, ^2 ^2 is equal to an expression d id of the groups (59). In like manner every term in (60) is 0IJI^\ to an expression in the second of the groups (59). Let the lU||qual terms in (GO) be til ^1 , etc. (61) in, by Pi'op. III., the terras in (61) are the roots of a rational lucible equati»^n, say fi (.r) := 0. Rejecting these, which are di||inct lonns in the second of the groups (59), it can in like manner IjHililhown that ceitain other terms in that g?*oup are the roots of a ig||onal irreducilile equation^ say /z (x) = 0. And so on. Ulti- HMI^lyi if y (a;) be the continued product of the expressions yl (a;), fii\«), etc., the teinis forming the second of the groups (59) are the roots of a rational ccpiiition of the (in — 1)"' degree. The ju'oof Wpplies substantially to each of the other groups. To prove the ipiDnd part, it is only necessary to observe that, in the first of the UfOups (59), the hist term is identi^'al with the first, the l.ist but one iHitb the second, and so on, m»i- 40 §56. Cor. 1 . The leasuuiiig iu the prupuyition proceeds ou t: assumption that the prime number m is odd. Should m be even, t: series J| , tt^l**':^, J2 = hi Ji, and Ji = /t2 J2J and hi Ji is rational. , y. '" As in §52, t being a certain fifth root of unity, each term in (55 fQ,.^, ,,(•// equal to a term in (39). The first term in {55) cannot be equal Ugug^, the tirst in (39), for this would make J2 = J; . Suppose if possi jjg^j,^,|"| that the first in (55) is equal to the second in (39). Then, latioimi equations (50), applied as in Prop. XVI.. \vl SI t i 41 I proceeds on t lid m be even, t: The law may trm. The piodi: (mi — 1)*** degrt .y Prop. XVII, er proof, that ti ion F{x) = is cewise. Fifth Dbgree. e ni.**» degree, wh . Then, by Pr^ the first or to t: ished, assuming t 4 if), t Ji = (u Jt , ''- ((■: Jj = III J,', ((•.3) '3 r, •.' n 1 theniforc Jj '^ti Ji , ci> J> = hi Ji , fi 3 '. 4 f, :i «^ J-i = J] , /(o J._, = gj J, . •I 2 fti J] , liciiig p(jual to Jj , is a rout of the eriuation <,'i (x) =^ 0. ai J] , involving only surds that occur in n , is in a siinide Thcrt'torc, Ity rroj). III., a^ J- is a root of the oquation ^^t fit) -- ('. 'riicid'i.rf' //I Ji , and tlici-ft'on; also /ij J^ or ci J| , are o:' tlia*' e.juution. Hence all the terms . the root, as tk Ji , «. Ji , 'ii Ji , hi Ji , (64) proots of the equation <,'•[ (x) = 0. I'ut r J^' , h>, J-> — hi Ji . But, just as it vvas proved in §56 are rational. Quadratic has a q^that, the roots of the sub-auxiliary ^'i {x) - being the c terms b^ J and ^i ^'^^ » '^'- ' ^'^'^"' ^^^^-^'^' '^^ "^^ l)articular cognate form of EJ that is not a teri|l in the series t\ A\ , f> .lo , . . . . , e^ Jc > it follows that, if . hi in a particular cognate form of //, there is no particular cognate , each term in (j5 fo,^ ^^C //j ti,,^^ j^ ,^„^ g,j„.j| ^^-^ „„,. ,>f ^\^^, umwv, hi Ji and h^ J^ . m ) cannot be equal Uauj,,, ^j,j^^. }^^ j^ ^ j^^ j^ ^ ^j j^.