.0^ (yC^^a-l^ CORNELL UNIVERSITY LIBRARY 924 068 868 DATE DUE GAYUOKD PBmTCPlNU E *,. The original of this bool< is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924068868151 gjhis ... is an authorized facsimile made from a microfilm copy of the original book. Further unauthorized copying is pro- hibited. Books on Demand is a publishing service of UMI. The program offers xerographic reprints of more than 130,000 books that are no longer in print. Some titles are also available on microfilm. The primary focus of Books on Demand is academic and professional resource materials originally published by university presses, academic societies, and trade book publishers worldv^ide. This on-demand reprint was reproduced from microfilm by printing each page on a continuous roll of paper. Since only one side of the sheet is printed, it is folded to give the appearance of having printing on both sides. It does, however, result in odd-numbered pages being in the left- hand position rather than in the right-hand position. Printed text and line-drawing illustrations reproduce well from microfilm. Half-tones (photographs) generally do not reproduce well, fe UMI BOOKS ON DEMAND University Microfilms International A Bell & Howell Company 300 North Zeeb Road Ann Arbor, Michigan 48106-1346 1-800-521-0600 313-761-4700 Printed in 1993 by xerographic process on acid-free paper THE CYCLOTOMIO QUINARY QUINTIC BY ERIC T. BELL DISSERTATION Submitted in Partial Fplfilment of the Requibkments for the Degree OF Doctor of Philosophy, in the Faculty of Pure Science, CJOLUMBIA UnIVEESITY Pitii or THI HE« era PRINTIKS COHPANf LANCASTIR. PA. 1912 PRECIS. § 1. (pp. 2-14). Two number imaginaries ^, x are defined, their properties sufficiently developed for application to the construction of the quinary quintic arising in the theory of eyclotomy for p=lOn+l, X = ix" - l)Kx - I); where note, p is at first a mere formal symbol, later its meaning is restricted to coincide with the arithmetical. The quintic is i^ (ao, • • • 0:4) , or simply i^j.the variables and coefficients are both complex; the form of F (which is of course real), in which all the variables and coefficients are exhibited as real, is quite unmanageable on account of length; the properties of f = kX , k a numerical constant, are however easily and rapidly handled in the ideal form. jF ha\ing been constructed, it is for the interpretation p a real prime =i IQn + 1 , identified, via the Lagrangian resolvents with X . Incidentally, properties of ^ , x j considered as complex factors of a real p in the field of fifth roots of unity, are given by means of Jacobi's theorems in eyclotomy. For- mulse implicitly containing the composition-theory (arithmetical) of F forms are written down. This section explains in considerable detail the internal structure and nature of F . § 2. (pp. 14-47). The equation F = 1, which is fundamental in this theory, is discussed from several standpoints. Transformations of F into itself are found (a first solution), and the equation is shown to be the direct analogue of the Pellian Equation (Fekmat's Equation). Next, a sufficient sketch of the general theory behind this and allied forms is given to indicate reasons for considering F forms at all; this relates to operators 0, such that /i, /a being two forms of the same kind, 0(/i, /2)=/3; viz., on/1,/2 as a basis, produces a form/3 of the same form. Throughout, the is multiplication; in the more general theory, not necessarily so. The coefficients in the first set of automorphics for F are shown to be connected with the units in a field of fifth roots of unity, and (by a species of accident not occurring in forms of this kind of higher order), the general form is written down; the coefficients are Leonardo of Pisa's well-known numbers 1,1,2,3, 5, 8, 13, • • • . Next, a solution oi F = 1 analogous to the well-known and useful cosh V, sinh u solution of the Pellian Equation is sought, and found; this is expressed in terms of certain functions of 4 arguments having 4 imagin- ary, conjugate-in-pairs, periods. In passing the general theorem corresponding iv pRfcis. to this is sufficiently developed for application to the n-ary n-ics, and proved. The solution of i^ = 1 is next considered from a strictly arithmetical point of view, although geometrical language is occasionally used for convenience. In connection herewith, are considered the properties of " 7r-numbers," viz., numbers in the field { '^PPI Pi , Vpp^ p4 , '^ppIps, ''^ppIpi], the Pi being prime complex factors (in Kummer's sense) in the field of fifth roots of unity, of a real prime p=10n+l. By a slight modification of Dirichlet's method in his (real) generalization of the Pellian Equation, the complex F = I equation is shown to have <=* solutions, which are discussed in several particulars. § 3. (pp. 47-75). The general object of this section is sufficiently explained at its beginning and end. Here, it may be pointed out that pp. 48 to 63 form one continuous algebraical argument, necessarily condensed. The con- densation is an absolute necessity; the work can not be carried on unless done briefly, and was actually done as written here, with occasional checks. E. g., if an " associated form " is written at-full length, it contains 3,125 terms of the fifth degree on 5 letters with complicated literal coefficients; if written as a function of all letters real, viz., as an explicit, instead of implicit function, it will contain about 101^ = 10,510,100,501 literal terms. Hence a condensed notation. The ordinarj' symbolic notation would seem to be useless in dealing with these forms, for constant reductions are being made in several sets of variables, and from the nature of the case, these reductions can only be made when actual products are seen, not symbolic. Attention is called to the dis- tinguishing and remarkable feature of these " associates," permanence of arithmetical form (through permanence of algebraic-number form) in the coefficients as well as in the variables, and in the linear factors of the form; the same permanence exists in all three under multiplication, division and linear transformation, evolution, etc. This " permanence property " is common to the n-ary n-ics of the same most general (non-cyclotomic) kind (see pp. 73^75); and is at the root of the existence of possible general theories of certain quadratic forms (e. g., the binary and quaternary, as existing) arithmetically considered. The problems of automorphism, equivalence and representation for " associates " are considered and briefly solved. A dif- ficult question regarding the separating of certain integral from fractional values in the coefficients of the automorphics is not solved, but the rational forms are all found, and the fractional very definitely limited. The general- ization is pointed out. In all this section the work has been cast in such a form PRfciS. V as to be immediately applicable, with a few slight changes of notation, to the general and important case of n-ary forms. § 4. (pp. 75-96). Here F is spoken of in geometrical terms; as in § 3, so many lines, etc., have to be considered simultaneously, that after the reading of the equation of the tangent surface, a symbol S is constructed which im- plicitly contains all the descriptive properties of a special 12G-point con- figuration connected with F = \, the configuration is only part of a 3,125- point, but a self-contained part. The whole of the " geometry " of this section is from a direct reading of S; to derive it by systems of equations step by step is impossible owing to the complexity. All that this section aims to do, is to point out the existence of symmetrical " surfaces " and configurations in space (here of 5 or 3 dimensions, according to taste), that are direct extensions to 3 , 4 , 5 , • • • , n space of the properties of the circle and in some aspects of conies; that is, the geometry connected -mth. the theory of the division of the circle is by no means confined to the inscription of regular polygons. ON THE CYCLOTOMIC QUINARY QUINTIC. Introduction. In the theory of the division of the circlcj there is the following well-known identity, G? = 4Z=P±pZ2; X ^ (x'-l)/{z-l); p prime. ElSEXSTEiN, in a great memoir, " Allgemeine Untersuchungen iiber die Formen dritten Grades mit drei Variablen welche der Kreistheilung ihre Entstehung verdanken," * studied in detail the corresponding expression for X as a ternary cubic, arising in connection with the trisection of the angle. Eisenstein's form is ^ s x^ -f ppi y^ -\- PP2 :? — Spxyz = 27X, where pi , p2 are complex prime factors of the real prime p in the field of cube imaginary roots of unity. It has been thought a matter of sufiicient self-contained interest to study by purely elementary methods some of the more immediate properties of the quinarj- quintic which arises in the same way as the ternary cubic in the theory of cyclotomy. As will be pointed out in the sequel, these three forms, the binary quadratic, ternary cubic, and quinary quintic have many properties in common, and some that are not shared by any similar (cyclotomic), quantic of higher order; that is, in some essential respects, these three cases form a small closed theory. That such a theory is included in the more general field of forms on n variables and of the nth degree which reproduce themselves in form with respect to multiplication, is obvious; but individually the consideration of the inter- connected sub-cases is not without interest. •Crelle, 28; 29. § 1. The Special Quinitc. The immediate object of this section is to build up the homogeneous form F (ao,aia2, astat) of the fifth degree which arises when (x^ — 1) / (x — 1) for p = 57J + 1 , prime is resolved into factors of the first degree, corresponding to the expression of X in the form f (vo) • f (vi) ■ ■• f ivi) , where / is a linear function of the roots which constitute each one of the five periods in the cyclotomic equation. A form F is first arbitrarily calculated from certain number-imaginaries as a base, and then identified with the required cyclotomic function, when the p of the calculation is made to coincide with the prime pot X. It is clear that the " p " of the calculation need not be a prime or any other number; it may be considered a mere formal symbol or abbreviation for the product of certain imaginaries (the x and ^ below); e. g., the completed F is equally intelligible whether the " imaginaries " be the operators of an abelian group, or as in the present case, the factors of the real prime p . Let ^1 , ^2 > 4'i, H> XI > X2 . X4 > X3 be a set of number imaginaries subject to the commutative law of multiplication. The order of arrangement is essential; the suffixes are ordered modulo 5 in ascending order of powers of 2; 2", 2', • • • ; in this order the ^p, x are said to be in " reduced order." Each set ^, X is closed. Hence from any function F {yp„^, ^„, , •■• ^u,; X„, , Xv,j ••• Xr,) is derived a "reduced" F{ ), in which every subscript has been replaced by its positive residue modulo 5. Again, if in F( ^„, , • • • x^ ) > each v,nls doubled and then reduced (mod 5), the resulting F' { ) is said to be derived by " doubling and reduction " from F { ) . This process may be repeated indefinitely, giving rise to a closed set of i^'s; the members of the set being derived from any one (except possibly the last) by " successive doub- ling and reduction." It is assumed that the doubling and reduction of an identity leads to an identity. The ^ , x are further subjected to the conditions of definii^ion: ^= xi^2, and ^ih=X2iz; also 4'2h=X2X3 (1). Whence by doubling and reduction, ^t = xz ^i] 4'i = Xi h; ^ = X3 lAi; \^2 ^t = X4 ^i; 4'i 4'3 = X3 h; h ^i = xi 1^4- (1') ^1 ^2 = Xih, whence xpi ^l = X2 fs 1A2; and From (1) or by (1') ^Pi X2 1^4 = X2 ^3 ii , whence f 1 \l/i =■ ^2 1A3 . (1") 2 ox THE CYCLOTOMIC QUINARY QUINTIC. 6 So from (1) ^i4'i= ii -h = XI X4 = X2 X3; say each = p (1"') Combining,
  • = Px,xz). (11) (^f'ihy = p'; (^.'W = P'). (12) {hhy = p^; (^i',(-.' = p'). (13) (^iv^i)^ = px?X2 Vx^xl = p-'(xiX2)--xi; (^.^^=' = PX2). (14) ((^1 ^I ^Pl)' = pxlx'. -fxlxl-r xt xf = P'{X1 X^r ■ (X2 Xs) xl. (15) (il4'z4'ir = fxlxl-pxlxi-p'xtxl^p'{xixi)-{^zx2y'xv/ ( ^1' /-|\^4 ^ pxi; px^; pxi', px3- Now put a + bp' + cp^ + dp" + gp" s/(5); p5 = 1, p :^ 1; then identically, expanding the circulant C ia,b,c,d,g) , /(O) -/(l) •/(2) ■/(4) ■f{3) = a'+¥+c'+d' + g^-5abcdg - 5 [a? (bg + cd) + 6' {ac + gd) -{- (? (6i+ ag) + d^ {ah + eg) + ^ (fcc+ oi) } + 5 [a {b-'g'- + c'd?) + b {0^0"+ gH'') + c {¥d^+ a} g"") + d {a^h^ + c'g-')-\-g {_¥(?+ a? d^) } = F (a, b, c, d, g) . It will be shown that F' (ao , ai ^i , aa 'As , ocz 4'3 > «4 4'i) is for p a prime number of the form 5w + 1 > a form to which (a;"— 1) / {x — 1) is reducible, wherein the as are complex integers of the field of fifth roots of unity, and the ^'s factors of p in that field. F' (ao, • • • ) is first reduced by means of (2) to a simpler form; from (2) at once: a^ (bg + cd) = al (aiai 1^1 4'i + "20:3 42 4'3) = pc^ (aiat + aiaz) . 6' iac+ gd) = al (aoa2pxi X2 + P^Xiazai) = PXi.-(«+<)lDd(l+A)_ /g'/\ [See Jacobi's memoir, Ges. Werlce, Bd. 6, pp. 254-274; or H. J. S. Smith's " Report" (" Papers " I), Art. 60]. AIso^ a well-known result, ^{h)-^{-h) = (-)*.p; so here which will be returned to presently. Putting t = s, denote is, s) by xis), then \^(*) = x(s)- ^{2s): s = 1,2, 3, 4: (mod 5). Passing over these details for the moment, the multiplication table is ap- pended for reference: — ^(1) ^(2) ^(3) ^(4) (2)\K3)}=p^(l)V(2)\K3)v^(4)-x'(l)x(2) or whence lAMl) = Px'(l)x(2). Similarly from the table (7) ^' (2) = px' (2) X (4); ^p' (4) = p^ (4) x (3); \AM3) = px'(3)x(l). Again, x(l) = v^^(l)/v^'(2); x(2) = \t^(2)/\K4); x (3) = v^' (3)/ ^J' (1); x(4)=^(4)/^(3); so that x{l)xi4:)=4^(l)^P(4:)/^{2)^Pi3) = pyp = p. and x(2)x(3)=.^(2)v^=(3)/vK4)^(l) = pV2' = P; and therefore x(l)x(4) = x(2)•x(3) = lJ=^?'(l)^(4) = ^(2)v^(3), which complete the identification for p = prime 5/+ 1 of ^ (i) , x(i) respectively with ^i, xt respectively, (i= 1, 2, 3, 4). For brevity write Tij V2', Tz', TTi for ^ppIp2; ^pplPii ^ppIpi; '^ppIps respectively, where as before indicated Pi = X.- = x(0, and the fifth roots are to be now and always so chosen that TTj X4 = vz T3 = p . The complex p-number, a+ bp-}- cp^ + dp^ + gp* being denoted by {a,h, c,d, g) , the solution of the equations (6) is 5»7o = ( — 1 + Ti + 7r2 + TTa + ir4 , , , , ) ; 57ji = ( — 1 , 7r4 , irj , tt; , Ti ) ; 5rfi= {— \, -Ki, Ti, vi, Ttz); 5i7j = ( — 1 , TTs , iri , TT^ , 7r2 ) ; ^'tt — ( ~ 1 > ""x , TTj , :r3 , :r4 ) , where the fifth roots are so chosen that TTl Vi Ts Ti = p^ . 