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Les diagrammes suivants illustrent la mdthode. n 32 X 1 2 3 4 5 6 r f' 51 Zi Y ■!l \i ( SOL\ABLK ()( INTIC !X)IATI(»\',S WITH (0.\I.\IL\- SLKABLK ('0KFF|(1I-;XTS. \\' (rEOIlGE PAXTON YOUNG, 1Tnivkk.sitv ('<.i,i,i.:.,k, Toko.ntu, Cawala. lifijH'inhd from Amirim» Jovrnnl »f .]f)ithriinit!rs, V„f. .V, X<i. ± 6-I-. ilM64 mi 5184^ Solvable Qulntic Equations with Commensurable Coefficients, By Georgk Paxton Young, Univemtij Collrijr. Toronto, Cnmuhi. OlUKCT 01' THE PArEU. §1. Some time ago, in the American Journal of Mat/tcmattcs (Vol. VI, page 103), the present writer sketched a genenil method for finding the roots of solvable irreducible equations of the fifth degree. The method was partially developed, and its application to certain forms of quintic equations was shown. It is now proposed to give the method the farther development necessary to make it applicable, by a' definite and certain process, and without any difficulty beyond the labor of operation, to all solvable irreducible quintics having commensurable coefficients. The following equations will be solved as examples of the application of the theory : 1. 2. 5. 6. 7. 8. 9. 10. a^+ 3x»+ 2a; — 1 = 0. x^— lO.*;"— 20x'— 1505a- x^ + -^ x+ 3750 = 0. 7412= 0. ^-''-^ n 2. y_x''2 . 11X89 _ 25^ "^ 125 ^ "^ "3125" ~ " " x^ + 20.r"' -f 20ar + 30x + 10 = 0. x^ + 320a;2 — lOOOx + 4238 - . a;^ — 20x' + 250a; X ,■5 — OX^ + 85 ; - X 400 = 0, 13 2 = 0. 5 , 20.1- , 21 ^ + 17+ 17=^- 100 Young: Solvable Quintic Eqmtions with Commensurable CaJ/icients. 11. jc" 4x . 29 13 10 + (i5=0- 12. a-» + ^.:+ ".= 13 13 13. x» 4- 110 (oa;" + 60j;» + SOO.r + 8320) = 0. 14. a:»— 20j;^ — 80x*— 150j;— 656 = 0. 15. a;"— 40^;" + 160x' + loOOa; — 5888 = O. 16. (,')-.o(|-)-e«o(|)'-.„„„(^) 11200 = 0. 17. x'+ 110 (5a;3_|.20i-'- 360a- + 800) = 0, 18. x'' — 20a;'^ + 320a;» + 540a; + 6368 = . 19. 20x3 _ jgQ^» — 420a; — 8928 writei quintic is of the form 20. x= — 20a;" + 1 70.7; + 208 = . The first equation iu this group was brought under the nc by a mathematical correspondent ; the fo.u-th has been treated by Lagrange • the others were formed by the writer with a view to the full illustration of his theory. The Method. §2. In the article of the Journal above referred to, certain principles were assumed, as having been previously established, or as being known to mathema- ticians. It was taken for granted that the root x of the solvable irreducible a;' -f 7 jja" + ^,33;' -f- jy^x +]h=0 ( 1 ) -]-(A| + A| + Al + Ai), or, putting n, for | Af , u, for i- a1 , ,^3 for ^ AJ , and u, for 1 A* , Wl+ W2+ II3+ «4, where u„ u,, n, and m, are the roots of a quartic equation, which, when irredu- cible, as it is in the most general case, that includes all the others, is a uni-serial Abelian. The expressions m, , lu, u^ and m^ are such that and u\u^ = Z; + cVz + (0 + ^^z) s^{hz + h^z) T Ulll^ = /,; + c^^z — (0 + ^\^Z) V{hz + hy/z) W|«, Z=zk — Cy/Z + (0 — ^y/z) ^{hz — h^z) r (^) ului ~h~ Cy/Z — {% — ^^z) A^{/lz — hs/z) i YouNO : Solvuhh Quinfh f!qnntiom with Cummmiumbh- duffivientH. 101 whore g, h, a, c, h, and ^ are rational ; and 2 = c'''+ 1, e being rational. It is readily seen that Pi U' io--^^-=-Jo Because u\ = ("'"^>)''"-'"'' , it follows from (2) and (;5) that v] =IJ+ BWz + (B" + B'Wx) V(f'-: + hVz) ] «•; = /i + B'^z — ( B" + B'Wz) V {hz + hs^::) vl = Yi - BWz + {B" - B'W:W{}tz — hs/:-) v\-B— BWz — {B"- B'Wz)V{fiz — hVz) J (4) where B, B', B" and B'" are rational functions of a, c, <■,/*, and <p. In like til "tl II II J/ I manner, because i<Jt<g= i-i^^i i^'', we have from (2) and (3) «J«2 = ^ + ^Vz + (^i" + A'WzWif'Z + hy/z), where A, Al, A" and A'" are rational. The value of yl is A = ^_^Jij(J''-''^ + '<zf'<'{P'-'t^z)\. (6) From these data, the six equations, involving the six unknown quantities a, c, e, h, 6 and <^, are (see Journal of Mathematics as above) obtained : 2>4 = — 20 A + bif + 15a*z jif, = — 4i^ + 40«C3 B" = 1 £'" = hz {(P + <?)'2 + 20(?,) = /.;' + ch — (j {(/ — ah) h {$' + ^h + 2d^:i) = 2Jcc — a {ff — ah) (8) Our business is to obtain wj, w', ?<3 and wj from these equations. §3. It will be found that ah is the root of an equation F{i/)=0, whose coefficients are rational functions of jj.,, jh, 2h ^^^ ih< find which, when 2h is zero, is of the sixth degree. Since ah is rational, it follows that, when the coefficients of the given quintic are commensurable, the equation F{y) = has a commensurable root. Let this be found. Then ct'^z is known. The formulae from which the equation F{i/) = is obtained give us, along with ah, the value of — . The remaining elements necessary for the determination of «f, ul, ttg and ul may then be obtained from linear equations, without finding 1 102 YouNo : Sohable Quintic E,puitioH. with Commaimrablc Coefficient,, It. w 11 be pon.ted out how u,c, e, h, and p can be found separately should wo desire to obtain their values. BH'araieiy, stiouKl Thk Pnoop. Onsfi in uhich p^ in zero. §4. When ^,, = the investigation is much Hnnplided. By beginning with th.s case, and present.ng a full description of it. we shall be prepared for giving Intel g.ble of the case in which p, is not assumed to be zero. When p, is zero equations (2) and (5) become ^ * ' and ;<?»2) (7) « Since B is the coefficient of the rational part. B' the coefficient of V., B" the coefficient of V (A. + /V2). and B" the coefficient of VW(/.2 + /J.) in the expansion oin\, their values are given by the equations ^ ^ >»' ^ ahB = 2k (l^ _ c'z) - a'cz' + 2czhe {6' - ^^z) ahB = 2c (^•» - ch) + a'h + 2l-he {6' - cph) ahB" = 2 (/<;» — ih) - ^^i^± £3M+.