^^ ^^^ particulai- cognate for Suppose ifpossi'diffepent in value from hi Ji . Therefore, bv Proi). III., Ai Ji ia in (39). Then, ,.atloi,al. 6 \2 i561. Pkoi'osition XXII. '('lie ii\ixiliitrv liiijiiiidintio r Jj nor J;i is c(|ual to Jj . They cannot both be e(]ual to !i'\ Jj . Therefore one of them is equal to one of the terms be equal eitlier to ^i Jj or to r\ J\ , all the terms in (G4) are roots of the irreducible equation of which Jj is a root. The same thing holds regisrding J;! . Therefore, when the series (54) contains more tlian two t(u-ms, the irreducil)le e(iuation which has J] for one of its roots has the four unequal terms in (tI4) tor roots ; that is to say, tw.' auxiliary biquadratic is irreducible. dei,'ree, §02. Let 5?M -■ Jj' , 5?/,,, = r/, J,* , 5»3 --. ci J[ , 5in = hx J? ; and, •)! being any whole number, let .S'„ denote tiie sum of the ?<"' jtowers of the roots of the equation /'' (.f) r= 0. Then .S'l = ; S> = 10 (/«, m + V, y/;. ); ,S,, ^. 15 | :' [m ?0 }; .S'4 = 20 ] 2' {n{ V, ) \ + .SO {u\ n\ + iCi i^) 120 m v. m ?«, ; Sr, = 5 I :• (ul) \ + 100 -; -{»;l u, uO i + 1 ")0 I 2- [a, w^ ul) ; where such an expression as 1' {n^ nr,) means the sum of all such terms as v^ >'■> ; it being undiirtood that, as any one term in the circle m , ?^2 , U4 , ii.-, , passes into the next, that next passes into its next, its passing into ui . The Roots of tick Auxim.\i{v Hiqit.\dratk; all Rational. §63. Any rational values that may be assigned to J| , «, , e, , aii-l hi in n , taken as in (G2), make r^ the root of a rational equation of the fifth degree, for M ;-y render the values of ,S', , A", , etc., in §G-' rational. In fact. ,S'i 0, 25 A'., .. 10 J, {/>, -f ay .'1 ), and so on.' of Ci J the san be inor ])rincij rti inv x/(/>- Cll = / fti bec( in the in n ,[ TlIE AlXILIAKY Hk^UADRATIC Willi A l{\ilC Si i.-Al\iuai V. §tU. Pkoi-osition XXII i. In oidci- tlmt /i , bikm as in (Ol'), may be the root of an in-niliicible tiijuation /' (./•) = ot" tlie tiftli (lt'!,'fee, who^e auxiliary bi■') where Ji and J> are the roots of thti int'ihii-iVile t'ljuatioii ^'■1 (x) = *« - 2 px -\- f/' = 0; and ai ^ b -^ d ^ { p^ - q^ ), a-i^ h — d s/ ( V^ ~ 1^ ) ') J'f ^ '"iJ d lieiiig rational ; and the I ,'S roots Ji" and J2 being so related that Ji Jj = q. By Proj). VII., when a quintic (equation is of the first (see .^JJO) class, the auxiliary biquadratic is irreducible. Hence, in the case we are considering, the ([uintic is of the stjcond class. The (juadratic sub-auxiliary may be assumed to be <,''| (x) - it- - 2 px -\- k = 0, p and k being rational. By Piop. XXI., the roots of the equation ^''l (x-) = are Ji and Ii'l Ji . Therefore k = (hi J\ )^ ; or, -Jutting 7 for hi Ji , ^ = q^ . By the same proposition, hi J\ is rational. Therefore .y is rational. Hence V'l i-^) ''«« the form specified in the enunciation of the proposition. Next, by Proposition XVI., there Ls 1 4 "iff" "^ a fifth root of unity I such tliat t J'z - lii Ji' . If we take t to be unity, which wc may do by a suitable interpretation of the symbol J2 , J2 = /'I -Ji ■ This iin[)lies