8 ox THE CYCLOTOMIC QnXART QUINTIC. Clearly when any one of the 17. (?' = 1 , • • • , 4) is known the others follow from it by dr. of either the exponents of the p's or of the suffixes of the -'s, which is of use in passing from functions of the periods to functions of the -'s . Since IJO + 1J1 + 1i + JJS + J?! + 1 = , any function of the periods such as a + 6770 + ciji + diji + gi]i + fvi. may be reduced to the form a' + 6' 170 + c' ri + J=i-=^^ / (x — 1 ) may be written as the product of 5 factors, each of which is the product of Ji factors of the form x — r"', where r is a primitive root of a:" — 1 = , and g a primitive root of p; that is, each product of n factors contains (as roots) all the roots constituting one of the five periods, and each of these products may be written as a linear function with integral coefiicients of all the periods, which is expressed in (8) . The actual cal- culation of the coefficients in (8) is tedious, as the only known method is indirect, \iz., by means of Newton's formula connecting the sums of similar powers of the roots and the coefficients of an equation. As such formulae in the present case, for p general, give merely formal results which admit of no (apparently) simple interpretation, they are omitted. By use of the sj^mbol [p/ p] denoting the power of p to which a given number is congruent ( mod p ) , the coefficients a, • • • , g ot (S) may be expressed in terms of the quintic char- acters oi p. As any results obtained in this way would be implicitly con- tained in Kummer's Law of Reciprocity, and as there seems to be no obvious simplification in this method of arriving at the results, the formal results for a, • •• , g are not given. In any case, if p is relatively large, the calculations are laborious, and are not needed for the present purpose. Again, referring to (6), ^ (z) is the Lagrangian resolvent; in the customary notation 4'{i) = (pS Va); and the usual statement of (6') is ^Ap)= (p, t) ■{(,-, r)fip-+\ r). tad"M-(n+I) ina (1+m> and ^n (P) = 2 P' and p = 5n' + 1 is to be prime, so that n' s 2?i, and (p, '?o)^ = pfi (p) • v^2 (p) • vC's (p) . Again, by Jacobi's theorem, (p, r/o)"= (p", )7o) ^i{p) •■• vJ'm-l(p), also (p. 170)* = {p\ vo) 1^1 (p) h (p) h (p); hence (p. vo) • (pS vo) = p. And, (pi vo)^ = (pS Vo) 4'! (p); (.p, 10)' = (p|, 10) fi (p) • 1^2 (p); 10 ON THE CTCLOTOMIC QUINARY QUIXTIC. and (1, ijo) = 2'j.= — 1. hence (p. 170)^ = — 1^1 (p) 4'2 (p) V^3 (p) 1^4 (p). The x's were defined as follows: x(l)= i^(l)/^^(2); etc. * or x(l)= (P, '7o)V(pS vo); x(2) = (p=, ,o)/(pS 'jo); x(3)= {p\ voT-Kp, vo); x(i) = (pS ';o)/(pS vo). From these x(l)-x(4) = (p, voY-ip', noY/ip', m)-ip', no), which by the foregoing is fi(p) • (po, voY/ 'Piip) • h(.p)h{p) = P'l'i{p)l4'3(p). Again h(p)/hip)=- (P, r)- (pS r)/{p\ r)-(p\ r); the numerator and denominator of this fraction have been shown each -equal to p, hence jD , . x(l) • x(4) = )j; also in passing and the equation is iri'educible, there are found two expressions for p as a quaternary quadratic; viz., p=iAl + Al + Al + AI- (A,Ar + AiAi + A,Ai) = Al + Ai + Al +AI- {Ao A2 + AoAz + Ai A,) the identical relation among the A's being Ao {Al -A2) + Ai{A2-Az) + Az {A, - ^0) = 0. In passing, the product of two (and therefore of any number) of primes of the form 5??: + 1 is written at once Ln this quadratic form, for multiplying either factor (flo, fli, 02, as, 04) of the first, p, with either factor (ba,bi,b2, bs, b^) of the second, g , then the product pq is of the above form, where Ao = Qo (bo — 64) + ai {bt — 63) + 02 (63 — 62) + 04 (61 — 60), Al = ao (61 — ^4) + ai (bo— 63) + 02 (64 — 62) + 04 (62 — 60), A2 = 00(^2 — ^4) + fli (bi — 63) 4-02(60 — 62) 4- 04 (63 — bo), A3 - ao (63 — 64) + fli (&2 - bi) !- 03 (61 — 62) + 04 (64 — bo); and when fl« = Oi — 04,- 6', = 6,-64, 12 • ON THE CYCLOTOMIC QUINARY QUINTIC. the conditions are flo a'l + fl'j "2 + ^2 03 = a'o a\ + 0003 + a'l 03 , ^0 ^1 + ^1 ''i + ^'2 ^3 = ^0 ^2 + ^0 ^3 + ^1 ^3 . A^Ai + ^1^2 + M A3 = A0A2 + A0A3 + A1A3. It is easily seen that there are four possible schemes for the ^'s, corresponding to each of two possible choices of factors of p , g to be multiplied together at first, giving in all, in this way, eight quadratic representations of pq; the process can obviously be extended. Some further properties of p will be taken up in considering the transformation of F (ao, •■•); before returning to F ( ), one simple result on the surface may be obtained. It has just been seen that p may be expressed in the form (ao+aip+a2P^+a3p') (ao-\-aip^-\-aip^-\-a3p-) , where the a's are integers; writing in the values of p", the coefficient of Vs in the simplified result must vanish identically, and the rational part of the product must be equal to p; whence, the values of p" being 4p= V5- l+i^lO+2^; 4:p^= - V5- 1 + WlO- 2V5; 4p4 = V5 - 1 - i^lO + 2V5; 4p' = - VS - 1 - i'^lO- 2^5; the above product becomes 16p = (4ao —01 — 02—03)^+5 (oi — 02 — 03)^+ lOof + 10 (02 — Os)** + 2'V5 { oj — (02 — 03)^ + (4ao — Oi — 02 — 03) (oi — 02 — 03) + 4oi (02 — 03) } - The quantity in { } must vanish identically (when for o's are put their nu- merical values calculated from a given prime lOn + 1 ) , or { j = at once, on writing 4ao (oi — 02 — 03) = 4 (oi 03 — oi 02 — 02 03) and expanding. Hence, if p is a prime of the form lOri + 1 , then always lGp = x'+ 52/" + lOu" + lOr , where x, y,u,v are integers. Example: 16 X 61 = 976 = 1 + 5 X 25 + 40 + 810. Recalling that the oo, • • • , 03 are respectively equal to Aq — At, Ai — Ai, A2 — Ai, Ai ~ Aif as above defined, also that Ao + Ai + Ai + A3 + Ai = p - 2, the o's satisfy Oo + Oi + flj + 03 = P — 2 — 5a<; ON THE CYCLOTOMIC QUINARY QUINTIC. 13 and X = 4ao — Oi — Qi — az; y = ai — ai — a^; u = ai; v — 02 — az. Combining this result with the formulas for the product of two primes of the form 10?i + 1 , it is clear that integral solutions of the equation a;" = M^ _|. 5j,2 ^ 1022 + lOw^ (for X, y, u, V, z, w) may always be found; the method of procedure to be followed has been sufficiently indicated; e. g., for ?/= 7, the simplest method would be to multiply together 7 primes of the form lOre + 1 , first factor of each by first factor of each, and then multiply the result by the conjugate expression as above, putting in the values for p" . Returning now to the form (4), and considering the manner in which it was written down, let {a, b, c, d, g), ( a' , b' , c' , d' , g') be the two complex 5- numbers giving rise to the forms F (ao, • • •) , F {a'o, •• •) , then if (a.b,c,d,g) ■ ia',b',c\d',g') ^(A,B,C,D,G), A = aa' •{■ bg' + cd' + dc' + gb' = ao «» + P (^i «* + "2 ot'3 + as aj + «< «i ) > B = cb'+ ba' + eg' + dd' + gc' = (ao a\ + ai Oq ) v^l + ("2 a'i + en a^ ) 4'i^i + ofsas lAl, C = ac' +bb' + ca! + dg' + gd' = (aoa'i + a2 «„) v^2 + aia\ \Pl + ("3 ai + a4 ofj ) yffi 4'3 I D = ad' + be' + cb' + da' + gg' — (as aj, + ao aj ) fs + (on «2 + "2 "i ) h 4'i + oci a\ \l'l , G = ag' + bd' + cc' + db' + g ' = (ao a\ + ai aj ) vt^ + (ai aj -|- ai a'j )\f'3 V^i + "J a'j ^ • Hence, if B=^,B'; C=4'iC'; D = izD'; G = ^^G' , the table (7) gives the following (12), which will be recognised later as con- taining the necessary results for the duplication (and hence multiplication) of F (ao, • • •); G' = ao a\ + "4 0:0 + Pi ("1 a's + 0:3 a', ) + P2 «; orj D' = ao aj + as a'o + Pi (02 a\ + ai a'^) + pt a^ a\ (12) B' = ao a, + at Oo + P* («4 «2 + «2 «! ) + Ps aj Oj C = ao aj + as aJ, -f Pa ("3 «'« + «< a'j ) + pi ai a. 14 ON THE CYCLOTOMIC QUIXARY QUIXTIC. Whence it is seen that D' , B' , C are obtained from G' by doubling and re- duction, as it is evident a priori that they should be. The remaining sections are concerned with the more elementary properties of the form F {ao, ai, ao, az, a^) which has been constructed, such that 3,125 (a:"- 1) / (x - 1) ^ F; some aspects being considered where the a's have the definite meanings assigned for p = lOra + 1 , prime, and some where the a's are perfectly general quantities. AH formulte deduced in accordance with (7), such as (12), and their consequences, are equally true for all interpretations of p, for they are formal consequences of the definitions of ^ , x • In the next section some simple properties of F ( ) = 1 are considered; this equation is of importance later. § 2. If Ti are the automorphics of F , Sj any transformation of F into F' , all such transformations are included in the group generated by Ti, Sj, say { Ti, Sj } . Denoting by T{u) the effect of the transformation T on u, let Xo, Xi, Xi, Xz, Xi yoi T= 2o, Uo, Wo, Ui ^ao + ai p + Qi p" + az p*' + 04 p*'; T(U,)=Ul, transformations T will be sought which are automorphic for F{ao , aj , aj , as , «< ) 4 U\ = Ea/ (a-, + P* yj + P-' zs + p" wy + p^'' Wj) . By S', a linear transformation, F may be reduced to the canonical form ao «i 0^2 0:3 «4 04 ao ai Oi az az ai ao ci aj az at ao ai ai 02 az at «o and = Fc{ao, ■■• ), ON THE. CYCLOTOMIC QUINARY QUINTIC. 15 and it will be assumed that this reduction has been made. Since in the ex- pansion of Fc, the coefficients of all the highest powers of the variables are unity, it follow that in order that T may be automorphic for Fcj and hence TS' automorphic for F , must r[(a-y+ p'2/; + p-'2,+ p''wy+ p''wi) = 1; 0'= 0,1,2,3,4). These five conditions are necessary; it remains to be seen whether or not they are sufficient. First it is clear that stated in this way the conditions are redundant; for if then / ( z , j ) = a-y + p'' yi + P^' Zj + f?' Uj + p^' Wj , (i=0, •••,4); n/0".^) = ii/(i.2o = n/o\3i) = n/o\4i); i i i i SO that in place of five fundamental equations i = fl/(i,i); (i = o,---,4), there may be put the same equation l = tlfU,i) in five different forms, if n/(i. 50 = n/0"' 0) i i be taken as the representative form f (j , ) •/(;', 1 ) ■ f {j , 2 ) • f U> 4 ) ' f U > ^) ■ It is therefore not necessary to consider five distinct \(f (i, j) JT forms; one only is sufficient; and this is the meaning to be attached to the " redundancy " of the first found five conditions. That the so-found sub- stitution actually transforms Fc into itself is obvious, for the result of the transformation may be written as the direct product of F (a) and n/(), and this product is Fc , since the last factor is unity. Hence finally F is trans- formed into itself by T" S' (S' applied first), where S' is the multiplicative substitution changing F into Fc , and T is of the form just found. The coef- ficients of T may be easily deduced from the equations of transformation. 5 = 1 : T = TT-l -F^ -V' n\ Wo Ml ■Ui Us Ui Ui Wo «1 Ui W3 Ui Ui Wo Wl Wj Ui Us Ui Wo Wl Ml U2 U} Wl Wo 16 ON THE CYCLOTOMIC QTJINARY QUINTIC. The problem is reduced to finding the solutions of Fc (Wo, Ml, W2, Ui, Ui) = 1, The coefficients Ui are a well-known set of numbers, " Fibonacci's Series," according to E. Lucas.* A moment's reflection will show that these numbers (being found), must occur in this problem; if it were a question of p = 7n + 1 and the corresponding 7-ic, the analogous numbers would belong to a recurring series whose scale of relation is (irreducible) of the third degree (or '"' order "). Putting ijo = P + P*; '71= p" + p' (as always), then Vo = 71 + 2 = 1 — jjo; >?? = 1 — ii; etc., whence 10 = 'lo; 10= — 10 + 1; lo = 2j7o — 1; r)l= — 3j?o + 2; jjo = 5?7o — 3, • - • , the coefficients being Fibonacci's numbers, and the general term in the series (found as usual from the solution of a linear difference equation),! is { ( 1 + a'5 )" - ( 1 - V5 )"} / 2" V5, whence Writing €„,. n= p^ vl for m constant, e^, „ is a linear function of ;;o , m with positive integral coefficients or of 170 or jji alone with integral coefficients, and. if n is fixed, «„, „ is of period 5 with respect to m . Putting ;„^ (-)-!{(! + V5)'- (1-V5yi/2W, in the notation of complex numbers used previously, and denoting by e^ ^ the values of p^n\> then <0,n=(;in-l, Mn> 0, 0, Iht ) '■ «0»n= (/in-1, 0, Hn, 1^. ; 0) <1, n = (/in, A^-1, /In, 0, ) : *'l,n=(0, /in-1, 0, /In, fin} «2,n=(0, /!„, /i„_l, Mn, ): <2,n=(Mn, 0, Mn-1, 0, ^) •Amer. Jour., I., "Th(?orie des fonctions numi^riqucs simplement p^riodiques." t Or more simply by Lagrange's method, " Sur les suites rdcurrentes," e u v r e s , t. 7. ON THE CYCLOTOMIC QUINARY QUINTIC. 17 «3,n=(0, 0, ll„, /In-l, t^n ) - «3.n=(Mn> M»> 0, fln-l, 0) <4jn=(;tn, 0, 0, /i„, JU„_l) : «4.n=(0, /:„, ;tn. 0, /tn-l). The ;i's are connected by the following relations:* a + 6=l; ah= — l, ;:„ = (o" - 6")/ (a — 6); ^.+2 = /v+i + /^; Po = 0, ^1=1. For n = 0, 1, 2, 3, 4, 5, 6, •••, /i„ = 0, 1, 1, 2, 3, 5, 8, ••• that is, (In is the nth " Fibonacci's number." If two or more T's are to be multiplied together, the formula,t Ps - M»-i J^+i = (- 1)""^ may be used to effect reductions. Or, among many others the n's may be used in the form v'osm(^2^°^r::^j- Now whence /(p)-/(p')-/(p')-/(p^) = C«ovi)'"=i; and 2m» + Mn-l = A^ + (;in + ;in_l ) = l^n+i', also '?0 = Mt. IJO + /ln-1 . Therefore it is easy to see that if Mo = t^^lMf^+i', Ml = "4 = Pn/V/i„+2; M2 = M3 = 0, ^e(Wo, Ml, tt2, W3, ''< ) = 1, and the u's are the required elements in T , where fin is the nth Fibonacci number. Also, it is clear that the question of automorphics is identical with that of finding the fundamental units in the field (p) , and that the above implicitly contains a solution of this problem; but the units are also easily proved to be p" jjS* independently, which is done presently. The above coefficients in •Lucas, Amer. Jour., I, p. 184 eqq. t Ibid., p. 196, Nr. 30. Jlbid., Nr. 5. 18 ON THE CYCLOTOMIC QUIXARY QUINTIC. T are not in general rational, (or integral) ; when these conditions are imposed on T the automorphics are shown later to exist by a method due to Dirichlet. At present another aspect of F ( ) = 1 , analogous to the foregoing is con- sidered; and it is asked, what functions serve to give solutions of f ( ) = 1 in the same manner that sinh w and cosh u furnish those of the Pellian equation? A general case is considered. Let a;" — pi x"~^ 4- p2 a;""' + • • • =^ po = be irreducible in the domain of rational numbers, its coefficients also being in that domain, its roots ai, • • • , «„; then anj"- rational function of a of degree m , where a is any root, may be reduced by means of the equation, to the form Zo + 2iq:+ ••• + Zn-iaT^, m < n . Putting Zi= Zo+ Ziai+ ••• + Zn-i o^"' , then since only symmetric functions of the roots enter in N^ILZi, it follows that N is a, (homogeneous) function of the s's with rational integral functions of the p's as coefficients; N is in fact the norm of the algebraic integer ( zo , zi , • • • > 2n-i ) , and is of the form N ^ (co,Ci, •••) (zo,2i, ••• 2„_i)" = F (Za, Zi, ••• Z„-i). Let it be required to determine the 2,- so that F(z9, ••■ z„-i) = 1; a perfectly general solution will be given if it is possible to determine At such that Zo + ziai + Zi<4+ •■• + 2„_i al'^ = Ao , 20 + Zi as + 2; ccl + • • • + 2n_i aj"' — A\, , 2o + 2l an + 22 a^ + • • • + 2„_i 0^~^ = An-l , AoAi ■ • • An-\ = 1. If the Ai may be chosen to fulfil the last condition, then c, may be found to satisfy F(2o, ••• z„_i) = 1; for, the determinant cannot vanish unless at least two a.