$.^ , z_{l+_ ^){4kc + ah) e -r —^ a*zB>" = 2{k^- ch)^ + 2 (!•* + ch)ie + ^) _ {e + fz){4kc + ah) e e ~ The equations (6), when p.^ is zero, become ^), = — 20^ + 1 5a'3 Ih = — 4fi + 40ttfz B" = 1 B'" = (8) Let ^2(0» + <?.2z+ 20<^) = Z;2 + c8a j A (0» + ^h + 26cpz) = 2kc + ah J and (9) (10) (7) (8) (9) YouNO: Solcahic Quint ic Binationa with Commcnmruile C<M'Jlicivnt«i. 103 SubHtitutiiig ill tho first of equations (8) the valuo of li ohtiiiiio.i froin tlio second of oquutioiiH (U), unci tlio value of /ie{0* — iph} obtained from (7), and milking u«e of (10), Ik!/ = — 8^- (^■* — Cy) + at I/* + ^tijA . Therefore, by the first of equatiouH (9), «V + (60y- li>^)t = U^-[-i,^. (11) §5. Again, from the last two of equations (9), because ? = ^■» -f l , h^ (0» + 4)'2) = (/.•» + th) - ( 2kc + ah) . Therefore, by (10), h^ (0» 4- <^'z) = {l^ + <V) - « ( 2/^< + Z/) • (12) Similarly, from the last two of equations (9), 2hz*?tiq> = az ( 2!ct + ^) _(/,•> + (\,j) . (13) And (0» — 4,«2)' = (0» + <?)»3)« — 4z (Oy ) . Therefore, from (12) and (13), (7ie»)»(0' _ 4,'2)» = { (/.•' + <V) - « (2A-< + 2/) r - ~ 1 02 (2Z:/ + y) - {L^ - t'y)\» .: zhV (0» - <phy = (/.■» + ^V)* - Z/ {2hi + y)\ Hence from the value of //e (6' — <^'z) in (7), (7i» + <'^)> — ^ (27rf + yf = 2a«^2 = yA\ But, by the first of equations (9), 4 20 (14) (15) Therefore, (/^ + <V)' - 2/ (2/rf + y)» = y (-f - ^J )'. And, from (11), Substitute in (15) the value of Ic^ + fiy here given. The result is t> {(^50y - ~2hyy-25G¥y] + <{- 2 (16P + i>5Z/)(50y |-iw)- 256/.v| = 64PZ/ (x - -^^y+ 64F//« - (167<^ + p^jf. (16) HoriH may bo written ».y^ + qijt =;) wn qimnti- urirl vvlicro 9 §7. The elinunution of t from fl... « *■ ^ F-n. the values o . , 1^ ^ " ^^^" ^, '"^^^^'^ " «^-^) = 0. "' w», n, w, <7 and nil Sg + {25^^^- 256/.V.„).y - 2048Z.^• 8kr = 800ZV- f^^'^^ y, P -L « 2/ ^ , /!'> W= _loo.» 3^/4 ^ ^ -S ^'^'^ - 2oG/.V,„U_ 2048/.'. ■.-....v-,..,,,,,„4i-X:r>;;--~ ^m = 126000y- 3000. .'4. 0. . , / ^ « ^ ^" ^ '^'J"/>.Z/ +24^.V-(l2800/c«+ \,8\,,612 ,, '' -^ + (^-5- ^:^>,/;„~ 14848/^^^^+ 512^,^^.4 8^.. = ^.|2500,.y4.(i9200/.^_^^^^^^^^^^ (18) V« — 8>t;- z= + (2|/'^V«-'-^%,/!^+8^.I/.)|, YoUNO: Soimftff Quintir EquiUiotiM iriffi f^omnirtiNiimh/r CiM^^cUittti. 105 qn — mr=j/ — 5000/.y + (— fiO;,? + 28 O//,/.-') //» vni — 8Zyy = // 1 1 25000^'' — 3000/,,,y» + ( 24//' + 80()/.7/„") // By substituting in (18) these values of rn — 8/r, qit — mr, on — H/.vy, wo got ^\ij) = m' + 'M' + ■ . • ■ + '/B/y + 'J, = 0, (19) where (yj = _ (525000000, y, = r)0()00000;;^, «/!» = — 200000 (200/7^5 + 1, l/,«) , J3= 102400000/.;'+ 128O000/.7),^,5+ 44800y,J, 54 = — (25()00/7;j^)5 + 409C000/.-V'4 + C40000^4/.-» + 448/>i), r/, = 8 192/.-»y.,;,» - 2048 X 185G/.V, + G4^' + "^^f ^ ky, + ''J." /ry,'^,, + (8 32 \* 125 ^'*- 2048/.-^ +X^7'4/'5). 25 6784 3125 1 I* §8. Assuming now that the coefficients />3. l)^, p^ are commensurable quan- tities, let the commensurable root // of equation (19) be found. Then a\ is known. Then, from (11) and (16), < or — is found. a §9. At this stage, as was indicated in §3, two courses are open to us. One is to proceed to find u\, m|, u\, ti\ without troubling ourselves to inquire what o, c, e, 0, ^ and /; are separately. This, the natural and the shortest course, we will now follow. Since ah and — are known, their product acz is known. And, by (14), A is known. Therefore czhe{0' — fz), which, by the last of equations (7), is equal to —aczA, is known. Hence by the first of equations (8), B is known. We might even more simply, acz being known, find B from the second of equations (9). The second of equations (8) gives us y (B's/z) = 2cV2 (^-^ - f^z) + ahk («^/■■) - 2kAaVz. (20) Now acz is known, and it is the same as {a^/z){cVz) ; consequently, the rigns 106 Vo.NO: Solvalk Quintio R^mtiom ,n,k C„^ural,,e Oo,M<.-i^. and "\ - f + ^ V^ + V(/*2 + iWz), . , . ^^^ ••• ('^^"^)^' = (^ + i?V.)^ - (/,. + /.^,) . And, by (7), n,u, = a^z. Therefore hz + 7,^, - ^5 + «' / ^s , . .» g'vesus ,^^^^^,^^^^^^/^^V^-(_^^V.)- This Hence ., + ., + „^ + ,^, ^^e root of the given quintic, i. known. To find a, c, e, 6, ^ and h separately. §10. If we desire to obtain the values of «... a ^ j , may first find « by means of a on!Z ■ ' ' ' ^ '''"^ '* «eparacely, we equations (7), ^"''^'"''^ '^"^^^""- ^^^ (^2) and tlie third of Yf + ^'^) = {k'+thj)-a{2lct + y) and Therefore, Also, by (13), Therefore, 2hze'e^ = az i2kt + ^) _ (p + f^y ± ^ {l±ly)~a(2H + y) ~ aeA <?