that e\ J\ ^ o-i J-2 , O) being what «! becomes in passing from Ji to J2 . Substituting these values 3 i of (?i Ji* and hi Ji in (62), we obtain ;iie form of ri in (05), while at I }_ the same time Ji' Jo' = hi J] = q. The forms of oi and a-y I'avo to 1. be more accurately determined. By Pro|). XIV., Jj* is the only ])rincipal surd that ri , as presented in (G2), contains. Therefore rt] involves no surd that does not occur in Ji ; that is to say, \/ ( p'^ — q'' ) is the only surd in (i\ . Hence we may put «i = f> -\- d.^ (p' — q^ ); 6 and d being ration-il. But n> is what rti becomes in passing from Ji to J^ . And Jt ditibrs from Ji only in the sign of the root y/ ( p' — q'^' ). Therefore «2 = h — d y/ { p' — q^ ). t^G5. Any rational values that may be assigned to }>, d, p and q in I'l , taken as in (05), make )•[ the root of a r.itional equation of the 41 fifth iU"^vw. ; to;- tln'\ rciuler tin- valiifs of -Sj , .V, , etc., in gO'J, mtional. In ff.ct, X, - (», 2") .S'j • 10{y + 7- /A' - 7- ./■^ ( ;/-J — q^ )} , 1111(1 so on, Tin: ArxiMAUv I'.ivadkatic Ikukducihle. §bf). Wlit'ii tlio aiixiliiiiy lii(|iia(lrati(! is iiiodiiciMe, tlie unequal ])articulMr oo<,'nato forms ot" J aie, hy V\o\). Ill,, four in number, Jj , Jj , J:( , J4 . As explained in §5.'), because the equation

h ■ Kence, putting hi = "», the first two terms in the ttrst of the groups (oO) may be written in the notation of (37), Jx J4 , J;j J '3 ; (6G) and the second and third groups may be written (jf J;; J^ J J:^ J 4 J2 ) I 3 1 3 .1 ,31 31 ( (Ji* J2 , Ja 'Ji > Js 'Ji" , Jt J3 )■ / (67) i^67. Proposition XXIV. Tlio roots of the auxiliary biquadratic e(|uatiou (f (.t) = are of the forms Ji = ju + n ^ z -j- ./ s, J2 = m — n ^ z -\- ^ »x , "J Ji = m -\- n x/ ^ — \/ '•'■, ^3 = »t — n ^ z — y/ si ; i where « = /> + 7 ^Z z, and si = p — q ^ z\ jh, n, z, p and q being rational ; and the surd .y s Ijeing irreducible. By Propositicms XTIF. and XIX., the terms in (06) are the roots of a quadratic. Therefore Ji J4 and Jg J3 are the roots of a quad- ratic. Sui)pos(! if possible that Jj J3 is the root of a quadratic. By Propositions IX. and XIX., J;} = ei Jf . Therefore ei J* is the root of a quadratic. From this it follows (Prop. III.) that there are not more than two unequal terms in the series, ei Ji , «2 J2 , «3 J3 , 64 J4 . (69) But suppose if possible that ej J* =62^2- Then, t being one of the fifth roots of unit), ^''i J I = r-.i Jo But, by Propositions IX. and XIX., Jo' = //, jf . Therefore, fc, jf = e, /fj 4 ^i' • There- ';$ 'J4 45 , ill !5;?<, ('1 = 0. 'I'lien-fun; onu nt llu- i««tts ut (he mixiiiiirv liiquii(h'atic is zero ; wliich Ix'cau.si* tin; :iii\iliary hiijuailrutio js MssmiK'd to ho irre(hicil)lo, is iin|iossil(l('. 'I'licrcfuni p\ j| luul co jl are uiu-qnal. In the siiiiio way all thc! terms in ((19) ran hf Mhuwn to hn unequal ; which, because it lias been proved that rlieri are not nioie than two unequal terms in (69), is inipossiljli!. 