-, ay are equal, which ON THE CYCLOTOMIC QDINARY QUINTIC. 19 contradicts the hypothesis that the fundamental equation is irreducible. Hence if the Ai are determined as indicated, so also are the z,-. It is assumed that all the Ai are finite, hence no one of them can be zero. The equation AoAi ••', A„-i = 1 is indeterminate to any degree; the simplest solution in which the restriction that the Ai shall be integers is imposed, is that in which the At are nth. roots of unity; hence this hypothesis is chosen. Moreover the Ai are to be distinct roots of unity, so that no condition may be imposed tacitly on the fundamental equation. This simplest solution gives rise to nl possibilities for the z,-, for the roots 1 , p , • • • p"~^ of p" — 1 = may be permuted in any way to correspond to Ao, Ai, •■• , An-i. But more generally, if / (0) , / ( 1 ) , • • • , fin— 1) are any functions whatever whose sum is zero, then Ai= /;^'«, (i= 0, ■■■,n- 1) furnishes a solution of ^(zo, •••)= 1. The choice of / ( ) is subject only to the condition S/(i) = 0; k is any finite non-zero constant. The function f (i) may be conveniently chosen as a complex factor of a real integer in the domain of nth roots of unity; one reason for such a choice being that the resulting /'s lead to a simple parametric representation of F ( ) = 1 in n-space, analogous to the parametric solution of the Pellian equation, with which the closest possible analogy is being sought. Choose therefore, with no loss of generality, k ^ e, the base of the Napierian logarithms. If w is a prime, any root p of P"-1 = 0, except p = 1 is selected, if n is composite, a primitive root. Choose the " coordinates " Xy, and/ (i) so that / (i) = Xo + Xi P-' + X, p=' + . • . +Xn-i pf"-"'. As i ranges from to n — I ,f (i) takes n distinct values; also I:p*' = 0; whence /(0 = E(Xi- Xo)p*'; 20 ON THE CTCLOTOlilC QUINARY QUINTIC. or changing the notation, this may be written in the form ii-i Also whence or say for *=o 11—1 «— I / n-l \ Z/(0 = Z X;E(P')'' =0, ■^i — V; ( Xl , Xz , • • • , Xn-l ) , *=0 In this the Aj are considered as functions of ra -^ 1 parameters Xjt; from the form of

    ;•( Xi , X2 ,•••, Xn ) , where the m are the periods; from the form of in order that the /j,- should be periods, it is necessary and sufficient that for some systems, not all zeros of the n's, the following equations shall hold: Ml + M2 + • • • + Atn-i = 2reo Tri p/il + P^ AI2 + • • • + P""' /in-l = 2wi wi - (i^V-l). P"-' m + p'<"-" t^2+ ■■• + P^"-"' M«-i = 2n^i « J To find /!,• multiply the equations in turn by 1 / 1 , 1 / p' , 1 / p'' , ••■,!/ p^"""' , and add; since p is a primitive root, all the coefficients in the result except that of fij vanish, hence Uftj = 2wo Tri-j- 2ni tt,/ p' + 27J2 «'/ p'' + • • • or ntij = 2wi ( p^"' ki+ ••■ + p''-("-»'- i-„_i ) ; where ki ~ rii — TJc, etc. For reasons of convenience that are made evident in the case of ti = 5 , the k's are now replaced by their values deduced from i'l + A"2 + • • • + ^'n-i — (mod n ) . It is clear that every solution of this congruence furnishes a set of periods. In the case n = 5 it is shown that all the sets so furnished are not distinct from a simple set, to be determined; non-simple sets being derivable from ON THE CTCLOTOMIC QUINART QUINTIC. 21 simple sets by multiplications. Not seeking here tie statement of the general analogous result, the simplest solution of the above congruence is (mod n). whence, denoting by (oq, • • ■ , a„_i) the complex number Oo + «! P + * * • + a„_i p"~'; the periods are given by n^y = 2«(±1, 0, 0, •••, =f1, 0, 0, •••)• Regarding now the particular case of the foregoing for n — 5, let (xo, xi, Xi, Zi, Xi) denote a point in space of the appropriate number of di- mensions, whose coordinates are ^0 = =^ 1 , =^1. • ±1 il = ±l, 0, • h = 0, =^1, • G kn-l ^ 0, 0, •• =f1 a:i; Fixa, •■•, Z4) = 1 is the equation of a particular surface in that space, F having always the sig- nification of § 1, the coordinates of any point on whose surface may be expressed as periodic functions of four parameters, as follows: If Xi = ^ (Xo -H p' Zi + p'' Xi + p'' Xz + p*' X,), (i = 0, .4); then when and only when F{zo, ••-, Xi) = 1 Xo Xi Xi A3 Xi = 1 (1) as is seen at once from equations (11) of § 1. Particular solutions of (1) are evidently 1); (Xo, ■••, Z4) = (i, p, p^ p', p^); or =(l,l,p^p^ or = (1, p, 1, 1, p'); or = (1, 1, 1, 1, 1); of which the first ■•• fourth give respectively 120, 20, 20, 1 solutions, by permutations; these special solutions are seen later to correspond to " planes," " vertices " of the " pentahedroid of reference " in S^ . But these are all special values of c^"'"'', and a general form to which they all belong is A^^''> , where /(i) is so chosen that i:/(t) = o. There is no loss in generality in choosing A ^ e, which is done in order to facilitate expansions, should such be wished. 22 ON THE CYCLOTOMIC QUINARY QUINTIC. Thus where 5iriXi = 2p"'e/f">; Z/(.0 = o. (2) A general linear solution, in terms of 4 parameters, of (2) is /(O = p'Xi+P='X2+p"X3+p*'X4; (i = 0. •■-.4) (3) or if the X's are to be put in evidence, /,• ( Xi , Xj , X3 , X4 ) . Writing ef^''> = Xi, or s Xi (Xi, X3, X3, X4), 5a;o = Xo + Xi + X2 + A's + Xu 5iri a;i = Xo + p* -^i + p' ^2 + p^ X3 + pX4 5T2a:2 = Xb+ p'Xi+ pX2+ p^X3+ p^X4 5t3K3 = Xo + p^'Xi + p^Xi + pXs + p=' X4 5^4x4 = Xo+ pXi + p^Xa + p'Xj + p*X4 J Multiplying in turn by p', p^*, p'', p*'", (i = 0, — , 4), and adding, (Xo, TTlZl, TTiXi, Tsa-J, T4.T4) 1111 (4) (5) So — (Xi, Xi, X2, X4, X3). (5^"' 5^/^' 5;rj^" 5;^^" 5^^V' is a solution of F {Xo, Xi, Xi, X3, Xi) = I. The Xi are periodic in the same way that e^ is periodic, viz., gt _- gl+2n«r». the Xi being functions of the Xi, it is required to express their periodicity with respect to the Xi; viz., quantities m must be found so that /i(Xi + w. X2 + M2, Xj + ^Js, X4 + ^'4) = /• (Xi, Xj, X3, X4) ON THE CYCLOTOMIC QUINARY QUINTIC. 23 with the condition Pi + m + 1^3 + /ii = 2niri . By (3) this requires P' Ml + P^M2 + P" /K3 + P" = 27lTi, where n is a real integer. Hence the m.- must be functions of p such that the left member is independent of p , and hence n,, must contain p^'"; {n = 1 , • • • , 4) . Or, using the condition that n must be real, there are four equations to find the periods: W + w + /'a + Mi= 2no -ri Pfii + p'^M2+ p' f3 + p*Mi= 2ni Trt p* Ml + p* /i2 + PM3 + p' M4 = 2ji2 Trt \ whence p* Ml + p' M2 + p* Ms + p;i4 = 2714 TX p' Ml + PM2 + p'' P3 + P^P4 = 27^3 ■^ , It will be seen presently that if p is a real prime of the form 5re + 1 , two (conjugate) resolutions of p into complex p-f actors furnish a set of periods; indeed this is evident from § 1. The above forms for the periods, on elim- inating p* , are reduced to 5mi = 2xi ( no jTii, Ks , 7i2 , ni ) 5a12 = 25rt' (Ko , «2 , «4 » Wl , 7l3 , ) 5m3 = 2x1 ( 7!o , 7l3 , Kl ,ni,n'i) bin = 2Tn (no , Til , rii , Tis , Hi) , 2rt , , , • n\ Ml) M2, M3, Mi — -c-(n4> «3» "2. ^i, U;; 2rt ("2,«4> k'i, n'3, 0); 2« , , . ■ n\ — («3, 72i, 7?4, 7l2, 0), 2xi (n's, ??2, n-j, n'i, 0) respectively, and since the ( ) must each furnish a multiple of 5 (otherwise there is no period), there must be the congruence or writing what is the same thing. "1 + n'2+ ti3 + n\ = (mod 5) , ni + Ui + ns + Hi = n , 4no — 71 = (mod 5) , which has the solutions nfr=0, 1, 2, 3, 4; n = 0, i, 3, 2, 1, («0, 71) = (0, I, II, III, IV), say which give: (I) no^l; and (IV) Tio = - 1; n = l n = — 1; no 1, ^h ^1, 1 n2^ 0, =F 1, 0, 7l3 = 0, 0, =F 1, Ui = 0, 0, 0, =P 1. 24 (11) no = 2; and (III) no = - 2; n = 2 ON THE CYCLOTOMIC QtriNARY QUINTIC. n s - 2; no = ± 2, =fc 2, ± 2, ± 2, ±2, ± 2, ± 2, ± 2, ± 2,± 2. 7ms=f2, 0, 0, 0, =F 1, =f1, =1= 1, 0, 0, 0, =f2. "2 ns 0, 0, =p2. 0, 0, =f1, 0, 0, =f1, =i=l, 2, 0, 0,=i=l, 0, =^1, 0,=f1 0, =p2, 0, 0,=p1, 0,=f1,=p1. n^s 0, 0, (all to modulus 5). In each set of 5 the top signs throughout go with the first written congruence; the upper and lower signs are distinct, viz., no value of fii deduced from an upper line can go with any m deduced from a lower line. From these are deduced 2S sets of values for the periods [5m/2in, 5}i2/^Tn, djis/^wi, 5fnl2iri] = l(no, Ui, Us, n2, ni); (no, nz, n*, ni, ns); (7io, ^3, ni, rii, ria); (no, ni, n2, riz, nt)], but it may be easily verified at once that of these 28 only 4 are distinct modulo 5; that is, all other periods are derivable from these 4 by additions of two or more of the 4; then by additions of the results, and the originals, etc. In the subjoined table, each column gives one of the four-sets of primitive periods. (2tz75). (1112) (1121) (1211) (2111) (1121) (2111) (1112) (1211) (1211) (1112) (2111) (1121) (2111) (1211) (1121) (1112), (mod 5). The mode of formation is apparent; the first complex number (1112) is written down, being a simplest number satisfying all the conditions; the remaining numbers in the same columns are derived from the first by cyclical permutations; the three so-found numbers become the first numbers in the remaining columns, whose remaining numbers are obtained as in the first column. Had any other (non-primitive) period number, e. g., (2, 2, 2, 4) been given, all the 4 periods could have been derived as indicated. Here ON THE CTCLOTOMIC QUINARY QUINTIC. 25 (abed) means ap + bp^ -\- cp^ + dp* . The four complex numbers in a column constitute a set of periods, all the columns the four primitive sets. E, g., from the first column, the values of the m's are: Mi= (1112) . 2«/5; ,i2= (1121) • 27rz75; /i, = (1211) • 27rj7 5; /X4= (2111) • 27rt/5. or Ml = (P + p" + p' + 2p*) 2«7 5; fii = etc. The above may be expressed in terms of the prime p-factors of 5, which are also of importance in the finding of units. Throughout these factors pi, Pi, pz, Pi of 5 are exceptional, playing the same role as 1 — i in the theory of primary- prime factors of o + ti numbers. Writing pi , ps , pj , pi for p^ — 1 , p' — 1 , P* — 1 , P ~ 1 respectively, the table of periods becomes Pi Pi Pi Pi (2«75) Pi Pi Pi Pi (mod 5). Pi Pi Pi Pi Pi Pi PZ pi . or finally. Hi/2ri fiif2n M3/2:rt inf 2iri' i/ Pi Pi Pi 1 / Pi P2 Pi 1 / Pi P3 Pi If Pi Pi Pi 1/pl pi Pi If Pi Pi Pi if Pi Pi Pz if Pi Pi Pi = [moI (mod 5). 1 / Pi P3 P4 If Pi Pi P3 If Pi Pi Pi if Pi Pi Pi i/p2 P3P4 if Pi Pi Pi If Pi Pi Pi if Pi Pi Pi Hence ^3 ( ^1 > ^2 1 Xs > \i) = Xj(\i-{- Sim, y^i + snii, X3 + 5/X13, Xi + ^Mu) — Xj ( Xi + 5' A(2I , Xt+sVzi) = = Xj ( Xi + s" nn , X4+5"/i34) = Zy(Xi + «"V41, \i + s"' ^U). or. (;■ = 0, 1, 2, 3, 4; s, a', s", s'" any integers.) Xj ( Xi , X2 , X3 , X4 ) = Xj ( Xi + sixii + s' p.ii + s" 1131 + s'" im , X2 + Sfin + s'fiii + s" P35 + s'" nn , X3 + s/iu + 5' ,^23 + s" na + s'" im , X4 + 5^14 + S' ll2i + S" H3i + S'" /X44 ) , for all integral values of the s's; which completes the setting forth of the A'j^ ) 26 ON THE CYCLOTOMIC QUINARY QUINTIC. as periodic functions, and hence the expression of the coordinates of a point on f = 1 as was required. Not pursuing the study of these functions here in detail, it is merely noted in passing that all may be expressed in terms of four simpler functions; viz., if any or all of the X's = 0, there ^nll still be solutions of F () = 1; also, since that is X,-(Xi,X2, Xs, X4) = X,(Xi, 0,0,0) -.Z.CO, X2,0,0) • .Y.(0,0, X3,0) X:{0, 0, 0, X4). From this, and allied relations, it is possible to express the Xi algebraically in terms of the simpler incomplete functions of a single argument. Again X ("Si, X2 , X3 , X4 ) = X ( Xi — oi , X2 , — 02 , X3 — 03 , X4 — 04) • X (Oi, 02, 03, Oi) . The addition theorem may be deduced without any difficulty; abstractly it is identical with the theorem which permits the duplication oi F {) , and the formulae requisite for either implicitly contain the other. Reverting now to the equation F = 1 , it is clear that the problem of iso- morphics (as treated in this section), and that of finding the fundamental units in [ p] , th« field of numbers constructed from complex fifth roots of unity, are identical. It is the form of this equation F = 1, and the necessary manner of its solution, that enables the present forms to be included strictly with the two correspondents 4 (a;"- l)(x- 1) = Y(xy- (- l)'^'' Zix)\ and 27 (x" — 1) / {x — 1) =w^+ p2h 1^ + ppi u^ + Spici'u . These forms of degrees 2, 3, 5 could all doubtless be considered, — as all possible forms of a " Kreistheilungskorper " — at once, but the general method or theori' is sometimes helped by the use of elementary methods. In the theory of any such forms, the solution of Fe = 1 (or the corresponding equation) is fundamental, aiid is identical abstractly with the determination of a system (if more than a finite number exist) of fundamental units for the field consid- ered, that is for ( x" — l)/(z— 1 ) for a given prime J) . For p = 3 the number of fundamental units is finite; the units are ± 1 , ± u, ± u-, co an imaginary cube root of unity. It p = 5 it is easy to show (see below) that there is but a single " system " of fundamental units that is independent; i. e., in terms of whkh all the other units can be expressed. But, as proved by Dirichlet, if p > 5 there is an infinite number of such fundamental systems. For ON THE CTCLOTOMIC QDIN.VRY QUINTIC. 