>3 az(2kt + y)-.(p^J,^'. But, by (9), B'" = 0^ Therefore, fro. the last of equations (8) e ^ 2e(l^^^ 2(F + 0'.) - . ^Uc + a'z) From (21) and (22), «^ (4*< + 2/) - 2^ (F + f^j^)—- • (!^Jy)~a(2l:t+y)-.aeA 2e(k^ - t'y) + 2(1^ 4- fi.A (21) (22) lis (21) (22) Young: Sohable Qnlntic Erpmtiom with Com memumhk Coefficients. 107 Put Then h = ay{Ald + i,-)-2ij{L^-^eu), T = P+fiy, ft -— ae,A _ 2aeT + a But (24) ••.ae{SA + 2yT) = ^6-ya. (23) /3a = - «V (2/.. + y)iUt + y) + ay (/.- + t^y)iUi + ,y) _ o, (;. + ,.^y and yo=- fV2kt + y)i4M + y) + ayi/,^ + t^y)^skt + 3>j)-^ Hl^+tW And y-a^ = ah ~a? = a' (. - 1) = «V. Therefore ./?a - ya = «V |y {2Jct + y){m + 7j)-2 {Jc' + t^yf^ Therefore, from (23), n\^,'^ 2>'f = «'^{^(2^'^ + y)(4/^^ + Z/)- 2(F + ^»n, .-. {hA + 2yrf = aV]y (2/.^ + .y)(4Z, + y) - 2 (I^ + ^^)^ p - (!/~a')\yi2fct + y){ikt -h y) ~ 2 {7.^ + t-^yfl^ (25) Now 7c y and ^ are known. And, by (14), A is known. Therefore (25) is a quoiratic equation from which a can be found. The quadratic has ts r ots ommensurable, and care must be taken in each case to select that one which sat^sfi. all the conditions of the problem. When a has been found. IJe «t and -are known, . and c are known; and, because . = .^+ i, the absolute ^24). N^t to find e, ^ and A, the third and fourth of equations (9) are 2i ~l,JJ _ . Hence, taking the values of B" and B'" in (8) - 2.(/.^ _ e^,)^ + 2 (;. ^ ^.^^(^ + ^) - (^ + .^j(4/.c + „^,) } (26) But e, 3, a, c, /<; are known. Therefore from thp «;m„lf e and , a,.e k„„„„. Therefore, from (7)', /rirkno;::"""'"™"" °'"''"°"' <''*' ^Ob Yovm: Solvable Oumllo V .- HumUc Equations with Com- .». ;/ ^r „ %ii. Jhe values of a c ^ rt ^ i , 8^'- IJien, becaiisG (see S9) ^* we have , ^^ + ^ ^2)' - (« ^2)=, and ^^^- = ^^ + (i^V.n Horetheq.antitie.towhioht~o:^^^^'~''^^^- /' and . are known. When . is .„ '" "'"'^'"' ""^ '^""^^"- ^'^-efore ""I'll 2 js known, because aV „ j " are known. Finally, and are ohf..' , ' «" '"'" ^"°''"' « '"^"d « ^. , 3 , ^ ,^, , '^' ^'^ "^ take the equation f>ecause k = ^' = ^^ ' (27) becon^e ^ '' "20 '/'^=2, ,,==, ^^ ,,,^ equations (n) ,„d (1,) T,, ^^ '^^ 200 40000 The equation (19), obtained by eliminating t from the • ^ J./,,)_/25xl2oS „ ,j25<x ° !"^'^^^^»^tion8(28),is ^^^-(-.0 )/-(^f)/ + (^XI25V^^ . /3209xl2r,x ..7.„/" ^ 90 7 109- (28) + ("^n^-^^"^ I?/ -d this has the commensurable root J-. Therefore Hence, from the two equations (28)^ V 130 ;•'/ + 22500 = 0, 125 a 4 In subsequent examples, when ,, and / i, u onee to fi„d ,, „„, ^^^^ ^^ ia'§, "„' ., rH,, :;" '•''""^. "« *»!! P-oeeed at "■ '" '""' "'^'""oe, the method of *• Young: Sohahh Qtdntic Eqmtiom mth Commmsumhle Coefficients. 109 obtaining the values of these quantities which was described in §10, and we will use the present instance for that purpose. To find . we t.ke Lhe equation (2o), {U + 2yTf:=z{y - ay,y{2ld + y){Ut + y) - 2(/.^ + /V/)^^ (29) 47 500* 4 20 ^y (14), , ^ 3// 4 Also, from the values of h, y and t above given, 2/.'C + y = _ \kt + y=z — ¥ + ey = Therefore 1000 ' 521 1000 ' 47 1000 ■ h = ay {U't + y) - 2y (lr> + fy) = _ •521a^+ 47 62500" Y = y{2M-\-y)-a (/,« ^ fiy) - " ^7 {125« + nj •^ 600" 125000' (30) Hence (29) becomes 125 (323a + 29)^= 36^(1 _ i2oa^). One root of this eauation is — ^^ r,,* +i • Inequation is -^^^, But this root proves on examination to be inadmissible. We must therefore take the other root, which is - ^^^^ Then, since c = -^ , and ah = ± , we have 26'xl3^' a 9439 25x4225' _ ^439 25=X 13' ~ P__7x9439 422500 ' _ 5x4225' ^ ~"9439«~ ' __ 398 ^ ~ 9439 • The sign of . is determined in the way pointed out in §lo. By means of the values of .,.,«., e that have been obtained, we get, from L two equals 2IT 19» ' '' 1 993^4"5 '•^-^'z = ~ j-^- . 110 Young: Sohahle Quintic Equations with Commensurable Coefficients. 47 Therefore, from (7), keeping in view that vl = — .-- , 500 j47^ / 9439 Y ~ 8 V13X 25X125/ Also, from the 6rst two of equations (8), B = --- , and B' = ' 500x4225 """"^ 7,3 + /,Vz= ^-^1^(21125+ 9439V5)} Therefore, < = m(' + 'v) - C25x/1 f (^»^' + »«'^^)} Therefore the root of the given quintic is known. §13. To verify this result, we have 9439V5 = 21106.2456. Therefore e^v/jl" (2^2^^ + 9439V5)| = .79696796, ei's/l'T' (21225 — 9439V5) [ = .01679462. (31) Also, and Therefore J13 100 13 + ^0 = .79696437 01696437. wf= 1.593932 , — «»= .00000359, — ul=: .00016975, — ul= .03375899, ?<i= 1.09773 — iU= .08147 — ^2= .17618 — ?