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 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)