27 p= 5 (see below) the single system is ± p^tiJ; ijo = p+ p*- Thus for p ^ 5 there is an essential distinction from the classes p > 7; the case p = 5 is intermediate between two very different species, viz., p < 3 and p > 7; in the first fall those forms which depend on a finite number of units, in the second those whose complete expression requires the use of an infinite number of systems of fundamental units, each system containing an infinity of units; the " dividing case " p = 5 is a link between the two others, for the number of systems is finite (one), and the number of units infinite. Nor does the sequence of coefficients in the case p = v admit of a simple explicit form for the automorphics, as was pointed out. In short, from this standpoint simplicity ends once the case p = 5 is passed. It may be worth while to indicate another reason for the detailed consideration of these and allied forms by strictly ele- mentary methods. Denoting by any law (or process) of combination, it is a definite question to ask what forms fijft, fs of the same kind, that is, on the same numbers of variables, and of the same degree, e. g., the/'s may be three m-ary 7i-ics, satisfy the relation 0(Ji,f,)=fz. If is X (multiplication), and the domain of the coefficients identical with the rational domain, the question is completely answered by Gauss' theory of composition, if the /'s are binary quadratics; if the forms are m-ary m-ics, Ka-GRAnge has given sufficient but not as yet proved necessary conditions, (0 still being X ) • In addition to these well-known solutions there are a few isolated examples of the same kind; e. g., Eisexstein's theorem on the trans- formation of the discriminant of the binary cubic into its third power; a result to which he attached great importance, especially in his triplication of class problems in the theory of binarj' cubics. It is possible to regard the arithmetical theory of forms from the standpoint of the operation (not necessarily multi- plication); clearly such a view presents new problems and interests, especially in the domain of quasi-algebraical number imaginaries. From this basis, the simplest forms arising for = X are the 2-ary 2-ic3; if either the degree or the order be independently extended, the theorj' becomes at each step more special. Thus for quadratic forms of 3 , 6 , 7 or more than S indetcrminates the composition-theory in any general form is non-existent, so that one point of analogj' is lost by merely extending the order; but if the order is constant ( = 2 ) , and the degree be extended, any theorj- becomes special to the vanish- ing point. But if both order and degree be simultaneously extended, complete analogj' is restored, and as the special form i^ + y^ was the basis (historically) for the composition theory of quadratics, so the 0-forms here are the guides, and the special theory for at least algebraic numbers of the nth degree, n being the order of the form, is approached, 'W'here = X the forms considered 28 ON THE CYCLOTOMIC QUIX.VRY QUINTIC. distribute themselves according to degree (and order), so that it is only neces- sary to consider the cases of prime degree. The properties that are easily approachable of such forms are in direct analogy to those of binary quadratics, owing to the existence of an analogue for the Pellian equation. The question of units, etc., is briefly considered in the next section (end) ; first the " divisors of F," and allied questions are taken up. In § 1, F was derived in the form F (oro, '•• , cii) = iolih^zU, where ^0 = (ao+ s'iai+ 'rsa2+ r-s aj -}" TTiOH, 0, 0, 0, 0); li = (ao, iriQii, TriaifTzaz, irioj); f 2 = (oro, Tsas, Tiai, xiofi, tjotj); f J = (oro, TTiOi, ■K^a^, xiai, rjofs); It = (ao, ""404, Traaj, X2a2j nai); where, e. g., to = a'+Ti(6' + c'p* + +pxi+ p= Z2 + p' a;3 + Pi 3^ is prime to p (and hence to pi and pz » for p = pi pz ) j and further let {/(p)P = PP?P2 (modg) which implies that ppl pn is a quintic residue of q . The congruence is equiva- lent to the following, that IS {/(p)-Ti)(/(p)-pn)(/(p)-p'Ti)(/(p)-p'n)(/(p)-p*n) = (mod q); say ffo (p) ■ gi(p) • giip) • gi{p) • gtip) = 0; that is, say, G(p)=0. and therefore by the usual theory, G(p)-(?(p')-G(p^)-G(p^) = 0. This last is the product of 5 factors, i = 0, 1, 2, 3, 4; (/(p) - P'Vi) (/(p^) - p«tO (/(p') - (p'-Va) (/(p^) - p«T4) = [i], say; consider now [0]. If (/(p)-^i)(/(pM-r4)=/'(p), then (/(p')-T2)(/(p')-7r3)=/'(p2), and IO]=/'(p)./'(p^); also f'(p)=f(p)fip*)-Tif{p*)- rif(p) + p, •Crelle, 30; 107 sq. t Ibid., p. 115. 30 ON THE CYCLOrOMIC QUINARY QUINTIC. whence [0] is of the form X + wi Xi + ^rj Xn + irj Xz +^4X4, where the X's are to be examined more closely. It is evident that {f {p) ^' {p^)) Xo= {f(p)-fip*) + p] ' {/(p^)-/(p') + 2>} Xi = pJip')-fip)-f(p*){fip-^)-fi(?) + p] ^i = P3f(p)S{p')-f(p'){f(p)-fip') + p] Xi = P2fip')f{p')-f(p') {/(P) ■fip') + p] X* = Pif (P*) ■ f (p') - f (p) {f (p')f ip') + p} , say Xi = h{p), then, as is evident a priori by doubling and reduction, Xi = h(p); Xz=-h(pn; X2=h(pn; Xi = h{p'), and Xo is a real integer, = {a + pno + yvi) {a + yrjo + firji) + p^ + p + p(2a— /3— y), (see below f or a , P, y) . Hence IIo [i] is of the form F (Xo, ■■• , Xi), where the Xo, • • • , X4 are functions of the same kind^the ao , • • • , ai'm F {ao, • • • , 04 ) ; for, as in i'' (a ) each of the variables a is a complex 5-number with real coefficients, and all four contain the same coefficients in different cyclic orders, so here the h(p)> •'• > h(p*) clearly satisfy all the similar conditions. Hence G (,p) • G (p'^) • G (p^) G (p*) is identical in all respects of form to F (a) and it has been so constructed that it is divisible by q. Writing for the moment f(p')^\, the above expressions for Xo, •••, X4 become respectively ai 02 03 at + p (0104 + 0203) + p^; PiaiUi — a^aiai — pui, ps ai a2 — 03 ai Oi — pat Pi 0.3 tti — Oi 02 Oi — pui', pi 02 04 — oi 02 03 — pa\ . Now these are all func- tions built up by multiplications by constants and additions of the results, of the essential variables in the G-form, whence if these essential variables have any common divisor different from unity, the above functions are also divisible by this common divisor. From these functions, by obvious multiplications and additions are constructed the following: Pi az a\ + pai 03 + p^; 2>2 04 of + pa^. 04 + p'; Pz di al + pai 04 + p"^; p\ 02 al + pan oj + p^ Let now oi, 02, 03, 04 be conjugate complex factors of a real number A^ , that is Oi 04 = 02 O3 = A, which is necessitated by the assumed congruence of = 5rf , and the thereby ON THE CYCLOTOMIC QUINARY QUINTIC. 31 determined three remaining congruences a% = vl; 0.1 = ^1; a\ ^ tj\ {moA. q) , and it be assumed as is legitimate on the assumption already made that ai is prime to -ppl pt , that A^ and p^ are relatively prime, then, by additions, etc., of the last found four expressions, the common divisors of the essential variables are all common divisors of pi 02 of; p2 04 of; Pi 13 ol, Ps ai a', and by what immediately precedes, these last are relatively prime. Therefore the essential variables in the G-form give a proper, i. e., primitive, representation of q in the form F . In other words, all letters having the same significance as above, the necessary and sufficient conditions that o be a divisor of the form F are //^7? Tz-f2^ jr^B pni^ any one of which entrains the others, and { ra / n } is the quintic character of m with respect to n. The actual reduction of G to the form F , and hence the primitive repre- sentation of an appropriate number, i. e., one satisfying the congruence con- ditions, is readily effected by means of the formulae for multiplication of 6-numbers, etc., given in § 1 or § 3; or the writing of G in the required form may be put forth db initio; it is however less laborious to write down the re- sults in accordance with § 3. As shown above, [0] is of the form X,o + tti Zi -(- t2 X2 -}- ttj X3 -|- ti Xt; and G is then F (Xo, Xi, X2, Xz, Xi);it is required to find the forms of the X's which must not contain any tt's explicitly, and must be as proved, with the exception of Xo which is real, complex numbers of the forms 6 -f- cp* + rfp' + gp^; b + cp^ + dp + gp*; b + cp'^ + dp* + gp; b + cp + dp'' + gp^ . Now /(/')-/(p')=«+P^+p'r+p'7+pV, fip') •/(p=') = a+PT'+p''^+p'/3+/r, a s xl + xl + xl+ 4 + xl; fi ~ XoXt -|- Xl X2 + X2 X3 + X3Xi + xo xi; y = Xo X3 + Xl Xl + X2 cci + 23 Xl + Xo X2 . Hence {/(p)/(p') + p}{/(p')-/(p') + p} = (a+ firio+ ym) ia+ 7'?0 + 0Vi) + p' + p (2a - /3 — 7) , which is a real integer; say I . Put now To- 4+XiX2+X2Xi + Xz Xl + X4 X3; r3 = Xo .T2 + Xi X4 + X; Xi + xl + X4 Xo; hence where 32 ON THE CYCLOTOMIC QUINARY QUINTIC. fi = XoXi + xl + X2 xz + ars a;o + Xt x^; u = xo xi + a:i 3:3 + X2 Xq + 2:3 a-j + yj; r2 = a;o 2:3 + a;i aro + arf + ars 3:4 + x, a;i; Pi = So + si p + S2 p^ + S3 p' + 54 p*; and -X'o = ro(5o —Xo) + ri{Si — a:s) + Tiisz — Xi) + rj(j2 — a;4)+r4(5i— a;2 )— paro, ^i = ro(si — X3) + ri(so — a;i) + r2(si — X2) + tjC^s — a:4)+r4(52— a:o)— pa-i, T2 = ro{s2 — Xi) + ri{si — Xi) + r2{so — X2) + r}{Si — xa)+ri{s3—Xz) — fX2 , X'i = ro(*3 — 3:4) + ri(*2 - a;2) + r^^si — Jo) + rsC^o — a;3)+r4(54— a;i)-2)a-3, XI = ra{Si - 2:2) + ri{Si ~ Xq) -\- r2{si — Xi) + T^isi — x{)-\-ri{s^-xC)-V^\', It may be easily verified that the coefficient of 7r4 in [ ] is ( Z; , X\ , Zj , X^ , Z; ) , a complex 5-number -ftith real coefficients, for the actual coefficient is ^(p) = Pt/(p^) •/(P^)-/(P) {/(P^) •/(p') + p}. The coefficients of jra, V2, tx are F(p^), II ip^), II {p*) respectively, or (ZqjZs, Aj, Z4, Zj); (Zq, Zj, Z4, Zj, Zj); (Zo, Z4, Zj, Z2, Z'l) , and the absolute term is I. Finally, putting 6 , c, J , ^ for Xp — Z; ; Z'j - Z^; ZJ — Zi;Zi — Z4, then [0] becomes 7+ {b,0 ,g,d,c) ti+ {b,d,0,c,g)T2 + (b, g, c, 0, d) 7r3+ (b, c, d, g, 0) Tn; or say [0] = ao + ai Tti + a2 T2 + 0:3 xj + 0:4 T4, then by the preceding parts G is F (ao, ai, orz, as, cii); that is the number 5 is re- presented properly in F by a system of variables of the same nature as those in the original F. In this connection note the distinction between F(a) and F (fi) , where in the first case the variables a are restricted to be five very special numbers, viz., ao a real integer and aj , • • - , a4 four complex 5-numbers conjugate in pairs, and not independent of ao; whereas in F (/3) the variables may be any quantities whatever. It is for this reason that b, c, d, g (and ao) are called the essential variables. When attention is fixed on the essentia], and not on the a-variables, the form obviously has a wider significance than its connection with the cyclotomic function out of which it arose, although in many respects the formal aspects of the two interpretations are identical. When F = k is considered as a locus of points in 5-space, the two interpreta- tions arc united; all possible systems of values satisfying the equation represent points on the surface, whereas only those points which belong to a certain 5-fold net-work (corresponding to the nets of unit squares in the plane, and of unit cubes in ordinary space) are relevant to F = k considered as a cyclo- tomic function. ON THE CYCLOTOMIC QUINARY QUINTIC. 33 Some further properties of the numbers occurring in the above are now considered. A " wnumber " is defined to be a linear function of vi, t^, va, T4 in which the coefficients are rational complex numbers constructed with fifth roots of unity, a special case being that in which all the coefficients are real rational numbers. It will now be shown that no ir-number with non-zero coefficients can vanish; or what is the same thing that iri, ttj, tz, -iti are linearly inde- pendent in ( p ) . Throughout, the fifth roots are to be so chosen that ""l S"4 = T2 ffS = P . The jr's are given by ir\ - pp\ Pi = 0; rl- ppl Pi = 0; ■^l - ppl Pi = 0; 7rJ-pp|p3 From these _I0 _ „« _5. _I0 _ ~.B ^S. ^1 — Pi ^2> ^i — Pi T'it whence < = Pi t|; _10 _ .-.B _6 ^3 — Pz^l' »Jt _16 _ „5 _10 Ps Tl — Pi ^3 and xi=p*p«7r|, ■Ki — Pi T2 — Pi Pi TTi , so that the ir-number N = a+bri + c' ir2 + d'ir3 + g'Ti= a+ bwi + — vl-\ tt? + -r" ^i Pi P1P2 PI Pi s o + Stti + CffJ + dTrl + JttJ , say, where, a, • • •, ^ are rational. Hence / = — ppf ps is a common root of a^ +/ = and a + hx + C3? + do? + gz'^ — . Eliminating successively the highest powers and absolute terms from the two equations, and repeating two relations with rational coefficients are found, R£+Sx+ T=0; R'x'+S'x+ T' = 0; the R, •• • , T' being rational integral functions of a , •••,/. Eliminating s?, iSR'- S'R)x+ {TR'- T'R) = 0, whence x being finite and determinate, neither of the () = 0,andarisa rational function of a, ■••,/. Hence VppJ p2 is rational, that is '^plpl is rational which is impossible, since pi , p2 belong to different resolutions of p into factors. Therefore N =^ 0; but N was any x-number with non-zero coefficients; hence the result is established. From this it follows that if two 7r-numbers are equal then they are identical; for let a+ biri+ • • • + girt = a' + b' TTi + •• • + g' ivi, 34 ON THE CrCLOTOMIC QUINARY QUINTIC. then (a-a')+ ■■■ + ig-9')7r, = 0, which is impossible unless a = a', ■•' , 9 = g'. Now, if possible let F (a) be resolved in two distinct ways into the product of five ;r-numbers. The coefHcients of the r's in the various factors are independent of the it's, hence any factor in either resolution cannot be the product of two factors in the other, for if it were such a product, this factor could be written in a form in which the coefficients of the r's would not be independent of the it's . Hence in some order the factors of the first resolution are equal, each to each, to the factors in the second resolution. Therefore, as has just been seen, the factors, in the two resolutions are identical. Hence F {a) is uniquely resoluble into the product of five ir-numbers. A further definition is now needed; the 7r-number a + ri/ (p) + ^2/ (p^) + ""s/Cp') + ■!^if (p*) in which a and the coefficients b, c, d, g in f(p)^b + cp + dp' + gp' + ep' are real rational numbers, is called a rational Tr-7iuviber; if in addition the coefficients are all integers, it is called a T-infeger. It will now be shown that F (a) is the product of five Tr-integers. For, bp + c + dp' + gp'^ ic-b)+{d-b)p'+ig- b)p'- bp^; bp^ + cp + d + gp' =(d-c) + (g-e)p'- cp' + (b - c)p^; bp' + cp^+dp + g^ig-d)-dp'+{b- d)p' + (c - cZ)p=; bp* + cp' + dp' + gp=-g+ib-g)p'+{c-g)p'-i-{d-g)p'. But F (a) =/o(a, b, c, d, g) •■■fiia, ■•■ , g), where /o ( ) = a + ^1 (6 + cp*+ dp' + gp") + tto (6 + cp' + dp-^ gp') + 7r3 (& + cp" + dp' + gp) + 7r4 (6 -f cp + dp-" + ?p') , and /i ( ), • • • ,/4 ( ) are obtained from this by changing ti, t^, tj, r^ into pri , P^ TTt, p' T3 , p* Wi] p' ri , p^ TTo , pVi , p^ Ti] p' Ti, pTTi, p^ ITS, p^ T4; p^ xi, p' a-2, p' Tz, pr^; respectively. Hence F (a) is the product of/o(a, b, c, d, g); fo{a, —g, b — g, c — g, g — d); fo(a, g — d, — d, b — d, c — d); fo{a,d—c, g- c, - c, b- c); fo(.a,c-b,d-b,g-b,- b); and by definition /o ( ) is a TT-integer. Clearly, the product of any number of Trrintegers is a r-integer, as is also ON THE CYCLOTOMIC QUINARY QUINTIC. 