<8= .50778. Therefore Wj + «*» + Wg + «4 = -3323 . Therefore x''= .004052 3x^= .331248 2a; = .66448 .99978 L Young: Solvable Quintie Equations with Commeiisurahh Coefficients. Ill §14. If we wish to exhibit the root as in §9, we find from (8) that 13 / 7^/5^ B + £Vz = Therefore, since aVx/z = — '^^-. , 100 ('' + Ti' G2")' "S=^,^('+¥)+v{©'(^+T)VS}. / 13 V/ uoo. with corresponding expressions for vl ul mJ. As a matter of fact, 3; l^ + -5 -; + 6^^= 6254 T (21125 + 9439V5)} . §15. It is interesting to observe the application of the theox-y to the equation x" — Sx* + 2a; + 1 = 0, (32) whose roots, with the signs changed, are tlie same as the roots of the equation (27). By reference to §7, keeping in view that Ic- - ^ , it will be seen that, wherever an odd power of p^ occurs as a factor in a term of any one of the coefficients of the equation F{,/) = 0, an odd power of p, occurs as a factor of the same term. It follows that, by changing the signs of both^>3 and ^5 in the equation (27), in other words, by passing from the equation (27) to the equa- tion (32), F{y) remains unchanged. Therefore the commensurable root of this 125 equation, which we have seen to be ^ , is the value of ah for the equation (32) as well as for the equation (27). To find tor — , the equations (17) give us yt = vn — 8h' vm — Mq ' The values of vn - 8kr and vm - U-q given in §7 show that, in passing from (27) to (32), vn-hr simply changes its sign, while vm — l-q remains unaltered. Hence t or — has the same absolute value for the equation (32) as for the equa- tion (27), the signs, however, being different in the two cases. Consequently, for Thus we get the equation (27), -- = — a 4 9439 25x4225 ' 7j<9439 '422500 ' 5x4225« 9439' 398 9439 112 -no: Sohahle Qnintic Eqrtntiom with Commemnrabk Coefficients, Hence, hy the first two of equations (8), Therefore, for the equation (32), the value, of »f, «■, v^ and „• are Pi not assumed to he zero. fn h! ''■ ^'^^f """^ T"'"^"" *''' "'''^ ^'"•^'•'^^ ^•'^^^ '" ^hich ;,, is not assumed ou!hr-l J^" ^^*^«^^'^'^* ^- b-" ^ll-*-ted above is'^ltill applicable, though the labor of operation, in dealing with particular instances, is i^Lased Putting, as before, y for ah, and . for ^ . we form equations corresponding to ai) and (16) , from these we obtam the values of y and t; then we find B and B^z from equations corresponding to the first two of the group (8): or B can of §9, when 5 + 5y. ,s known, .,„, orf-y being also known, the root of the given quintic is known. §18. The values of B and B'^z which correspond, when g or - ^1 is not zero, to those given in (8) for the case in which g is zero, are obtained from the quations 2) and (3) by keeping in view that, according to the first of equation (4), B IS the rational part and B the coefficient of v/. in the expansion of «?. Put Then and P=2h{k'>-ey)-.{g^- y){gk-ty)l Q= 2t{1^-fy)-^g^-y)g,_gt^ ^ .^S'~ ^ri" ^^l + ^^ ^ "^ '^^^ + ''^'^'\iiff' i- y) + 2^^-} I {(f-yfB'=.a\2gP+ {g^ + y) gj + 24' { 7. (.<;» + y) + 2<;^^ } I (33) (34) Young: Sohahh Qumtk Equatiom mth Commemnrahk Co>:fiioimt.. 113 By (5) and (6) and Tlierefore ^ (<f — .'/) = [/ (A^ — i'ff) + azA', j>i = — 20A-\.5ff + l5>/. ««^' = ■ ~^ (5^' + 15// - p,) _ ;j (/,■-• _ /a_y) . (35) §19. From the second of equations (6), i'5 (r/' - y)' + 4B {,f — yy _ 40/// (y3 __ ^)2 _ _ Therefore, from the first of the two equations (34), iP^ - 4%)(r/' -y)' + 4 (^^ + u) P ^B<j!jQ + ^azA'{t[r ■\- y) + 2yk\ =0. Puttmg for uzA' its value in (35), (/-Z/)[(A-40/,/)(,^-,) + 2(,/+ 3^_|,,).;,(,. + ,,)^ 2^,_^^j + [l(^^ + Z/)P+8^«e(2_8^(/,._,.^){,(^.+^)^,^^^.,j^^^ ^^^^ But, from the values of P and §, Therefore, rejecting the common factor .r/'-y, (36) becomes 0'.- 4o^y)(rr-.^) + 2(,^ + a,; -!,,){,(,. + y) + 2^;,^ - 4 (/ + /y)(i/Z; - t!j) + 8r7.y (/,• _ gt) _ g (/, + gt){¥ - Z^) = . Arranging according to the powers of /y, This is one equation between the unlcnown quantities y and t §20. From the last two of the equations (6), hc^ (0^ + ^^z) = (7.^ + fy) _ 2/,a, +'(^3 _ )(„ _ ) and 27«^ (e^) = 2/.a/ - (l, + ,ft) - if -•,)V.. i,) . Therefore ^ «/.V (6^ - ^Hr = . ] (Zr + ,V) - 27.«^ + (^ _ ,,)(« _ ^)=j - i2i'2«<- (7^ + ^2/)-(i^^-//)(«2- j,)2;.. (37) 114 Young: Solvable Qidnlic Equations with Commemurahle Coefficients. Tliereforo But A' = he {$' — ^h) . Therefore, hy (36), |""20^^(^i/'+ 1S.'/-A)-<7(/''-'-<V)[=(/^"-<V)'+ (,'/-?/) or, arranging according to the powers oty, ' ^^ ' ' ^ i/ U + ' .V) f I -i/*(i/^- 5-i'4) + 8^7.^(./_.l^,)_ 16/,.= o. (38) This is the second equation between the unknown quantities y and t. §21. We may now either eliminate t from the two equations (37) and (38) so as to obtain an equation F{y)=Q whose coefficients are rational functions of the coefficients of the quintic to be solved, or we may eliminate y so as to obtain an equation T^ (/) = whose coefficients are rational functions of those of the quintic to be solved. In the former case, let the commensurable root y of the equation F{y) = be found. Then, by (37) and (38), t is known. In the latter case, let the com- mensurable root t of the equation ^ (0 = be found. Then, by (37) and (38), y is known. When y and t have thus been found, we find B and B'^z, exactly as in §12, from the equations (34), or B can more readily be found from the second of equations (6). Then ii\ = B + BWz + V {(5 + B'^zf - («iw,)'} = 5 + £V3 + V K^ + ^V2)^ - (<7 + iWzJ \ . Therefore x=iu^-\- Ut,-r u^-^u^ = [^ -t- B'sfz + x/ { (i? + £V2)^ - (^ + aV2)»f]i + [5 4- B'^z — sf\{B-\. B^zf - (^ + a Vz)= }]* + [B-BWz-\rV{{B~B>^zY-{g-a^zf\'\^ + [.B-BWz-s^\{B-B>^zf-{(j-a^z)^\-[K It need scarcely be pointed out that since y = ah, a^z is known. % Young: Solmbk Quinfir Equations loith Commenmruhh' Cofmeuts. 