'riicrcforo Jj J;, is not the root of a qtnidratic equation. Therefore the product of two of the roots, Jl and J4 , of the auxiliary bi(|uadratic is the root of a quad- ratic equation, while the ])rodMCt of a ditlerent paii-, Ji ami J3 , is not the root of a quadratic. Hut the only forms whioli the roots of an irreducible biquadratic can assume consistently with the.se conditions are those given in (68). §68. Proposition XXV. The surd ^/ sy can have its value ex- pressed in terms of \/s and \/v. By Propositions XIII. and XIX, the terms of the first of the groups (67) are the roots of a biquadratic equation. Therefore their fifth powers Jl J;j , '^'i "Jl J3 J4 j: J, , (70) are the roots of a biquadratic. From the values of Jj , J> , J-, and J4 in (68), the values of the terms in (70) may be expressed as follows : J? J3 = iP + Fi ^/ ^ 4- {F-i + t\ n/ =) n/ « + {l'\ + Fs^ z) ^ .S-, + (/-o + I'l s/ :) n/ .s v/ .-1 , J^ Jl = /' - F, y/ z + (/'% - F:, s/ ~) x/ .^'i - (Fi - F,^ z) ^ s - {F, - F, ^z)^s v/ «i , J^ J., =. F- Fi ^ z - {F., - /'V v/ c) v/ •'•■1 + (^4 - Ft^z) s/ s ~ (/'o - .'■'7 n/ ^) n/ *• n/ n , jlji=F+ Fy s/ Z - (/'2 + ^3 n/ ^) n/ « -{Fi + F,^ z) s/ sy + (/''o + F, v/ ^) v/ s v/ .^i , (H) where F, F\ , etc., are rational. Let -(Jl J3 ) be the sum of the four expressions in (70). Then, because tliese expressions are the roots of a biquadratic, - (J? J3 ) or \F -f 4/^7 >/ s s/ Sy , must be rational. Suppose if possible that y/sy cannot have its value expressed in terms of ^/^fn^d x/s. Then, because >/s ^Z si is not rational, = 0. By {(S9>), this implies that n = 0. Let {A\ J3 f= L + Ly^ z+ {L; 4- />3 x/ .-) s/ s -f (A, -f A, v/ ■•-) V/ Sy + m + L; y ^) v/ .S' v/ '^'l , 1*1 46 wlicru A, Ai , elf., me iiitioii.il. 'I'lion. hh ul»oV(', A; = 0. Koopinj^ ill view that /* = <•, tliis iiiciiis tliit hi' 7 = U. Miit 7 is nut zero, for tliis woiilil iniikt! y/ « = ^Z "1 ^ wliicli, t»cciius« wo lire rt'iisonini^r oil tlie hypotlu'His tliiit ^Z ■'* zero. And it wn.s shown thill n i,« zno, 'llicrctoic Ji = \/ "» '^"'' -J3 = — \/ /<. Therefore Ji J:j = — y/ ( /'" — 7'-)'. wliich, because it has been pKivcil that J| J;; is not th(! root of a (|iiailratic ('f|iiatioi>, is impossible. Hence v/ ''i <'i lieing rational. Therefore >/ ( «»i ) = \/ ( y- — '/■■' •) = (''I + '"^ \/ •') (y" + '/ \/ -) = (''1 />> 4- «J 7 -) + \/ = ('"i 7 + '•-' /') = P + Q y/ ^' Here, since /j^ — 7- :; is rational, either /' = or Q =: 0. As the latter of these alternatives would make y/ (/-''" — 7' ^) rational, and therefore would make >/ ( /> + 7 \/ -) ^»' >/ •* reducible, it is inad- missible. Therefore cj ;) -|- f.) 7::; = 0, and >/ il'' - 7'^,^ = ('•17 + CiP) >/ ^ Now q-. is not not zero, for this would make y/ («.v] ) = ^ [) ; which, becaust; y'' s is irreducible, is impossible. Therefore c^ = 0. But, by hypotlu'sis, c'l = ; theiefore y/ s\ , which is equal to (<*! + c-i y/ z) ^ K, is zero; which is impossible. Ifenco ci cannot be zero. We may therefore put re = I , and h (I -{- e^ ) = p. Then s = p -f 7 y/ ^ = A (1 -f «2 ) + A