35 the sum or the difference. Hence any algebraic function of 7r-integers is a s--number, for it is easy to see that the quotient of two 7r-integers is a tt- number. ' In certain cases, to be examined presently, the quotient of one TT-integer by another is again a Tr-integer. If TTi and T2 are two ;r-Integers such that ti / ^2 is again a ir-integer, then vi is defined to be a T-unit. These units are of fundamental importance in the theory of these forms, and may be investigated by a method, or rather slight modification of the same, due to Dirichlet; " Verallgemeinerung eines Satzes aus der Lehre von den Kettenbriichen nebst einigen Anwendungen auf die Theorie der Zahlen,"* also " Sur la Theorie des Nombres." In passing, it is noted that the basic idea in questions of this sort is as old as Leonardo of Pisa's identity concerning products each factor of which is the sum of two squares. Dirichlet himself points out that his theorems arose in considering questions of this genre given in Lagrange's theory of the multiplication of forms. There is a well-connected chain of theorems from Leonardo, on through Fermat, Euler's four-square result, and Lagrange's wide gener- alization — (in one direction only, however), up to Dirichlet. Of this general kind of theorem is Lagrange's quaternary quadratic identity, so fundamental in the theory of these forms. But the full importance of self-reproducing identities, of which the Lagrange-Dirichlet kind form only a special type, was fully and clearly recognized by Eisenstein, " Uber eine merkwiirdige identische Gleichung." f It is suggested that the general theory underlying all these isolated phenomena, may be a natural method of approach to the more arithmetical and less algebraical consideration of the properties of integers. That such has proved to be the case in the existing theory is obvious upon slight reflection. The most general theory of this sort would probably relate itself to operators , as suggested above. Applying now Diricpilet's methods (as cited), it will first be shown that it is possible to assign values to the variables in a :r-number which will, render this number less than any assigned finite non-zero quantity. The application may be made only on the fact already proved that no 7r-number with not all zero coefficients can vanish. By an adaptation of the method, it may be shown that there is at least one integer which may be represented in an infinite number of ways integrally in F; next, from any two of these representations a solution of F = 1 is found, and it will then be shown that all solutions, as in the theory of the Pellian equation are derivable from a certain determinate fundamental set. The first part of the proof consists in showing that if Pi = ao + P' cii TTi + P^' Oil T2 + p'' as Ts + p*' at tt^ , * Werke, I, Nr. XXXV. t Crelle, Jour., 1844, especially the remarks at the foot of p. 105. 36 ON THE CTCLOTOMIC QUINARY QUINTIC. then integers a.- may be assigned which will render TLUo I Pi I less than a finite positive constant (to be determined), in an infinite number of ways. It is essential to note, that since x,- is irrational, then for | a,- 1 or a,- selected, ao may be assigned as an integer so that < | Po [ < 1. Consider the set N of ( 2 1 n I + 1 ) integers, -In|, -|n-l|, ... -1, 0, +1, ■■•+\n-l\, +\n\^N, and assign to | or,| (i = 1, 2, 3, 4) any one of the (2| k| + 1)^ possible sets of 4 values selected from N . Now if ai is given, so are implicitly 02,03,014, for ai, a4 are conjugate, and either one determines b , c, d, g , the essential variables, that both a^ and as are given. By the assigning of values to | cti | is meant the assigning of values to the essential variables in a,-, and the sub- sequent determination of | a, 1 . There are clearly ( 2 | n | + 1 )^ — 1 possible Don zero sets. It is immaterial whether the values of | on | , 1 0:2 1 , \az\, \ai\ together are spoken of, or whether the values of the essential variables in a single a< ( 1 , . . • , 4 ) are spoken of, so long as the exact meaning of the assign- ment is borne in mind. It is also to be noticed that ao is a real integer. Let I tti 1 be such a set of four values, then ao may be assigned as an integer so that «o + I ai 1 1 iTi i -f [a2 I k2 I + I 03 1 1 X3 1 + I ai 1 1 T4 I = Po (0< Po< 1); that is, Po is a positive proper fraction, and the a's are all integers. There are (2|7i|-f- 1)*— 1 such positive proper fractions, for if |ai|^ ... =|a,| = 0, then ao = . Now divide the interval from zero to unity into some number of parts less in number than the total number of positive proper fractional values of Po, so that at least some one pair of these values for Po must have their end-points in a same interval; it is convenient to divide the — 1 in- terval into 16 I n \^ parts. Let P'q and PJJ be a pair whose end points lie in the same subdivision, then the absolute value of their difference is less than l/16|7iP. Put Fo-P'o = Po=(|a;i-ia;i) + (|a;i-ia';i).|nl + (I«2l-k';i)-U2i + (kal — l«3l) • U3I+ (|a'J — lai'l) • I ^r^ | = ao + i ai I • I Ti H" I a2 i ■ I 7r2 1 + I as [ • | tts i + | a4 1 -1^41, then [a.|^2|n!, (i=l, ■■■, 4), and < | Fo - FJK 1/ 16| n|^ the first part of the inequality coming from the fact that the difference is positive, and cannot be zero, for if it were the two non identical rr-numbers would be equal, which is impossible. Again, since \ai\^2\n\ (i=l,---,4), l/|ai|^Si/16ln|*, ON THE CYCLOTOMIC QXnNARY QUINTIC. 37 hence < I ao + ai Ti + aj ir2 + as 7r3 + a4 T4 I < 1 / 1 a^ I , where | a ] is the greatest of a,- , the a,- being chosen as above. Consider next |Pil = ko + aip (1 + X)^ which is represented by an infinite number of sets of values for the a's. Again, the congruence a< s a.' (mod m) has at most m incongruent roots; hence there are at most m^ incongruent sets (a,), {a\) for i= 0, •••, 4. Let two such sets (of 5 values each) be selected from the co sets which satisfy F (a) = m, say (a), (a'); and put /,• (a ) = ao + ai p*' TTi + ■ ■ • + at p*'' TTi. 38 ON THE CYCLOTOMIC QUIXARY QUINTIC. Prom (a ) and (a) may be derived ( as Dirichlet, " Sur laTheorie des Nombres," 1. c), a solution oi F {a) = 1 in the following manner. /o(«)//o(«')^/o(a')-/l(a') •/2(a')-/3(a')-/4(a')/F(a')-.-(A) By reductions which have been frequently used in former sections, the nu- merator on the right is of the form Ao + ^i tti + A^ t2 + Aiir3-\- Ai -Ki, and the denominator is m by hypothesis. It \d\\ be shown that ^4.- = (mod m) and hence that/o {a) I fa {a!) is an integer. Dirichlet remarks that if the variables (a) be replaced by the (a') , the resulting A\ mil be congruent to the Ai, since (a) = (a') (mod vi) , and deduces the result in question therefrom, but without some changes (replacing ttj, ttj, ta by their corre- sponding TTi expressions); the conclusion cannot be drawn in the same manner in this case; it is easily seen directly hovrever; for m =/o («') •/!(«') -h (a') -/3 («') -U W), Aq + ^1 :ri -f Az TTi + /I3 ""3 + Ai -Ki ^fo(cc)-Ma')-fAa') • /,(«') •/4(a'), •whence Ao + AiTri-{- •••-{• Ai TTi — m = (/o(a)-/o(a'))/i(a') ■ h (a') f^ (a') fi {a') , and. /o (") — /o («') = («o — a'o) + Ti (ai — a'l) + :r2 (ao — a'a) + tj (as — q's) + T4 (ai — ai ) = 7« (org + ^1 '^i + "^'2 '^z + as TTa + a^' tti ) , since a,- s a't ( mod 7;2 ) . Hence Ao + -"^l TTi + Ai 172 + ^3 TS + -^4 T4 — m = mvlo + mA'i TTi + mylj ttj + mA'i its + w^-^i ^4 , but if two TT-numbers are equal they are identical, hence every coefficient Ai, Ai, Ai, Ai is a multiple of m, so also is Ao — m, and therefore Ao is divisible by m. Therefore the right hand side of ^ is a r-integcr; so that if (a') . (a") are any two sets of a-values as in the above, then /o(«')//o(«")=/o('«); similarly /o(«")//o(«')=/o(«), where the coefficients in /o (a) and /o (a) are integers; and clearly, by the same process exactly, ON THE CICLOTOMIC QUINAKY QUINTIC. 39 hence f or i = , • • • , 4 , n/.(a')//.(«") = n/,(a), that is, or, the values of a determined as in the foregoing are a solution of i'' (a) = 1 , and there is evidently an infinite number of them. It may be noticed here that Xo is real; for Xi , Xi also Xj , Xz are conjugates, either member of a pair being derived from the other by changing p into p~' wherever it occurs, and therefore Xi X2 X3 Xi is real, that is, 1 / Xo is real, therefore Xo is real. [This is also evident when the transformations of F are considered, where the actual formulee show the result.] In order to bring out the analogy betv/een F(a) = 1 and the Pellian equation, it is necessary to consider a quantity Introduced into Arithmetic by Eisenstein, the " regulator," or rather considerations of a similar nature. First it will be shown that the ratios of the real logarithms of A'o and any X,(z4= 0), are incommensurable. Throughout, unless the contrary is expressly stated, the solution or " point "(l,0,0,0,0)oni^(a) = l,is excluded. If possible let Xj* = Xf , then by successive changes of -ai into p-jri , etc., and from the expressions of the A^'s as 7r-integers, it follows that Now Xo is rational only for the excluded point, and XS'---Xr= {F(a) }"= 1 for all values of m , and thence Xq = 1 , which is impossible (except at the excluded point). If possible, let X" = A'?", then as before A-i — A.2, ^2 ~ -^3) -^3 ~ ^*i -^« ~ -^0- Raise each side of the respective equations to powers m^ , m? n , m^ n^ , mn^ , n^ , so that from all, A"^"' = ^t. , fn, n independent, which is impossible. Hence the ratio of the real logarithms of any power of Xo and any power of another X is incommensurable, for otherwise some two powers would be equal, which has been shown to be impossible. Considering still the equation F (a) = 1, it will next be shown that any expression of the form A'" Xj A'3 X| where m, ■ • • , s are any integers may be replaced by a corresponding expression in which all the exponents are positive integers, and which expression may be written in the form Xo ( ~ Xo , " equiva- lent to Xo" ) . For It has been shown that Xo , • • • , X^ being the ^-factors of f (a), then A'l, Xi, Xz, Xi ~ Xo, since all are 7r-integers. Also A'5 is again of the form A'o, and the product of any two Xi{j , i = 1 , • • • , 4) is of 40 ON THE CYCLOTOMIC QUINARY QUINTIC. the form X; and therefore ~ Xo; all of which is seen directly from the early- part of this section; or may be inferred from the formulse for multiplication given § 3. Therefore in particular when vii, mj , rris, m^ are positive in- tegers, XJ" ••• X^ " Zo. Let now Xo be the first factor of an F (a) that is equal to unity, then Zj is the first factor of some other F (a) which is also equal to unity, for Z'J ~ Xo . Whence XjcoZi; •••J2"Z4, so {F(a) }" ~ {F(a) } = 1. Let n be negative, = — m; then zs = 1 / xs* = zr x? x^ xr ~ xs- , for (XoXlX2X3X4)'"=l. Similarly a negative power of any other X may be replaced by a positive power of an equivalent X by multiplying by a suitable power of 1 ^ F (a) . Hence any expression XJ" • • • Xj"* where the n's are any integers, may be replaced by a corresponding expression in which the m's are all positive, and which ~ Xo . Having reduced Xj" • • • X^ in this way, the resulting Z'o deter- mines a solution of F {a) = 1, either for thea.orfor the essential variables. mi = m2 = TTls = 7?l4 = , If the excluded case is refound; hence the totality of solutions is found by letting mv(i= 1, 2, 3, 4) assume all integral values from — » to + «> ; that is, there are oo* solutions derived in this way. Next it will be shown that Zp Z^ A'J" A'f' may be made to approach as nearly as desired to 1, when mi, mj, mj, mi are arbitrary national numbers, the quantity mi log Xi + nii log X2 + mz log X3 + mt log X4 = I, the logarithms being real, is of the form I = mi h + mih-h mz h + m^ li where no lil Ij {i "¥ j) is commensurable. For any A^• ~ Xo as has been shown, also it was proved that hlh, is incommensurable, and the same conclusion follows in the same way if for /, any I different from 7o is put. Since lot h is irrational it is possible to find integers mo, mi such that I mo 7o + mi Zi I < €, where t is a previously assigned small positive quantity. Similarly for j mo 7o + m,- /,■ | , (i + ) . Also the ratios of h / h and /i / /< are incommensurable, hence integers mj, m\;'in^, m\ may be found for e', 5' previously assigned, which make | m\ h + Tn\ Ii\ihl; \S'\>h\S\. Hence log | Xf • - • Xj" | may be made to approach zero as a limit, when integers mi, • • ■ , TO4 are properly chosen; that is, Xmt...xr= * 1+ «, « = o, for TOi , • ■ -, mi some integers. Hence izr---xri ~iXoi may be made as nearly equal to unity as desired, and since the m's are all integers, the values of the variables in Xo will be integral; therefore integral values of the variables may be found which make |XoI = l, and at the same time F{a)=l. Further on, F (a) is briefly considered as a surface in 5-space; it is ad- vantageous here to examine some o*' the simplest properties of such a surface. First, it goes to infinity in the real direction given by Xq = 0, or Xo = is a real asymptotic " plane "; also each Xi = is an asymptotic plane, but these last are imaginary; they intersect In a single real line, for X.- = Zo + P-' ii + P" U + p="" U , where L's are real, and X.-=0(i=^ 1, -••,4) have in common Lo = Li = L2 =^ Li = 0; hence F(a) = l has one real asymptotic plane Xo-1, and one real asymptotic line; also (1,0,0, 0,0) is a point on the surface. A point whose coordinates are real integers is called a "net-point"; and the surface is considered as being immersed in the net work formed by all planes parallel to Xo, Xi, A2, X3, X4 = 0, at unit distances apart. Considering the sheet of the surface on the positive" side of X'o = 1 , that is, the space which includes net-points whose coordinates 42 ON THE CTCLOTOMIC QUINARY QUINTIC. will make F (a) positive, the volume bounded by the surface, a plane parallel to Xo = 1 , and three planes parallel to and at finite distances from them, does not contain an infinite number of net- points (if any), for this bounded part of the space has in it no portion of the surface extending to w . Hence, since Xo = 1 cuts the surface in a space which does not lie wholly (if at all) at oo , the volume Xo=l, Xl = Cu Z2 = C2, X3=C3, 0 ^ , where n is arbitrary, and it does not follow as before that n may be so chosen that/o=0. But if -n = E{hlT,) then unless IqI I'o is an integer. Hence when io = «-*Jo k an integer, n cannot be found so that /q' = 1; that is, if (a;', y' , z' , u' , v') is the minimum solution, viz., the net point which lies on a plane such that there is no other plane containing a net point between this plane and A'o = 1 , then any other net point lying on the surface is found by taking the successive powers, positive or negative, the latter being replacable by the former, of this minimum Xo . This is in entire analogy to the deduction of all solutions of the Pellian Equation from a fundamental set. Finally, in tliis connection consider the limited volume cut out by the four planes and the surface, and put \X\ = \b + cp + dp'+gp^; 6, e, (i, jr being real. In all cases !p''xp = |zp. Whence, multiplying in succession by p^ , ^ , p^ and bringing the results to the same form as | X\^; \X\^ = \{c - h) + {d- h) p-^ {g -h) p^ - hp^^ = l(<^ - c) + (? - c) p - cp5 + (6 - c) p' P = \ig - d) - dp-\- {h ~ d) p^^- {C - d) p' P = |6 + cp+^p'+jP»P Putting 4ai = 1 + VS; 4a2 = - 1 + VS; iPi = ^IQ-2^1>; 4^2 + <10-2^15; and |ZP< £, the above give {b-Ca,+ {d + g)a2V+{ch-id-g)fi2V< e. {(c-b)-id-b)ai+{g~2b)a2]'+{{d~b)p,-gp.