115 §22. Second Emmpk.—Afi an illustrative example, lot a;» — 1 Ox' — 20j,-» — 1 Hq:,,,- —7412=0. Here y = ^- = 1 anrl /• — /'» _ 1 rpL •' 10 ^. «"« /■ 20-^' Theretoro the equations (37) and (38) become 50y/ + y(8/^ + 8/« + 528/ + 7412) + 596t - 6216 = 0, and 25y-(16^'H.56/'-64^-1771),/-(2384/»_S0384),/-88804 = The co,.mensurabIe values of , and f which .satisfy these equatio.. are y = 2 ^- - 4. Ihen, we can get the vahieH of B and JJ'^z from the eraiations ru/ keepmg ,n view that a.A' is known by (35). As L as B . m it r=;=-t^ TiLir ^^^^" -' --"- ^^^' ^--^ ^^ -- ^- 4i?= 7412— 320, .-. i?= 1773. In order to obtain the value of i!-^,, „„ „,„t find P. Q and „U'. By (33) and (35), and Therefore ^=-62 4-9 = — S3, y= 248 + 5= 253, rt2yl' = — 77 + 31 = — 46. i?V2 = G53(Wz + 2J Vz = 653aV2-?l(^'i-^* = a-s/2(653 + 598)= 1251^2, .-. -e+5V«= 9(197 + 139^2). Hence, since m,m, = g, + ^^g =1 + ^3, a:= 9S[(197 + 139V2) + V j(197 + 139^2)^-^(1 + ^2)*]* + 9i[(197 + 139V2)_VUl97 + 139V2)«-gl-(l + ^2)«]^ + 9*[(197 - 139^/2)+ VU197 - 139V ^f - ~^^ (1 - V2)»] ' + 9i[(197-139V2)-V](197-139V2)^-^^(l_V2)0: 110 YouNO: Solvable Qitintic Equations tdth Commttutirable Cofffu-'writa. To verily this result, 81 (1 + -v/-)°= 1.012490, -^- (1 — V 2)" = —.000150536, 197 + 139V2 =393.57568, (197 + 139V2)*= 154901.8, 197 — 139-V/2 =.424315, (197 — 139^2)' =.1800434, ^{(197 + 139V2)»— g^j (1 + V2)»| = 393.57436, /{tl97 — 139V2)' - g\- (1 — ^/2)''| = .180194. Therefore «i= 6.888, «4= .412, xii— 1.602, «3=— .276, .-. a; =7.526. Modification op the Method to Meet Special Cases. First special case : When p^ and p^ are both zero. §23. When p^ and p^ are both zero, a modification of the general method is rendered necessary by the circumstance that the equations (11) and (16). from which y and t are to be found, are then virtually one, and so are insufficient to give us the values oi y and t. In fact, they become 2 ■W^ and t(^50y'~-^-p,y)-Piy = (39) §24. In an article which appeared in No. 2, Vol. VII of this Jownal, the present writer showed that, when pt and ji^ are both zero, ^ and p^ have the forms 571* (3 — m) ^ 16 + m* l'i = _ n° (22 -f- m) i''-' 16 + m» (40) f«. hod is , from ent to (39) il, the ^e the (40) and and YouNO: SoloaMo Quinli,^ Equatiom mth Commemunib/e Co,:fficimta. 117 These expressions for p, and p, f,.r,n..h the criterion of solvability for the qnintic The root of the equation is ''"' ^ ^'""^ ^'^ = "• W where ;i is a root of tiie quartic equation ^' — in/J' — 6X' + mX + 1 = 0, M.J. CGhtsha,. of Ottawa, m "Notes on the t^.intio," gave tho relation between the coefficients of the solvable quintic a?+Pi3?-\- p.ar + p^x + i^j = , and, in his wider formula3, the forms of p, and p, in (40) are included. They were subsequently announced by Mr. Emory McClintock, who had discovered h m '"dependen ly It is to be regreUod that Mr. Glashan has not made pubHc the method by which his conclusions were reached. §25. From our present position the criterion of solvability of the quintic ^ ir. t T" ^'^"'''' ^"^ '^^ ««^"*^«" «f ^''« ^q-^'^tion effected mo e tT For, ;; ' '""" ""^'^'^l ^° ''' ""'''''' "^ ''^ Journal jns^LZl in = and ?i = 2/ ; then the first of the equations (9) may be written Ih = % (3 — wi) . Also, by the second of equations (9), Pi=: ~ AB -{■ 4,0ty . But, from the first of equations (8). 5 = - < (y + 24) . Therefore Pi = Uty + »At = 2/// (22 + »,) . And, by (15), in connection with (14), Id • • y — (42) (43) (44) 16 + jn» 16 +m' 118 YoirNO : Solmihie QaliUic Rt/nntioni with C nitnauinrabfe CofJpcienUi, Tiieroforo alHO 2ty = 2fn« lO + m* 10 + m'' By the HubHtitution of tliOHO vuIuoh of /y uud 2(i/ in (42) and (43), the foimulflB (40) aro ol)laine(l, « in (40) being what wo Imvo called 2<. To find now Mie root of the eriuation (41), cliniinato m from the ocjuations (40). Tiie rcHult is u Boxtlc eqnation, i^ (n) = . When the cooflicientH of the (piintic (41) .ire cominonBtu'ablo, the .soxtic iI'(m) = hua a conunensurablo root. I,il Uiis be lound. Then n is known. Conso- qnently, since m=:'2/, t is known. Then .// is known from (.39). Then B is obtained from the second of "iquations (9), and B'\/z from (20). Also Therefore «i»4 is known. Therefore, as in §9, the root of the given quintic is found. §26. T/ilid Examfilc. — As an illu.strative example, lot j?->r 4^+ 3750= 0. , Here the equations furnishing the criterion of solvability are 025 _ 5n*(3 — m) IG + w' ' n»(22-fm) 10 + m'' • 4 3750 = These are satisfied by the values wj = 2, n=.b. Therefore '=4- Therefore, by (39), w = ^^^ 4 Therefore, by the second of equations (9), B- 025 4 And, by (20), ijBWz = - 2 (A)(cV2) = - 2 {t'!i){t^!j), 025 .