y/ (1 -f e2 ). Having obtained this form, we may consider z to be identical with 1 + e^ 7 with h, and p with k (\ -\- e- ). §70. The i-easoning in the preceding section holds good whether the ecpiation /'' (,*;) = be of the first (see §30) or of the second class. If we had had to deal :;iin[)ly with ecpiations of the tirst class, the proof given would have been unnecessary, so far as the form of z is concerned ; because, in that case, by Prop. Vlll., Jj is a rational functio'i of the primitive tifth I'oot of unity. \'i §71. PiiopnsiTioN XXVII. lln fuini ^ivA: •■■"i.- 5J- 1. 1 3 ;' J J„ J? Then J;' J J Jf J.J = Jj (J; Jj Jj K ) is an identity. Therefor.^ ^ Jj = ^1 (Jf J* Jj j;^ ). Similarly, J? 1 J? = .13 (jj A J? 3 ) 4 = A, {j\ j\ 4 A )' '"'^1 J' = ^1 (J' J' j^ J' ). 1 •"' \''4 2 1 3 ' Substituting these values in (G2), we get 3 4 n = Ai {Jx -I3 Ai J.2 ) -f A2 {A'i Ji J3 Ji ) + A^ {aI a1 aI j?) + Ai (J4 j| Ji"' JJ). (74) This, with immaterial differences in the subscripts, is Abel's expression; only we need to determine A\ , A-i , J 3 and J4 more exactly. These terms are the reciprocals of the terms in (73) severally divided by 5. Therefore they are the ronts of a biquadratic. Alsc, no surds can appear in A\ except those that are present in Ji , J2 , J3 and J( . That i:3 to say, Ai is a rational function of y/ s. ^/ s\ and v/ «• But it was shown that ^Z ^i \/ s '= he v/ ~. Therefore Ji is a rational function of v/ s and y/ z. We may therefore put Ax = K + A" J, -f- K" Ax + K'" J, J 4 , IH A', A". A"'himIA"" iMiiiir iMfinM.I. I'"it til.' Ir.inis .(,, .1., .1 ,l„ cm ul.itf \\ itii J. . J . , J. , J I rii 'i".^' I'litioiml values that may lie iissij^'iicd to III, ti, *', //, A. A". A' aii'l A" make >'[ , as picscntid ill (71), tlic I'dot (i!'aii niiiatiiiii nt' the lirih (le;,'ree. For, any ratiitiial values (if///, //, etc., iiiake llie values of >'| . Xj . etc., in §'Il,', rational. ^'•i. It may he noteil that, not only in the cxpres.sioii for I'l in (71) the root of a fniintic ((|iiatioii whose anxiliaiT l)i(|iia(lrati(; i.s iire- ihieililo, l)nt (in the uiKleistaiiiliiiir that the snrils ^ h and ^Z - in Ji may lie rediicitile, the e^•|avs^illll lor r\ in (71) contains the looi.s ei|uations of the liflh deL,'iee whose auxiliary lii(|iiadratics ith of lia\o their roots ralional, and of ail that liavo (jnadratic siili- auxiliaries. It is iniecewsaiy to oiler proof of this. ^74. Tlic ei;nation ,/•' 1 (),-•■■ 4- n./;- 4- lO.i' 4- 1 = is an (•xaiii|pli' of a solv.ilile (jnintic vvith its auxiliary liiipiadratio ii'K (lueii)le < Mie of its roots is 4- wor' 4- 1 ..5 + w hciii},' a )irimitive tit'th root of unity. It is ol)vions that this root satisfies all the comlilions that have heen |Miii,ted out in the |irecedinj,' analysis as ncces.sary A rout of an c(juation of the .seventh dci^rec of the same charactei' i-! + + 4- ..T...^ -f + ..6...7 o bcini; a juiiiiitive seventh root of unity. 1'lie ,L; |ir(HM'(Uiig Mith (le-reo form undor e cycle that {■2m + ir ''/-a)-