}'< e. 44 ON THE CYCLOTOMIC QUINARY QUINTIC. {(d-c)- (<7-c)ai+(6-2c)a2P+ { (g - c) Pi + b^^}^ < «. {i9-d) + dai-{-(b + C-2d)a2}^+[-d^i- ib-c)p2P -47, in absolute value, thus if ei, tj, eg, ei are each > €, the inequalities may be satisfied by taking 2 sets of equations of which the first is cPi- (d- g) Pi = «i, (d-fc)^i- <7/32= C2, (5'-c)/3i+&/32= a, -dpi- {b-c)p2 = a, the determinant of the system is (i3!-j8l+^ift)^-0, so that this set is not independent; in fact €4 = — £3 ft / /3i — £2 . It is now required to actually assign values tob, c, d, g which will satisfy the inequalities. It may be verified such a set of values is o< rx - ' ■ — r+ , r- , — ~ r; c< V2(H-a2) 2V2(/Si+ft)' V2(/3i+ft)' 2V2(l + a2) MPi+M' 2^l2(l + a2)' Numerically these make each { J'' < e/2 in the first, and therefore also in the others, since these were obtained from the first by multiplying by quan- tities whose modulus is unity. Put l2V2(H-a2)i l2V2(/Ji-|-ft)l where E [ } means the greatest integer contained in ( j , l(l+a2)V2J l(^i+ft)V2j hence b < t' + e"; c< t"; d < t' + «"; g < f' . That is, values have been assigned to a, b, c, d, g which make | A'p < e, a previously assigned constant. From this it follows that there is only a finite number of solutions of F (a) = 1 for which R = \Io- Ph- P'h- P'lz\'< e. ON THE CTCLOTOMIC QUINARY QUINTIC. 45 where < t < oo , For the planes Xo ^^ n^o! -^1 ^^ ki', Xt = ft2j -^3 ^^ «^j and the surface, bound a volume which for k's finite positive or negative quan- tities contains, if any, only a finite number of net points; which is a direct consequence of the foregoing results. Again, it has been shown that X^---X^>^ Xo^X'o. and that Z' TT" V "F^ "I^" — 1 0AjA2A3A4 — 1 when ^0 ■^l X2 X3 A 4 = 1 J increasing the suffixes by 1 (mod 5) while the indices remain in the same places, X[ = X^'X^XrX^, also (Xo---X4)~=l, whence Xi = Xo"" A'J"-™ X^-^ X^"-^ . Writing r„ for log 7„ , from this and similar relations, Ti = — OT3 /o + (wzo — ms) I1+ (mi — viz) h + (n^j — m.}) Iz, ^2 = ('"^3 ~ ^i) h — mj Zi + (mo — w^z) /a + (mi — mi) I3, I'z = (m2 — mi) lo + {mz — mi) h — mi h + (mo — mi) h, Ti = (mi — mo) Jo + (mi — mo) Zi + (mj — mo) h ~ m^Iz, I'o — J^o 7o + mi 7i •{• m^Ii -f ms J3 . From this, among others is the relation between two solutions: — Zq 7o /i Jj 7j z; /i /2 /3 -(/0+/1+/2+/3) r^ U h -(/0+/1+/2+/3) /o I'z h -{Io+h+h+Iz)Io h r. -(7o+/i+/2+/a) 7o 7i 7, Since Xo, ••• , Xi give a solution of F (a) = 1 , it follows that 7o+ ••• + 7, = 0; = 0. 46 ON THE CYCLOTOMIC QUIXAEY QUINTIC. from this, adding all rows of the preceding determinant, if /q + 0, 0, or (/i + 73)^= ih + hy or (7,-/3)2=- (7, -J,)2. Finally, the statement that all the units used in this section are contained in the formula =fe p"* tj^ is verified. A unit is a o-number whose norm is unity; hence if = pT rj^ is a unit, 7i h 73 h h 7, h 7o h h 7o h h 7o h h ± p"'p2'"p3'»p'^(,(,,l)n= 1; pUi ptm pSm pirn ^ ^lOm "no VI - 1; hence p" tj, is a unit. It may be verified as in the following, that all units belong to tiiis form. Next, it is required to resolve p into factors. prime. Put and hence so p = 5n-\- 1, = {a' -g') + ib'-g')p + etc., = a + bp+ cp'' + dp^; f(p)-fip') = A' + B'vo+C'ni; f(p')-f(p')^A' + B'm+C'vo; p= iA + Bvo)(A + Bni) = A' - AB - B'; hence p is to be represented by the form x^ — xy — y^, which (as usual) is considered as half the form 2z? + 2xy — 2y^; of determinant 5; hence 5 must be a quadratic residue of p, denoted by 5Rp . If 5 is a non-residue of p, this is denoted by 5Np. (See for this, and the ensuing reductions Mathews, " Theory of Numbers," pt. I, Chs. II, III.) From the usual theory, 5Rp if p s ± 1 (mod 5), but 5Np if p s ± 2 (mod 5). Again, the only non-equivalent " reduced forms " of determinant 5 are (2,1, — 2), (1,2, — 1); the first is that appropriate for the representation ON THE CTCLOTOMIC QUINARY QUINTIC. 47 of p s =*= 1 ( mod 5 ) . Hence p has been resolved into conjugate p-factors. In the next section further properties of F are considered, especially the transformations. § 3. If the properties of the simple quadratic form 3? + i^ , or oi 3? ->= ty^ are discussed, it soon becomes evident that the complete theorj"^ of these cannot be given unless the more general form ax' + 2bxy + cy^ is considered. In particular, if linear transformations of either of these forms into itself or into the other, are sought, the general form arises. As the problem of repre- sentation of numbers by a given form can always be reduced to that of finding the automorphics and a problem of equivalence, in other words to the theorj' of " associated forms," the finding of these latter is a step of fundamental im- portance. The construction of an " associated form " is arbitrary to a certain extent; the form here adopted will be justified by showing that this kind of form which arises first for quadratics, is also essential in the theory of forms of n variables of the nth degree. In brief, the associated form is derived from the given form by multiplying the latter throughout by a sufficient power of the leading coefficient to make that coefficient an exact power; if n is the degree of the form, the leading coefficient is to be made an exact nth power. This having been done, the representation of the new leading coefficient in the form is considered, this being possible (in the case of binary and quaternary quad- ratics, as in the present and all similar cases) by elementary methods because the transformation has reduced the problem to one of forms which reproduce themselves in form with respect to multiplication. For this reason the treat- ment of quadratic forms in the fifth section of the Disqmsitiones Arithmetical appears perfectly natural and simply intelligible, whereas the more elegant " improvements " of the theory by other writers only obscure the funda- mental reason for the mere possibility of the theory. For a similar reason the theory of triplication of classes can exist in binary cubics; and it was possibly this fact with relation to the binary cubic discriminant that led Eisexstein to construct such a theory. The more elementary parts only of the trans- formations-theory are touched upon here. Some formulae are collected for reference. (I) F ^ F(,afs,ai.,a2,az,ai) =^ al+ VlAVi'A. + VVlVi(A+ VPlVif^l '+ VPI Vi °I ~ 55p' Qfo ai Ui as ai — op (aj Vq + pi af Ui (1) + Pi(4 "2 + Pi ^3 «3 + Pi l^l "4 ) + ^P"^ ("0 yl + Pi ai Vl + Pi or; M^ + Pi U} til + Pi cci nl) 48 ON THE CTCLOTOMIC QUINARY QUINTIC. where «o = «i or4 + 02 az] «i = pz ao 02 + pa^ at; Mj = poci as + P4 "o on; M3 = pi ao ai + pai 04; U4 = pai a2 + pa ^o as • (II) 3125(a;''-l)/(a:-l) = F; p = 5ri+l; pip4 = p = P2P3, Pi = So + «i P + a2 p^ + ors p' + "4 P^ » P2 = ao + 03 p + ai p'^ + a4 p' + 02 p% (2) ^3 = ao + 02 P + a4 P* + ai p' + as P% P4 = ao + a4 p + as p^ + a2 p' + ai p^ , where ao , • • • , a4 are real integers among which by (2) there are the following relations ao ai + ai 02 -|- a2 as + aj a4 + a4 ao — — n, ao 02 + 02 0:4 4" a4 as -f- ai 0:3 + as ao = ~ n, (3) 5{al+5l+al+ai+al) = ip+l, 1 + ai + a2 + as + a4 + ao = , (III) ffi = '^ppl vz; T2 = Vppfpl; irs = ^ppIpi; 1^4= ^pplpy, whence the following table that will be frequently used: TTl ITJ ITS Vi Tl pi TTi pz ITS pi W4 P T2 Pi 7r4 p P4 Tt\ (4) X3 Ps Tl Pj X2 X4 P4 T3 Whence, writing a, 6, c, , z, 0, u); at = {y, z, u, v, 0) The formulae for transformation of F now follow; in any case where the work is not given in full, the result is evident from the formula themselves. (VL) F will be transformed by x=kX+moY+noZ+roU+ioV y = hX+ ■■■ z=hX-\ I- (9) u=hX-] v = UX+ ■•■ 50 ON THE CYCLOTOMIC QriNARY QUINTIC. the transformed ^.- will be denoted by ?',; the determinant of the transformation (9) is S; (9a) and ki = liX+ vu Y+niZ+riU+ U V, (10) whence fo = h+Ti (A-i + pU-2+ p'A-3+ p2A-4)+ TTj ih+ P'h+ Ph+p'h) + T3 ih + p" h +pU-3 + ph)i- Ti ih + phi + p-h + pH-i)- • • = X { Zo + ( n + TT, + 7r4) /i + (:ri p^ + TTi p' + 7r3 p2 + ;r4 p) ?2 (11) + ( Tl P' + T2 p -|- TTJ p'' + 7r4 p^ ) /j + ( ITl p^ + TTj P^ + ITj p + T4 p') Z4 } + y (ml + 2 { n } + r { r } + F (0. where { ?« } is what the coefficient of X becomes with m written in place of I; etc. Whence, writing ^o = LoX+MoY+N,Z+RoU+ ToV; ^,= L,X+ ■•■; i\ = LiX+ ■■■ + T,V, (12) Lo=lh+^iUiOhhh) + ^2(hhOhk) + ^zihhkOh) + ^,{hhhkO)] (VII) "\Mien Lo, • • • , X4 are given, Mo, • • • ; • • • , Ti are found by writing m , • • • , < in place of I . Writing for brevity Xo = Zo; Xi = (h Oh h k); X2 s (/i /3 o?2 /.4); \z = {hUkOh); \i=(hhhhO), then io = ( Xo I- iTi Xi + :r2 X2 + ^3X3+ 5r4 X4 , , , , ) ; 2/1= ( Xo , Xl Xl , X2 X2 , TTs X3 , X4 X4 ) ^2= (Xo, ^3X3, TTlXl, ^4X4, 3'2X2) (13) L3 = ( Xo , ^2X2, Tti X4 , TTl Xl , X3 X3 ) Li = ( Xo , X4 Xl , ITS X3 , X2 X2 , Xl Xl ) with which compare (7) ; whence i^ ( Xo , Xi , X2 , X3 , X4 ) = Lq Li L2 Lz Li , (14) or simply F (X); then in the same way are found F (fi) , F(v), F(a-), F (t), corresponding to the M, N, R, T; and F (X) , ■■■, F(t) are real integers. (15) If this last is not obvious, the equivalent of it is proved later. Again, lFi\)y-F'= {Fi\)^o/Lo] • lFi\)^yL,\ ' {F{\)^JL2} • {Fi\)^'JLz}.{Fi\)^JLi], ON THE CYCLOTOMIC QUINARY QUINTIC. 51 where F' is the transformed F; but the left hand side is also the product of (XoX+ •••+ ToV) -UUUW, {UX+ ••• + TiV)UULiLu -••; {LiX-\- ••• -\- TiV) • LaLiL^Li. Hence the product of these five is divisible by { F (X) j^; therefore the total coefBcient of the term X'Y^Z-^U'V, «+••; + € = 5, is exactly divisible by { F(X) }*. In the same way there are congruence relations for F (>:),-••, F (t) . It will be sufficient to obtain the relations for the first product only; the others are symmetrical with them (by \Ii), and may be written down therefrom. In order to see in detail the relations between the structure of F and the associated F' , the following considerations on 5-numbers are needed. Above, (in IV 6) the coefficients of the two 5-numbers multiplied together were all single letters, with no distinction as to real or imaginary; the form of the result shows that they were implicitly supposed real. Such a 5-number is called " simple," those 5 numbers in which the coefficients are simple 5- numbers are called " complex." A complex 5-number is again a 5-number, and a single such, but not conversely. The most general form of a complex is M= { (aoGiaaasQi), {bobibzbzbi) , (C0C1C2C3C4), (dodidtdzdi) , (9091929^9*)] ^ [ao+bi+ cs + do + gi, ai+ bo+ Ci + d3 + 92, (16) Ci + bi+ Co-\- di + gs, a3 + b2+ Ci+ da + g^,, cn+bi + C2 + di + ga]. Whence M is the sum of simple numbers in 2^ — 1 ways; of simple numbers none of whose coefficients is zero in 3,125 (=5^) ways. The theory of the linear transformation of F forms, and more generally of the forms considered in § 2, is identical with the theory of multiplication of complex 5-numbers, or complex n.-numbers in the general case. Similarly for the associated forms. The immediate object is to find in simple form the products Lo Li L2 Lt , •• • , etc.; for this X,- Xy must be calculated. \^Tien X,- Xy is known, X2» X2j follows from it by doubling and reducing; also if /(p) = (abcdg), then/ (p*) is found from/ (p) by means of the substitution (bdgc) , as may be verified at once. From the values of the X's given in VII, by applying (6), find f{p)^\i\2 = {n+hh+hh, hh+hh+n, P2+hh+hh, hh +l2h+ h h +hh, h U +l2h+ ID g{p)^\l=ill+2hh, q + 2hh, 2hh + 2hh, ll + 2hh,ll + 2hl2). 52 ON THE CYCLOTOMIC QUINAKT QUINTIC. A(p) = XiX«= (/J + /i+/|+/J, hh + hh + hh, hh + hli + hh. ^ , , . hh+hh+hh, kh+hh + hh)- Or for brevity Hp) = {yiyiViZiyz); 9{p)= {xiXiZiXiXi); A (p) = (wziZaZszO (17) Performing the indicated substitution, /(p) = XiX2= (yiytyiZiys) ; g{p)='Kl= (xiXiZiXoXi); hip) = \iXi=(wziZ3Z3Zi); f(.P^) = X2X4= (yiZiyiysyi) ; ?(p^) = X^= (xiXiX^XtZi) ; /[(p2) = XjXj= (wZiZiZiZs); (IS) Sip*) = X4X3= iyiyzZiyiyi) ; g{pi) = Xj= (a'ia-4a:2ZiX3); /(p') = X3Xi= (yiyii/sy^i) ; gip^) = X§= (a;iZiar4X3a;2) ; where Pi + 2hh = xi;P, + hh + h k = yii 2{h U-\-klz) = zi n + 2li h = Xi;ll + h k + Uh = y2i h k + hU + h h+ U k = Z2 Jl+2hh = Xz;ll + hU + kh^y3',hh + hh + hli=Z3 ^ ' P^+2hk=x^;ll-\-hh + h h=yi; h h -\-hk + h U =[z, II + II + n +ll=^w among these are the linear relations, a:i + a;2 + 3:3 + a;4 + zi = w + 2z3 + 2z4 = yi + t/2 + Jfa + 2/1 + zj and the quadratic relations '^1 y\ + 2yi Zi + 22/3 2/4 = a:f + (2:3 + ^4) (2-2 + \i) yl + 2^4 Z4 + 2yi 2/3 = a-| + {xi + 2:3) (2:4 + ii) t/§ + 2yi 24 + 2^2 yi= xl+ {Xi + 2:4) (2;i + :i) y\ + 2y3 24 + 2yi 7/2 = a-* + (a^i + Xi) (2:3 + :i) which come from (18) by lfip)V = 9ip)'g{p'); and then doubling and reducing with respect to the 2:'s and t/'s which is per- missible by (19). The L products which will be required and which are now being constructed are: LiLiLsLi', Lo Li L3 Li', LoLiLzLi; LoLiLiLi; L1L1L2L3 (21). By (13). LlLt = ( Xo , Xl TTl , X2 X2 > X3 TTS , X4 ^4 ) " ( Xe , X4 Wl , Xj Tj , X2 TTi , Xl Tl ), ON THE CTCLOTOMIC QUIKABT QUINTIC. 