-. B>y/z = - 2<3V.'y = — ^ V 5 . o 4 \ ■rt icnUt, ft f S 4 1 fui-inulffi :' now 'he rcHult is a ^(«) = I. COHBO- 'hen B is Voing: Hdvabk Quintic Eqmtiotw with CommimuraLle Confidents. Ill And «.M, = aV« = ^^'"\ Thorofore ^■=[-t6+0+^((T)'(>+^7-(T)1]'. +[-"r'('-^v^'((Ty('-t7+cfy}]'. + [--:(-tV^{(T)"('-f)+(T)l]- *Ss.vwjd «;,f eta/ ccwe; IK/ic/i »i«^= »,»,. §27. Intiiecu . u. ludi«,u, = „,»3, '' = 0. Consequently, if cis diHtiuct from zero, / or ' ,(iMite ; while, if c i« zero, t asnumes the form ^ . As we cannot here proc. ^■ UnUing t, the general method ha« to be modified. §28. Because ■ . O, ^ = «'« =o, a,.J .^ = ( ^ )(„.,) ^ q. Also ^^ = c'z. Equation (5) becomt gA = /i> — c',:. Therefore, from the fii st of equations (6), and, from the second of equations (6), (45) (46) From (33) and (35), P = 2k (/.- - ^z) -{rf- ah)(,jh - acz) and aQ=2at (/.•» _ c'z) - {g^ _ a\){ah - yc) = cfo . Therefore, from the first of equations (34), taken in connection with (46), ^9*P, + /^{2{k>~<J>z)-.fl + 2czA'=0. (47) ot^ilt'^t" '" ^°^^^^"-^'^)' - '-^' '-- ^^« value of .M.(._,.)^. z (Ay = (P _ ^,y + ^. __ 2^3 (^. ^ ^,^)^ ^^^^ 120 Young : Solvable Quintic Equations icith Commensurable Coefficients. Therefore, from (47), by the elimination of A', 4c»a = -; [4/P» + M2(A'-o'^)-^|] (49) (/;'i _ (fizf + / — 2if» (i' + c'z) But, by (45), c'g is known in terms of the coeflBcients of the given quintic. Therefore the equation (49) gives a relation necessarily subsisting between the coeflBcients p^, m, p^ and p^ of the solvable quintic in which UyU^ = u^u^, in order that UlU^ may be equal to u^u^. To find now the root of the quintic, ch is known by (45), and B is known by (46). To find B's/z, making use of the values of P and ciQ and (J.'v's)^ obtained above, we have, from the second of equations (34), g" {B'^/z) = c^z\2 {h^ — c'z) + j/' } + 2A; (^Vz) • (50) Now, by the first of equations (34), keeping in view the value of azA! in (35), g'B = (f \ 21c (7^ — c\) — g'l: ] + lazA! (^) = (f\1h {1^ — &z) — (flc\ + IfczAl. As B and <?z are known, this equation determines the sign of czM or (c\/z)(-^'v'z) , and therefore determines the signs with which a^z and ^xjz are to be taken relatively to one another. Hence z(J.')'' being given by (48), B'^/z is given by (50). Therefore 5 + 5Vz is known. And MiM4 = gr. Therefore, because xc{=iB-\.B^z^^\{B->rBWt)-g% the root of the quintic is known. §29. Fourth Example. — As an illustrative example, let 7? — 1]_ ; 11 X 42 , li><_89_ 25 ^ "^ 125 ^'^ 3125 """ Finding the value of &z from (45), and substituting in (49), we find that (49) is satisfied. Then, as in the preceding section, ^„ „ 11X89 Pi= — '^B — 5 The value of (?z is 16 m B= ~ Therefore 4X6= Jc" ch ~ V25V V400 16 )~ 100 ^ 25 y ' and ^+<'-(1)U+b)= 100 V 25 63 / 11 Y 200 V, 26 y ' Young: Solvable Quintic Equations with Commmmrabk Coefficients. 121 From (48), (^^e)' = r''' ^ 'Y The value of czA', obtained from (47), is negative. Therefore cs/z and ^Va must have different signs. Put and 25 4 Then, from (50), dividing by g^, B'^/z = — -'^ Audi UjUi=.g. Therefore This is, in a different form, the value of mJ given by Lagrange. §30. Fifth Example—In the instance just considered, c is distinct from zero. It may be well to give an example in which c is zero, as the mode adopted above of determming the sign of A'^z does not apply in that case. Let jb" + 20a;» + 20a;« + 30a; + 10 = 0. • Here g= — 2, and 7^ = — l . Therefore (46) becomes 20(l-c«2)=_ 2(20-30). Therefore t^z=0. This value of c'z satisfies (49). Then, by (46), B = -A From 48, {A'^^zf= 1 + 64 + 16 = 81 .-. A'Vz=9. ' ^ ' Because c is zero, we cannot determine the sign of A's/z relatively to that of cVz . But the root is the same, whatever sign be taken. Having found A'Vz, we have, from (60), 45 Vz = —2x9 .-. B'^/z = — A ^ - 2 2 2 And UiVi — gz= — 2. Therefore «!= — 7 -r- <v'j49 — (— 2f\ = 2, «'=-7-V]49-(_2>^}=_16, «i= 2 + V{4 + 2»}=8, v\= 2 — V{4+ 2')=— 4. Therefore a; = 2^ — 2^ -f 2* — 2^- Therefore 5+5V.= -^--i = -7, 5-£V. = _A + JL=^2. 2 2 122 Young: Solvable Quintic Equations with CommensumUe Coefficients. §31. In the article of the American Journal of MatJiematics (Vol. VI, page 103) referred to in the opening paragraph of this paper, the solvable irreducible quintic in which ii^Ui is equal to j^'s was discussed, and the roots of the equa- tion were shown to be determinable in terms of the coefficients ^Jj, i^si ^tc, even while these coefficients have no definite numerical values assigned to them, but remain symbolical. The solution that has now been given is much simpler than the former; equally with the former, it is applicable to equations with symbol- ical coefficients, the assumption being of course made that the coefficients are related as in (49) ; and it possesses the advantage of being part of a general theory. Additional Examples. §32. Sixth Example.