53 which by (6) is m+ >f tJ+ XItt^H- ^f ir|+ X|:rJ, Xq X4 tt* + Xo Xi TTi + Xj Xi TTl Tj + Xj X3 TTi TTa + Xj X3 TTi TTz , Xq X3 TTs + Xl X4 TTl Ti + X2 Xq TTi + X3 Xj ITS Tl (22) + X2 X4 ^2 Jr4 , Xo X3 TTs + X2 X4 7r2 TTi + Xl X3 Tl TTs + Xfl X2 7r2+ X4 Xl TTt TTl , Xo Xl TTl "1- Xl X2 Xl 5r2 -}" X2 X3 T2 ^3 "T X3 X4 X3 7r4 + X4 Xq X4 ) which by (4) is, upon rearrangement of terms with respect to resulting r's: (Xq, X2X3P, X1X4P, X1X4P, X2X323) + Ti (Xjps, XflXi, X2 X4P4, X2X4P4, XflXi) + T2 ( Xf pi , X3 X4 Pa ) Xo X« , Xo X2 , X3 X4 p3 ) (22a) + jr3 ( X| , Xl X2 ^12 , Xo X3 , Xo X3 , Xl X2 P2 ) H-X4(X2jl2. X0X4, XiXsPi, XiXsPi, X0X4) Putting as usual p+ p^^ 10; p'+ p^ ~ m, the last is (X2+ I30PX2X3+ J?lpXiX4) + TTl (X3P3+ '?oXoXi+ 5;iP4 X2 X4) + T2(XiPl+ J702'3X3X4+ 771X0X2)+ 5r3(XiP4+ '?02?2XiX2+ JJlXflXs) (226) + T4 ( Xf P2 + 5)0 Xo X4 + Vl Pi Xl X3 ) Li Li is obtained from ii Li by doubling and reducing, which is equivalent to an interchange of rjo , 171 in (22&) whence X2 is = ( Xg + ?7i pXs X3 + m P^i X4 ) + TTl ( \lpi + ijiXo Xl + 770 P4 X2 X4 ) + 51-2 (Xf pi + 771 P3 X3 X4 + 770 Xo X») + Jr3 (X4P4 + 7J1P2 Xl X2 (22c) + 770 Xo X3) + 7r4 ( Xj P2 + 771 Xo X* + 770 pi Xl X3) Rearranging (226) with respect to 770, 771, the coefficients of the it's in the coefficient of 771 are found from those of 770 by doubling and reducing, ( Xj + TTl X| p3 + T2 Xf Pl + X3 X4 P4 + 7r4 X' ^2 ) + 770 (pXj X3 + TTl Xo Xl + TTi PZ X3 X4 + TTs ^2 Xl X2 + ir4 Xfl X4 ) {22d) + VI (p>n X< + ^i Pi ^2 X4 + X2 Xo Xj + 773 Xo Xs + TTi Pi Xl X3 ) ; similarly for (22c), 770, 771 interchanged, whence Li Li Lz Lt is of the form (A + £770 + C771) {A + Bm + Cno) = A^ - B^ - C - AB + ZBC - CA; where (22(f) is written A -{- r]a B -\- iji C (23). The remaining 4 products in (21) may be similarly found. They could also be obtained by division from (14), but it is less labor to build them up by multiplication as above; or the (24) 54 ON THE CYCLOTOMIC QUINARY QUINTIC. product of two numbers as (16) may be used directly. It is easiest to form first the symmetric function of the L's, which is done. The results indicated in VIII are to be finally exhibited in the form ^q X -\- ■ • • , where the coef- ficients fo ■ • ■ are numbers of the same form as fo • ■ • ; hence at each step in the reductions, functions of the it's are to be replaced by their equivalents from (4) as soon as they arise, and final results ordered according to tt's linear. From (13) and (4) Lo XI = pXi + TTl Xo + X2 pi Xl + T3 P2 Xa + TTi Pl Xj , Lo TTi = pXs + Xl Pl X4 + X2 Xo + X3 P2 Xl + X4 P2 Xl , Lo TTz = pX2 + TTl ps X3 + X2 Pl X4 + X3 Xfl + X4 Pl Xl , Lo ITi = pXl + TTl Pl X2 + X2 P3 Xs + X3 Pl Xl + Xi Xo , But Lo Li = ( io Xo , Xl Xl Lo > X2 X2 Lo , X3 X3 Lo , xi Xi Z-o) » Lo being simple, and so for Xo Lo , Lo L3, and io Li . Putting for the moment A = Xo ^ + ( X2 X2 + X3 X3) -B + ( Xl Xl + X4 Xl) C, B == X4X4^+ (xiXi+ X2 X2)5 4- (Xo+ X3X3) C, C = X3 X3 ^ + ( Xo + Xl Xj) B + ( X2 X2 + X4 Xl) C, (25) D = X2 X2 ^ + ( Xo + Xl Xl ) jB + ( Xl Xl + X3 X3 ) C, E = Xl Xi/1 + (X3X3+ XI X4)i?+ (Xo+ X2 X2) C; th en LiLiLz= (ABODE), i3iiXi= (ADBEC), L2LiLi= (ACEBD), 11X2X3= (AEDCB); (26) follows from (13), (22d) by (6); one result being found the others follow as in (18), or directly. From (26) X0X2X3X1 = (AXo, BXo, CXo, DXo, EXo), and so for the others; but first the values of A, • • • , E are written down from (27), (which could be omitted, except for convenience); the columns are the coefficients of x.- in the functions at the left. As a check, the weight of the X's in the column x.- is constant and equal to i modulo 5. (The table is on next page. ) Directly from (27), ^^Titing ON THE CYCLOTOMIC QUINARY QUINTIC. 55 Ao = \l + p (pi Xf X3 + P2 Xi X| + Pi Xi X4 + Pi X2 \l) Ai = p\i (Xi X4 + X2 X3) + P3 Xo Xf-f 2p4 Xo X2 X4 A2 = p\2 ( Xi X4 + X2 X3 ) + pi Xo Xj + 2p3 Xo X3 X4 A3 = pXs ( Xi X4 + X2 X3 ) + 2p2 Xo Xi X2 + Pi Xo Xf A4 = p\i ( Xi X4 + X2 X3) + 2pi Xo Xi X3 + P2 Xo Xi Bo = 2X0 Xi X4 + Xo X2 X3 + Pi Xi Xi + pz X| X4 B2 = pXi X3 + Xg X2 + 2^1 Xo Xf + pi Pi Xi X| + Pi Pi X| B3 = 2p\i X3 X4 + Xg X3 + pi Xo Xi X2 + P2 Pi Xi X4 (27a) B4 = p\i ( Xi X4 + X2 X3) + 2pi Xo Xi X3 + 232 Xo Xi Co = 2X0 X2 X3 + Xo Xi X4 + pi Xf X3 + P4-X2 Xf Ci = 223X1 X2 X3 + Xj Xi + 233 Pi X3 Xf + 234 Xo X2 X4 C4 = 23Xi Xf -I- X? X4 + 231 Vi Xf X2 + pi Pi Xf + 232 Xo Xi Di = pXf X4 + Xi Xi + 232 Pi Xi + 233 Xo Xf + Pi Pi X3 Xf D2 = 223X1 X2 X4 + Xi X2 + 23i Pi Xi Xf + Pi Xo X3 X4 E3 = 2^X2 Xf + Xi X3 + Pi P2 xf + p2 Pi xi X4 + P4 Xo xf (27) 1 Vl W2 TTs TTi Xo A Xg p3 Xo Xf pi Xo Xf p4 Xo Xf p2 Xo Xf Xi Ti A PP2 Xi xi Xi Xi Pi p3 Xi xf Pi p2 xf p Xi xf i X2 TTi A ppi X2 xf P2 P4 xi xi X2 p X2 X3 Pi P2 xf X2 2 X3 iTi A ppi xf X3 P3 P4 X3 xf p X| X3 xi X3 Pi ps xi 3 X4 vi A ppi xf X4 p xf X4 P3 P4 xf p2 P4 xf X4 xi X4 4 Xo -B p Xo X2 X3 xi Xi p3 Xo X3 X4 P2 Xo Xi X2 xi X4 Xi ;ri J5 p Xo Xi X4 pXi X2 X3 pi Xo Xf p Xi X3 X4 pi P2 Xf X2 1 X2 :r2 B ppi Xi Xi p4 Xo X2 X4 p Xf X3 p2 Xo Xi X2 p X2 X3 X4 2 X3 Vi B ppi Xf X4 p Xi X2 X3 p3 Xo X3 X4 p X2 Xf pi Xo Xi Xj 3 \iiriB p Xo Xi X4 P3 P4 X3 Xf p Xi X2 X4 p4 Xo Xf p X2 X3 X4 4 Xo C p Xo Xi X4 P4 Xo X2 X4 Xo X2 Xq X3 pi Xo Xi X3 Xi «•! C ppi Xf X3 2) Xf X4 p Xi X2 X4 P2 Xo Xi X2 pi Xo Xi X3 1 X2 7r2 C p Xo X2 X3 p Xi X2 X3 p Xi X2 X4 P2 P4 Xf X4 P2 Xo Xf 2 X3 T3 C p Xo X2 X3 Ps Xo Xf pi Pa Xi Xf p Xi X3 X4 p X2 X3 X4 3 X4 Tfi ppi X2 Xf p4 Xo X2 X4 P3 Xo X3 X4 p Xi X3 X4 p Xi Xf 4 56 ON THE CTCLOTOMIC QUINARY QUINTIC. From (27o), (26), (25), A = Ao + iri Ai + ir2 A2 + Ts A3 + Ti Ai, B = pBo + Xl Al + TTJ B2 + T3 B3 + Tt B4 , C = pCo + Xl Cl + T2 A2 + T3 B3 + X4 Cl , D = pCo + XI Di + X2 D2 + TTj A3 4- X< B4 , E = pBo + Xl Cl + X2 D2 + X3 E3 + X4 A4 . (276) These are not the same as (25), for Ao , • ■ • , E3 are independent of x's . Notice also that these functions are isobaric, modulo 5; e, g., Ai is thus of weight 4 in the X's. Finally, using (24) and (276), taking the product of the two following matrices: Ao Ai A2 A3 A-i pBo Ai B2 B3 B4 pCo Cl A2 B3 C4 pCo Di D2 A3 B4 pBo Cl D2 E3 A4 Xo, Xi, X2, X3, X4 pXt , Xo , pi ^1 > Pi ^2 ) Pi ^3 pXj , p4 X4 , Xo , Pi Xi , pz X2 pX2, P3X3, ^3X4, Xo, pi Xi pXi , p4 X2 , Pi X3 ,^4X4, Xo . (multiplication by rows of first, columns of second), find Afl Ai Aj A3 A^ Bq Bi Bj Bj B4 Co Cj Cj C3 C4 d; d; Di D3 d; . E» E'l E|t Ej E4 (28) whence ZoA = AJ, + A'lXi + A2X2 + A3X3 + A'jxi; • whence, from (26), Lo Li L3 Li = (AJ, + • ■ • + A.\ n; LoE = E'o+ ••• + E;x4 (28a) ..; Ei+---+E;x4) (286) ioiii2i3= (Ao+--- + a;x4; •••; b;+--- + b;x«) Consider now ^oLiLiL^Li-, Ld'iLiLzLi-, LoLi^^L^L^; LoLxL,aU; L^LxLiL^^,. (29) and (see 11) in particular the coefficients of A', • • • , T^ in the last four. In ON THE CYCLOTOMIC QUINARY QUINTIC. 57 all (by (14)), the coefficient of Z is f ( X ) • The coefficients of Y in the last 4 are respectively Lo Mi L- L3 L^; Lo Li M2 L3 Lu La ii 1-2 M3 W, La In. Li Li M i] each of which is a complex 5-number, e. g., LoMiL^LzLt being the product of (moj/^i ti, 112 t^, us tz, fn iri) and the first- written number in (286). This number will now be put in the form (12), as required, the reductions being effected directly from (286) and (4). It is convenient in working to use the principle of constant weight (mod 5) already noted. The form of the result before actual reduction to form (12) is (A" B" C" D" E"); but in this it is not necessary to calculate all the letters, B" , • • • , E", in virtue of (286) follow from A" by permutations, represented by the substitutions [ABCDE], [ACEBD], [ADBEC], [AEDCB]. A" = {fxo A'o -f pfii E'i + PH2 Dg + p;x3 C2 + pm B\ ) + Ti («) A'l + ;ti Eo + Pi 112 D4 + P3 m C3 + Pi Hi B2 ) + X2 (po A'a + pi Hi E'l + /i2 Dq + p3 Hi Ci + p3 Hi Bj ) + X3 (m A3 + p2 Hi E2 + Pi Hi D'l + H3C'o + Pi Hi K ) + ri (ao A4 + pi Hi E3 + Pi Hi I>2 + Pi Ms C'l + Hi'B'a) . Finally, adding all so found results (without writing them down), LoMiLiLsLi = {//o(AoB;c;d;e;) + PHiiKKKc'iD'i) + pm2(D3e;a;b;c3) + pH3iC2ViElk'2B'2)+ pHi{B\C[B\E\k\} I + iri[Ho{ )i+ fii(. +P3H3{. + 'r2{^o( h+PiHii + P3H3i + T3{M )3+P2Hii + H3{ + '^i{l^o{ )i+PiHl( +PllJ-i{ where in any column the ( )„ means the same arrangement of A, • • • , E as at the head, with the indicated change of suffix. From (286), referring to (26), the value of Lo Li M2 U3 Li is written down from (30) by means of the substitutions [BCDE] and (see 13) [ hi i^z y-i Hi] , and in the same way LaLiLiMzLi by [BCED] and [wm: A'l Ms]; io^ ^2 is Mi by [BE] • [CD] )o+P4W( )i )3+P4M4( h\ )l+ M2( )o )i+P3Hi{ )s| (30) )l-\-p2Hi{ )i )a+PiHi{ )«} )3+P2Hi{ )2 )l+ M*( )ol 58 ON THE CTCLOTOMIC QUINARY QUIXTIC. and [fiiin] • [fiifi]- It now remains to evaluate J/o 11X2X3X4 in the same way, i. e., arranged according to it's. The result will be derived in a slightly different form to those shown above, but finally all will be reduced to a common form, i. e., corresponding to ^0 , 6 , h, h, ?4 , (see (7a)), where the letters corresponding to a', ••• , g' all denote real integers. By (23), Xi LiL3Li= A'- B^'-C^- AB + SBC - CA , where ^ = Xg + iri Xf j?3 + T2 \l Pl + TTS Xj Pi + TTi Xj Pt] B = p\i X3 + Tl Xo Xl + TTi Pi X3 X4 + TTs p-l Xl X2 + Ti Xo X4; C = 2)Xl X4 + 2-1 Pi X2 X4 + TTJ Xo X2 + TTZ Xo X3 + Vi Pl Xl X3; or say ^ s ao+ ai+ 02 + 03+ 04; B=ho+ 1- 64; C = Co + V U; then after reductions, collecting together terms of same weight (mod 5), (the self-explanatory table is appended for verifications) XiX2X3X4= {(00+20104+20203) -(62+25164+26263) -(c«+2ciC4+2f2C3) + 3(6oCo+ 61C4+ 62C3+ 63C2+ 64C1) — (0060+0164+0263+0362+0461) — (CoOo+Cl04+C2a3+C302+C40i) } + 4 similar { ) derived from this by successive augmentations of all suffixes by 1; and by the principle of weight used throughout these 5 in the order indicated furnish all terms which on reduction contain respectively (tto = 1) , "•2, T4, Xl, Ts; call, the result o' + 6' iri + c' Tr2 + P^2 ) f'lMs) 1'i(mi , fiO , Pif^i y P3t^3 , Pil^:) + c'(pA'3,P4M4,M0,P2Ml,P2A'2) ■'^2(1^2 > Plfil , 1^0 , Plt^i , Pi Ml) (31) -{-d'(j)fi2 , PzfiS ,Pif^i,I^O, PlMl) ^3(^3 , PiM2 , P2,"l , Mo , Pitii) +9'(Plil . PiJ^i > P3P3 , Pi^ii , Mo) T4(P4 , Pll^i , Pil^i , PlMl ) Mo ) , Oo bo 0061 Qo 62 Oo 63 Oo 64 1 4 1 1 1 1 2 1 3 2 3 2 4 2 2 1 2 2 3 2 3 3 3 4 3 3 1 4 1 4 2 4 3 4 4 4 0= 2.01 2.02 2.03 2.04 2.14 P 2.12 2.13 2.23 2.24 22 32 2.34 42 ON THE CYCLOTOMIC QUINARY QUINTIC. 59 where on left v multiplies each number in its column, similarly for a' , ••• ,g' on right. (Compare (30).) The nature of these coefficients will now be examined, and it will be shown jBrst that the term free from tt's is a real integer. That it is an integer is obvious, for only multiplications have been performed in its derivation; the reality follows with some work. On reducing by (4), a' = XJ - p2 (Xf X| + Xi X| + 4Xi X2 X3 X4 ) - 2p\l ( Xi X4 + X2 X3 ) + 3p (pXi X2 X3 X4 + pi Xo X? X3 + P3 Xo Xf X4 + P2 Xo Xi Xi + Pi Xo Xi Xi ) — P ( Xg X2 X3 + ps Xo Xi X4 + Pi p2 X? X2 + pz Pi X3 X| + P2 Xo Xi xi ) — P (Xg Xi X4 + P2P4 X| X4 + p4 Xo X2 X| + jh Xo X? X3 + Pi Pi Xi xf ) or a' = XJ - p^Xf ^l + Xi X2 X3 X4 + Xi X|) - 3pXi (Xi X4 + X2 X3) + 2p {pi Xf X3 + Pi x§ X4 + Pi Xi Xi + Pi xi X4). "X^ — P (Pi Vi xf X2 + Pz Pi X3 Xl + Pi Pi \l X4 + Pi pt Xi xf ) . This could be shown to be real by actually expanding, but a reference to the values of the p's and X's shows that the labor involved is prohibitive. How- ever, referring to the values of h{p) , h (p^) given in (IS), it is evident at once that the coefficients of — ^ and ~ 3p\l in a' are real. The entire coef- ficient of -j- 2p may be generated by doubling and reduction from any one of its terms; e. g., from pi Xf X3 are successively derived ji2XiXi, Pi'S.lXi, Pi Xf X4 . But pi Xf X3 is a function of p , say

    (p'')isa real integer. Similarly the entire coefficient of — p in a' is a real integer. Hence finally, a' is a real integer. In the same way is found c' = ipi Vl X| + 2pi Xi Xf + 2ppi Xi X4 ) - (pi X5 Xi + 2ppi X2 Xi X^ 4- 2^X0X1X2X4) - (pXi X| + 2p Xo Xi X2 X4 + 2^3 Pi Xo Xi Xi ) 4- 3 (pXo Xi X2 X4 + ppi Xi X3 Xf + PPi Xf X2 X3 + Pz Xi X3 X4 + pXo xi X3 ) (32) — {piPi Xo XiXf -f- ppi Xf X2 X3 + Pi Pi Xo X^ + PP2 Xi Xi + Pi xi Xs X4 ) — (PP3 X2 xi X4 + xg X2 + pXo xi X3 + PPi Xi X3 xf -^ ppi xf X4) , that is, the coefficient of X2 is of weight 2 (mod 5), (as it should be), g' is found from c' by doubling and reducing all suffixes (including those of p), d' similarly from g' , and finally b' from d' in the same waj', as is evident from (4) and the manner of writing the a, b, c expressions, or it may be verified 60 ON THE CYCLOTOMIC QUINARY QUINTIC. directly. Hence if c' be written as a function of p, it follows as above that 9' = fip'); d'=f{pn: b'=f(p'); c'=/(p), and therefore b'ui,-hc'ni + d'^2 + 9'Mi=f{p')-9(p*)+f(p)-g{p'') +f(p')-9(.p')+f(p')-g(p), where / ( ) , g {) are rational integral functions, for by (VII) the /I's have values of the nature indicated. If ff (p)/ (P^) = Go+Gip+G2p' + 6'3 p' +G,p\ a form to which the product of any two rational integral functions of p is reducible, then clearly b' m + c' ^, + d' ^2 + g' HI ^ iGo - Gi- G2 - Gz- Gi, a real integer. Therefore, summing up, the absolute term is a real integer; i. e., a' lie -\^ p {b' Hi + c' tiz -\- d' H2 + g' /ixi ) is a real integer. It will next be shown that the coefBcients of the it's are imaginary, of the form (a", b", c", d", g") . Writing for the moment M.-=?(p'); Pi=h{p'); a'.---,g' as above, this being permissible by (VII) and (2), the coefficients of ri , • • • 7 ^4 respectively become: i^i) : a' 9 (p) + f (p') g il) + f ip) h {p') g {p') + f ip*) h (p') g (p') + f(p')h{p')g(p'), (ttO : a' 9 ip') + h (p)f (p^) g {p) + f {p) g (1) + f (p*) h ip') g (p*) (33) +fipnhip')gip'). (n) : a' 9 {p'^) + h {p-')/ (p') g (P') + f (P) k(p') 9 (p) +f {p') g (I) + f(p')h{p')g(p'), (n) : a'gip*) + A (p)/(p') g (p') +/(p) h {p')gip') ^-/(p^)A(p)?(p)+/(p^)?(l) which are of the form vip) + s(p') + cc(p) + Hp') + y(p') r}(p') + P(p') + Hp) + yip') + ^ip') (33o) 1? (p') + « (p') + 7 (p) + Up') + /3 (p') V (p') + y (p") + Pip) + « (p') + 5 (P-), (336) ON THE CTCLOTOMIC QUINARY QUINTIC. 61 )7 , • • • , 7 rational integral functions. In any line it is clear that the coefficients of the various powers, of, p sje, ,not iAen^ca4 for general values of the letters involved, and thereHir-i iiv gejie^aj .^ach coefficient of t's is a complex 5-number. Writing a(p) in the fcrm a.,-\- ••• +«4P*, 7(p) = etc.; ••• the final form of the coefficient ic ibeordjer written in (33a) is ( a" h" c" d" g"), ( a" d" h" g" c"), ( a" c" g" b" d" ) , {a"g"d"c"b"), which by eliminating g" as usual become ( a'" b'" c'" d'" ) , ( a'" d'" b'" c'" ), (a'" c'" b'" d'" ) , {,a"'Od"'c"'b"') all the letters being real integers. All the foregoing considerations lead up to the construction of the associated forms, which latter are now examined. The factor" composing an associate are finally to be of the forms (34) [0] = {UX-\-Mo Y->rN^Z+R, U+ ToV). UUUU = ^'oL\ULzU, [2] = {Li X -\- • ■ • ) . Lo Li L3 Lt = Lo Li ^2 J^3 Li , [ 3 ] = ( is X + • • • ) -LoLiLi Li ^ Lq Li L2 ^'3 Li , l4:] = (LiX+ ••• ).LoLiL2L3 ^LoLyL^L^^^. It has been shown that the coefficient of K in [0] is expressible in the form indicated by (336); but if 6" had been eliminated by means of 6"(1 + P+--- + P^)^0, instead of g" , the form would have been, a'+ ;ri (r.orjj^) + T2 {rj.orj^) + t, (rjj,or,) + ^4 (fjjj.o), with which compare (12). In this it has been shown that all the letters Ti , • • • , r^ denote real integers, and it is clear that the t' are functions of the a ,b,c,d,g in (5), and of the I's and m's in (9). Since a, • • • , g occur similarly in all the coefficients, there Is no need to put them in evidence; the coefficient of Y being denoted hy tpo (,1 , m) , then those oi Z ,U .V zre ifa{l,m) , a{l,T) ,ipo{l,t) , and the absolute term may be put = ipo {I, I) . Similarly in [1], • • • , [4], so (34) becomes [i]^ 'r2a2, ir4ai, TTiofi, tts as) 7 -f- • • • [4] = kX + {ao, Viai, Tjas. T2 «: > tiqii) F -{- ••• (The as are distinct from those in (1).) [0], • • • , [4] are of the form (7). ON THE CYCLOTOMIC QUINARY QUINTIC. 