— het • x^ + 320x^ — lOOOcc + 4288 = . Rereg = 0,k = —lG. Because gf = 0, we use the formulae (11) and (16). The commensurable values of y and t which satisfy (11) and (16) are 7/=8, ^=6. Also, by (14), A =56. Therefore, from the second of equations (9) and from (20), .'. 5-f BVz= — 16(37-f 20V2). And Mi».4 = — 2\/ 2 . Therefore X = Ui + Ui-\-tli + U3 = [— 16 (37 + 20V2) + V |256 (37 + 20^2)' - (- 2V2)''}]* -f [— 16 (37 + 20V2) - V i 256 (37 + 20V 2)^ - (- 2-/ 2)''}]^ + [_ 16 (37 - 20V 2) + V{ 256 (37 - 20^2)' — (2^2)^^' + [— 16 (37 - 20V 2) - Vi 256 (37 - 20^2)^ — (2^2)^}]^ §33. Seventh Example.— Let (^y+^»(fo)'-«» (to) +-'="■ or, x^ + 40000x'* — 690000x + 10800000 = 0. Here g=0, h— — 2000. Because <7= 0, we use the forraului (11) and (16). Young: Solvable Qumtic Eqmtiom ivith Comvicns arable Coefficknts. 123 The commensurable values of y and t which satisfy (li) and (16) are, 2/= 2000, <= 50. Also, by (14), ^ = 36000. Therefore, from the second of equations (9) and from (20), 5= -1700000, ^Vz=-400000V5. And uiUi = — 20V 5 . Therefore = [- 100000 (17 + 4V5) + VI 100000^ (17 + 4V6)^ + (20V5)ni^ + [- 100000 (17 + 4V5) -V] 100000^17 + 4^5)^ + 20^5 "^ + [-100000(l7-4V5)+V{l00000Vl7-4V5)«-(20V5 4 ^ + [- 100000(17-4^5) + V{l00000Hl7-4V5)^-(20V5)«}]i §34. Myht/t Example.— Let a;" — 20a;3 + 250.r — 400 = 0. fll\7nT(^i^^T''jT'" ^ '" ^'''^'' ^'""^ ''''' ''' "«« ««t the formulae (11) and 16) as m the two preceding examples, but (37) and (38). The com niensurable values ofz/ and ^ which satisfy (37) and (38) a4 y=2, t = — 2. Therefore, from the second of equations (6), 5=60 The valuP nf A' ■ thovalueof^V..sknown. Then, by the second of equations (34), 5V.= 44V ' Therefore B + B'Vz = 4(15 + 1W2). And Mi«4 = ijf + a Va = 2 + V2 . Therefore a = Mj -(- t^^ + „^ ^ ^ = [4(15 + llV2) + V|l6(l5 + l,V2)»_(2 + V2)=jp. + [4(15 + llV2)-V{16(15 + nV2)^_ 2 + V2ni + 4(l5-llV2) + V{16(15-nV2)^- 2-V2»m + [4(15-11V2)-V]16(15-11V2)3_(2-V2).|],. §35. Ninth Example.~hQt Here g = X bx -\- ~~x~ — - = 0. -, /^- 0. Because g is distinct from zero, we use the formute (37) 124 Young: Solvable Quintic Equations with Commensurable Coefficients. and (38). The commensurable values of y and t which satisfy (37) and (38) are y=^,t = -i. Therefore, from the second of equations (6), 5= - - . Finding A'Vz as in the immediately preceding example, we have, from the second of equations (34), 5Vz = -|-V2. Therefore i? + 5V2=|(l+V2). And UyTii = — (2 + V2). Therefore x = ni + W4 + U2 + tis = [|(l + V2)+^{^(l+V2)'--^(2 + V2)»}]' + [|(1 + V2)-^|A(1 + V2)^-A.(2 + V2)»|]* + [|(l-V2)+^|A^l-./2)^-4-(2-V2)^}]* + [|(l-V2)-y|A(i_V2)^__i,_(2_V2)^|]*. §36. TentJi Example. — Let R ■ 20a- . 21 This and the next two examples are intended as additional illustrations of the method to be followed when 2h ^^^ 2h are both zero. The equations furnishing the criterion of solvability given in §24 are and 20 _ 5n*(3 — m) 17 16 +m' ^ _n^(22 + m) l7' iiS^TO^"- ' and they are satisfied by the commensurable values m = — 1 , n = 1 . But, by 1 1 §25, n = 2t. Therefore t=: --. Putting k—0 and <=—, equation (11) YOUNO: Solvable Quintic Equatmis with Commmsumhle Coe^fficients. 125 becomes i ./ 2 \ Therefore also cVz = t (aVz) = ^^ . Therefore, by (20), 1/B'Vz = — 2 (ch)(cVz) .-. 5Vz = — ^^^ 68 • And, by the second of equations (9), B = ^ . Therefore 68 5 + 5V«=-4(l+V]7). And «,,«, = ^^i^ . Therefore 68 X = + [- OT (- ^-) -V{(' 1?^)'+ (>(f7}]l §37. Eleventh Example.— Let 4x 29 The equations furnishing the criterion of solvability are — i — ^n*{S~m} 13 "l6+m2 ' 29^ _ n'{22 + m) 65 16 + m^ ' and they are satisfied by the values m=7,n=l. By §25, n = 2.. Therefore t = -^. Therefore, from (11), y=-l_. Therefore Therefore, by (20), y£V. = - 2 (c^.)(cV.) . Therefore B'Vz = - ^^''^ And by the second of equations (9), S = ~~. Therefore B + B's/z=~^-+y^^ 260 • 120 Young: Solvalk Quintic Equations with Commensurable Coefficients. And Mit<4 = ~r . Therefore 260 ' V I ^ 260 65 +[-'-tr-vi("-tor)'-(i?y)]' +[-"^:^V{C---ir)'K^r71]' §38. Tialffh Emmp/e.— hat „ lOx 3 ^+13 + T3=^- The equations furnishing the criterion of solvability are 10 _ 5?t'(3 — m) 13 3^ m 16 + m' ' 7).'(22-fOT) 16 4- m' " ' and they are satisfied by the values m = — 7 , n = 1 . By §25, n = 2<. There- fore <= ~ . Therefore, from (11), // _ . Therefore And, by (20), yB'Vz = - 2 {ch)icVz). Therefore B's/z =-^. And, by the second of equations (9), B =z — — . Therefore 52 B + B'Vz='-y'' And iiiVi =■ - . Therefore 00 260 _r5-V65 (/5 — V65Y /V65y) -,i ' - L' "266"" + V / I V^260^ ; - K-er) \ J r5 - V65 _ ' f /5— V65Y /yesy I -li "•"L 260 vlV 260" 7 '^ 65 y j J "•" L 2 5 + V65^'' YODNO : Solvable Quintic Equations icith Commemnrahle Coefficients. 127 §39. ThirteentJi Example.— hQi a-" -f- 110 (5a^ -f- 60x« + 800x + 8320) = 0. ^7-VZr^l'/''^~^^^' """^ ^''"^ commensurable values of y and < which satisfy (37) and (38) are //= 5 X ll^ t=i 10. Therefore, from the second of equations (6), /i= - 220 x 765. Finding A'^z as m the second example, we have, from the second of equations" (34), 5V8=-220X337V5. Therefore ^ ' B + B'as/z = — 220 (765 + 337V5) . And «,«, = _ 11 (5 + ^5), Therefore x=[- 220(766 + 337V6) + Vj220^(765 + 337^5)^- (_ 65- IIVS)'}!^ + [- 220(765 + 337V5)-V] 2203 (765 + 337^5)^- (_ 55- 11V5 H i + [- 220 (765 - 337V5) + V{ 220^ (765 - 337^5)^ - (_ 55 + 11^5'* + [- 220 (765 _ 337V5) - V\ 220»(765 - 337^/5)^ - (- 55 + 1 1V5)»|]1 §40. Fourtcaith Example.— hat x^' - 20a;»- 80.t' - I50.c- 656 = 0. Here ^= 2, Z- = 4, and the commensurable values of y and t which satisfy (37) and (38) are y—2t=2 J ^ > Therefore, from the second of equations (6), 5 = 204. Finding A^z as in the 7:^Zjr^^ZT'"^ '- '-^ '- ^'^ — ^ - «-^^- (34)! ^=12X17, and 5V3= 144^2, .-. 5 + 5V2!= 12(17 + 12\/2). And u,v^=z 2 -\-V2. Therefore Xi = U^ + Ui -j- ?/2 + «3 = [12(17 + 12V2) + V|144(17 + 1 2^2)'— (2 + ^2)^? + [12(17+ 12^/2) _V|144(17 + 12V2)^_(2+V2)»? + [12(17 - 12V/2) + V{144(17 - 12^/2)^ _ (2- V2 4]^ + [12(17- 12^2)-V]l44(17-12V2)3_(2-V2>^.[j!. §41. Fifteenth Example.~Let a;" — 40a;3+ l60.^•^ + 1000a-— 5888 = 0. M 128 Young: Solvable Quintic Eqiuitiom with Commmsumhk Coefficients. or Here g-i, k 8, and the commensurable values of j, and t which satisfy (37) and (38) are y=8,<=_4. Therefore, from the second of equations (6), B= ii52. Finding JV^ as in -B + ifV2 = 48 (24 -f 17-V/2). And /f,M^ = 4 -f 2V2. Therefore X = [48 (24 + 17V2) + V]48»(24 + 17V2)'- (4 + 2^2)^}!^ + [48(24+ 17V2)-V{48»(24+ 17V2)»_(4+ 2-v/2)»n^ + [48 (24- 17V2) + V{48«(24- 17V2)»- (4- 2V2)' ]i + [48 (24 - 17V2) - Vi48«(24 _ 17^/2)^ - (4 - 2V2)^}]S. §42. Sixteenth Example.— Let (f ) - '' (-2 ) ~ «00 (-rj_ 2000 (I) - n 200 = 0. a'— 200a;'— 4800a;'— 32000a; — 3200 X 112 = 0. Here i/ = 20, /. = 240, and the commensurable values of y and t which satisfy (37) and (38) are ?/=80, <=20. Therefore, from the second of equations (6), 5 == 640 x 165. Finding A's^z as in the preceding examples of the same type, we have, from the second of equa- tions (34), iJVz= 640 x 73^/5. Therefore 5 + 5V2= 640(165 + 73^5). And MiM< = 4 (6 + V5) . Therefore X = [640(165 + 73V5) + V{640'(165 + 73^5)'- (20 + 4V5)''n^ + [640(165 + 73V5)- VJ640M165 + 73^5)' -(20 + 4^5)^1* + [640 (165 - 73V5) + V] 640' (165 - 73^5)' - (20 - 4^6 "li + [640(166 - 73V5) - V{640'(165 - 73^5)'- (20 - 4^5)''}]*. §43. Seventeenth Example.— Let ic" + 1 10 (5a;3 + 20a;» — 3e0a; + 800) = . ^TJT^ r ^^' ^ "^ ~ ^^^' ''"'^ *^^ commensurable values of y and t which satisfy (37) and (38) are ^ .'/ = 5 X 11", t=z~io. loUNO: Solvable Quintk Equations with Commensurable Coefficients. 129 Therefore, from the second of equations {ii), Ji = ~ u x 7500. Finding A'Vz as in preceding examples of the same type, we have, from the second of equa- tions (34), B'^/z = — n x 2700-v/5 . Therefore Ji + Ji'Vz = — 3300 (25 + 9-V/5). Andultt^=~n{5 + ^/5). Therefore x=[~ 3300 (25 + 9V5) + ^|3300»(25 + 9V5)» + n»(5 + ^5)^1^ + [- 3300(25 + 9V5)- V{3300'(25 + 9^5)^+ li»(5 + ^5)^]* + [- 3300(25 - 9V5) + V]3300«(25 - Wdf + 11»(5 - V5)^ni + [- 3300(26 - 9V5) - V{3300»(25 - 9^5)'^ + ii''(5 _ V5)'}]1 §44. Eighteenth Exampk.—Let a;" — 20x« + 320x» + 640x + 638 = 0. Here gz=z 2, k=z - lo, and the commensurable values of y and t that satisfy (37) and (38) are y = 8,<=_5. Therefore, from the second of equations (6), and the second of equations (34), Bz=- 12X166, B'Vz = ~ 12 X 117^/2, •■• ^ + B's/%~ — 12(166 + 117^2). And «ii«, = 2 ( 1 + V2) . Therefore a; =[- 12(166 + 117V2) + VI 144(166 + 117V2)»- 32(1 +V2)»m + [- 12(166 + 117V2)-VM44(166+ 117^2)^ _ 32(1 + V2)»m + [- 12(166 -117V2) + VM44(166-117V2)^_ 32(1 -V2)»li + [- 12 (166 - 117V2) - V| 144 (166 - 117V2)» - 32 (1 - V2)»|]J. §45. Nineteenth Example.— Let a:'*— 20x»— 160a:«— 420a; — 8928 = 0. Here (/= 2, /.• = 8, and the commensurable values of y and t which satisfy (37) and (38) are y=72, t= — Therefore, from the second of equations (6), 5= 562. Finding A'^z as in preceding examples of the same type, we have, from the second of equations (34), 5Vz=-284^/2. Therefore B + B's/z = 4(138 — 7 ls/2). 180 YODNQ: Solvable Qumtic Equations with Commeimirabk Coefficiaits. And Mi«4 = 2 — 6\/2. Therefore x = [4(138 — 71-v/2) + ^116(138— 11^/2)* — {2 — W'2f\Y^ + [4 (138 — 7 W2) — VJ 10 (138 — 7W2)» — (2 — 6V2)»i]i + [4(138 + 71V2) + \/il6(138 + 71V2)«— (2 + 6V2)''i]i + [4(138 + 71V2)— Vil6(138 + 71V2)'— (2 + 6^2)' j]'.. §46. Ticentitth Example. — Let «"— 20j;8+ 170a; + 208 = 0. Here gr = 2, h = 0, and the commensurable values of y and t which satisfy (37) and (38) are y = 2, < = 2. Then in the usual way we get 5+ ifV«= — 12(1— -v/2). And r^Mj = 2 f \/2 . Therefore a = [— 12 (1 - V2) + VJ 144 (1 — V2)» — (2 + Va)"}]* + [- 12(1 —V2) — vi 144(1 —V2)»— (2 + V2)''|]i + [—12(1 +>v/2) + VJ 144(1 + V2)»— (2 — V2)»ni + [- 1 2 (1 + V2) - V^ 144 (1 + V2)« - (2 - V2)» j]i. ^^K mta. ■:l isfy (37)