63 The last four are obtained from the first by use of (36). Among the quantities in (38) many relations are now apparent, of which a few are written down before continuing with the associates. The set (38) being further shortened to (38a), hX + Ai Y +BiZ+ Ci V-\-DiV^ [i], (t = 0, .- . 4) then AoAiAi AzAi = F (ao, CKi, 02) «3, cci); Bq •• • Bi= F (/So, • • • , Pi); Do--- Di = Fi5o, --, h); Co--- Ci = F{yo, ■-, 74). Again, from the form of (1) and from (V), (38), (3Sa), it follows, first that if •f(?Oi fi, fs, fs, fi) is a rational integral function of a;, ••• , v, say G {z, y, z, u, v); then also must FiAo,---,Ai) = G(ao,---,ai),---; F(Do, --- , D^) = G{So, ---,5^). Further if there is any relation of the form Fi^) = Gix, --■), then there are at the same time FiA.---) = G{a);----.F(D) = GW. Comparing (11), (12) and (37), and remembering the value of fc, = Oii) ao Tpoci poci paz pai «1 ao P4Q;4 ^3^3 PiCC2 at piai ao pzoci P3013 az pocci p^ai ao Pia.\ I CK4 p\oiz p^ai piai ao i where the notations mean!^ that the coefficients in the matrices go by rows with the m's and a's respectively, and (the signs all plus), corresponding rows are to be equated; in this way the above stands for 5 equations for the m'i. In exactly the same way there are 5 equations for each of the n,r ,i\ the system for the Ti's is deduced from the above by writing n in place of m on the left, and j3 in place of a on the right; similarly the t , i sets are written down by replacing in the above vi and a by r and 7, by i and 5 respectively. Thus, (with (50)) there are 25 equations to determine the constants of transformation (9), there being 5 sets of 5 equations each, and the determinant of each of the 5 systems is Hi h\ C\ di 02 62 C-2 di az bz Cz dz = k = 5/,V(l-P k a\ jSi 7i Si ai P2 72 ^2 OC3 Pz 73 ^3 04 Pi 74 ^i S") = I. and S-S-'S" = S"; also if SS"'=-S". S"' = S-^S", and if S'" is an S, (S^iS") = + l, • Papers, I; especially ON THE CTCLOTOIIIC QTHNAKT QUINTIC. 71 which requires that a determinant of identical form to D shall = + 1 , hence the S'" is not distinct. Thus all the required transformations are found uniquely. If S is any one of the transformations, and for Z) = 1 , l\ , /^ , /j , f^ any integers, SS' gives all transformations, where S' = S" S'" , and S" = 1 m'l ?«2 ^3 m\ "'i «2 "3 n\ •■i ^2 J-s r'i A U ^3 h S"' = \l h h \ h\ I 10 10 1 Put \a^hxCiiz\ = k, the cofactor of 1 in S" is 2) = 1 . Then <» solutions of this for a^, bo, Co, do may be found; hence also further solutions from the multiplication of J^ (as a determinant) and D , say Z> • i ao > • • • j <^3 1 ; multiphing by rows of first and columns of second. The greatest divisor of ao,bo, Co, do is a factor of Jc, and by the general theory ( S" ) may be reduced to the form in which the principal diagonal of the (4th order) determinant is composed of factors of k , the numbers below it of zeros, and those above it of positive or zero elements less than the corresponding /c-factor in the diagonal, — in fact by the usual theory of the re- duction of a matrix to canonical form. Now if any set aQ,bo, Co do be chosen as above, which make I Co &i Cidil = k, then 00 sets are found on multiplying by the « values of D, as above, but there is in each of the classes so determined always only a finitie number ob- tained by use of the reduced matrix; hence in each class is a " reduced " form from which the others in the same class may be derived, being equivalent thereto. As the entire idea of reduction and classes is worked as well for n-ary n-ics arising from Lagrange's kind of equation in fact more clearly, the special case of this paper is carried no further, but will be recurred to when the general theory is given. One result may be stated which (if new) is of interest, the number of classes cannot exceed the number of divisors of k" , where k is an integer represented in the n-ary 7i-\c. Finally in this connection it may be readily shown how from one transfor- mation of a [ 0] • • • [ 4] into a [ 0'] • • • [ 4'] all may be found by means of the F = 1 equation of § 2; as this generalizes at once to forms of nth order and degree, it is given. 72 In (9) ON THE CTCLOTOMIC QUINARY QUINTIC. = S ^ Z / lo mo • ■ • to let the result of priming all the letters, thus /J,, • [4]; F'=[Q'] , f^ be denoted by S', [4'] similarly for S"; and let F^ [0] be such that S'{F') = F, viz., let F' be transformed into F by means of S'; let also S'S ^S", and further let SiF) = F so that S is any (or all) of the automorphics of F , then all the transformations of F' into F are uniquely determined by { S"\ . As usual, in multiplying the two matrices S' and S, the multiplication proceeds by rows of S' and columns of S. Let the corresponding coefficients to k, A, ••• , etc., of F heinF', k' , A' , etc. Then from (49) it is seen that the S" are determined uniquely by k% + A'ji + B-ji + Co /3 + z'o /;- = I'o/o (ro,r„r„r,,r,) k'm'o + A'oVj'i + Fomg + Com'^ + D'om'i = Xo/o( m'o , ■•■ m\) (52) A-Yo+--- =A'o/i(''o, ••• f<) where the/' are the correspondents in F' of the/ in F , (see 46, 47), and (see 49) Xo is the first factor of the equation F {_ao, ••• . a^) = 1, where note, F has nothing to do with F of the two forms, it is the symbol of the equation considered /^ = 1, in § 2; there can be no confusion. For it may be verified, on expanding, or directly from the formula; in (30), (34), (37), that the necessary and sufficient conditions for S' (F') = F are kk'^^^foir,, r„ •••, r, )./;(/'o, Ti --o-'-fiCo Aok'* = fo (wi'o, m'l, m \)-fAro,n 5oi''=fo(«o. Co K == Jo \ ''o I Dck'*=fAfo, , n\) -fiir^, Ti ■■■) •)---fAro- ■fAro---). ■fAro---). ■fATo---), ), ), (53) ON THE CYCLOTOMIC QUINARY QUINTIC. 73 and these, combined with (49) give Vo(^o. ■■■ri) + hfo){m'o, ■•■ m\)+ [-lifoifo, •••O ~ Xofo ( • • • U) (54) fofoil'o, ••■ ri) + tifo{m'o, ■■■ m\)+ \-tifo(.fo, ■ • ■ Q = Xofo (4» ■ * ■ U) • and by direct substitution, or by inspection, (52) follows from (54). Re- writing (52) in final form, all transformations of Fi into F are deduced from any one such transformation by means of the formulae /o ( '4'. w'l . I'h I'z, ^4 ) = ^ofo ( Wo . w'l . ^2 ,^<'3'^''^)> (55) where u stands for the letters 1 , m, n, r, t in turn, and as pointed out in the automorphics, equations such as (55) implicitlj' contain 25 linear equations. In conclusion of this section, a very brief indication of the forms in the general theory analogous to those of this section, is given. It is seen that the present paper treats only a very special case of those described now. The form [0] • • • [ 4] here considered has arisen in the theory of the division of the circle, or what is the same thing, in relation to the irreducible in /J [ 1 ] equation a;"-i + a;"-2+ 1- 1 = 0, where n= (p- l)/5, and pis a (real) prime of the form lOn + 1 . The equation F = 1 of §§ 1, 2, has been seen to give rise to the automorphics of [0] ••• [4](= G , say ) , also to the theory of (Gaussian) equivalence of (r-forms, simllarlj- to the com- position of such. But in the most general case, where all numbers considered are to be ordinary algebraic integers, the irreducible equation ^dll be of the type x" + aiX"-^+ l-a„= 0, where the a's are real integers. The most general case, where the a's are al- gebraic integers, is treated in the same way precisely as this (apparently) more simple case. Call the equation here corresponding to the i^ = 1 of §§ 1, 2, the " F equation "; it is constructed as the norm of the general alge- braic integer of the nth degree (belonging to the above equation). This is in fact {F) & Lagrange's duplication form, and the resulting forms are of the nth degree on n variables. However this is not strictly the generalization of the § 3 forms G , nor is the F derived in this way so general as that of § 1. 74 ON THE CYCLOTOMIC QUINARY QUINTIC. To find this general F, proceed as follows; let p be any real integer in the field, whose factors are da, i>i, d^, d^, ■■• , t?^i; so that the O's involve no ir- rationality not in the field. Then if Ai= U+aidJ{ai) + aU2f{ao)-{ + a",-' t?„-i/ (a„_i) , where / (a,) = Xo 4- Xi a, + a-o a^ + • • • + Xn-2 oci"'^ , the " i^-equatlon " is JlAi =1, (i = 0, •••, h - 1), where the a:'s and u are the n variables; by a slight modification of Dirichlet's method, it may be shown that this F-equation has oo"-i solutions, as in the case of § 2, the f -equation had oo-* solutions, and in the same sense. Next, to find the G-forms. Subject the variables u, x; to a linear transformation » — / '00 'ol • ■ • ^On-l , ^n— Ij In— I, 1 ■ • " 'ti— 1, n— 1 i then if before the transformation Ai was F (u, Xg, xi, •• • , x„-2) , let it after the transformation become A\ , so that A', = F (looJl, 0, • •• , J-n-l, o) U' + aiF (loiylii, • •• , In-l, l) X' + ■ • • + \- ar' F(lo,n, ■■■, In-l, n-l) ^'n-l , and let Z,y denote what Ai becomes when the jth column of 5 is put for the variables, (each for each), and let Li ^ Ltjj Llj • • • i/n— 1, J , then the general factor [ /] is A\ Lj Ltj (i = 0, • ■ • , n — \); and j = 0, •• • ,n — 1 gives n such factors as a " base." For j fixed, the product ir[i] is constructed; and gives the most general G-form. It may be shown by the methods of this section, that [ i] = ku' + Ail x'l -f yi,2 a-2 + • • • + A^n-i a-^-i , where the A,j are all of the same form, and for thc/values of i = , • • • , n — 1 , the/orm5 of the A,j run through, in some order, the forms of the At . That is, there is a principle of permanence of form with respect to w-adic number coefficients, as well as with respect to the variables, in the G forms considered. The stu<3y of these G-forms seems to constitute the most natural extension of the properties of binary quadratic forms; for the theory is not specialized ON THE CYCLOTOMIC QflNARY QOTNTIC. 75 by extension of degree, that is, it exists for these forms in toto. The G forms are rational integral functions of their variables; the coefficients are numbers in the n-field; when expanded in full, real also in coefficients of ar" + ai x"'^ • • • = . §4. A form being given, it may be interpreted as the embodiment of arithmetical or of geometrical facts. The two interpretations are not mutually helpful; thus few, if any new arithmetical truths have been discovered by geometrical reasoning. But often the statement of an arithmetical theorem or its proof, in geometrical language tends to conciseness; e. g., Eisenstein's proof of the Law of Quadratic Reciprocity, Poinsot's proofs of Feeii.vt's and allied theorems, or Catlet's proof of Wilson's theorem, also the works of PoiNCAKE and Klein in the theory of forms, and Sylvester's theories of Residuation and Partitions. But if analogous methods are applied to higher forms, one essential feature is at once lost, viz., space-intuition; thus in place of the simple and obvious fact that the diagonal of a rectangle divides the rectangle into two congruent right triangles, which observation obviates tedious arithmetical calculations, there is a complicated question regarding the " super-lines " of symmetry of a regular solid in ?i-space. Sight intuition is useless here, as it was not in the plane, and all that is accomplished is a suggestive simplification of language. On the other hand, forms which present themselves in the study of numbers are frequently those of intrinsic geometric interest. As an example that has possibly not been explicitly stated before, the triplication of class property in binary cubics, due to Eisenstein, has its analogue in the geometrical theorem that the equation of the general develop- able of the fourth degree in tetrahedral coordinates is the binary cubic dis- criminant = 0, the two theorems are formally equivalent. This theorem is not isolated; its generalization to n-space is involved implicitly in Lagrange's n-ary n-ic forms; the discussion of this is reserved; there is a difficulty in stating the degree of multiplication. Here it is simply pointed out that the geo- metrical interpretation of F = k is not likely to suggest any new arithmetic; however, the properties of so simple and symmetrical a surface, it being in many respects the 5-space analogue of the sphere, may not be without interest. The complete discussion of this " surface " is too extensive to be undertaken here; only those properties which result from a direct reading of its equations are considered. Although the interpretation is in terms of Ss, all may be carried out in ordinary space, for the 5 coordinates of the points considered as belonging to Ss, may by the usual convention be regarded as the "coor- dinates of a sphere." The points particularly considered are those which form a self-congruent net work under the automorphic transformations. By reason of che Imaginary nature of the coordinates in Si, the immediately foIlo\\'ing 76 ON THE CYCLOTOMIC QUINARY QUINTIC. configuration of 120 points reduces in S3 to points, not spheres. For, the coordinates of any point of the set are projectively transformed into (p°, p*, p". p''> p") , a, ■ • • , <7 all different, and the (120) spheres (xi, • • • , ars) in S3 which correspond reduce to points, for xi + xl + xl + zl+4=0. But there yet remain 3,005 special points (as seen presently) whose coordinates do not satisfy this equation, and these correspond to actual spheres, though possibly of imaginary center or radius in S3; in general one x at least is imag- inary. The whole surface must be interpreted with reference to these ideal elements; if F be expanded into real form, the G of § 2, a different configuration is obtained, containing in its equation over 3,000 terms, and obviously such a collection of symbols is quite unmanageable. The sphere interpretation is only pointed out; the natural method of discussion seems to be based on considerations of Ss . One special surface is of such frequent occurrence that it is designated by a particular name, viz., " 5-oid "; the rest of the nomen- clature (where needed) is taken from Cayley's " On the super lines of a quadric surface in space of five dimensions." * The 5-oid is ul+ul + vl + vl+ul = 5, and bears much the same relation to F = k that the director circle bears to conies, or the director sphere to quadrics. Let = (p (mo, • •• . W4) be the equation of a surface in S5 which when referred to homogeneous co- ordinates Wo, ••■ , lis becomes $ = 0, then the tangent surface at the point ( w'o , • • • , «i ) of ¥> is defined to be (WO *„.„ + h M5 *„'.) = 0, M6 = 1 , where the notation means that u' is to be substituted for u in the partial derivatives of # ^ith respect to «o, • • • . "5, and finally ws is to be put = 1 . This definition is in accordance with usage. Let F^^l^UoUiUiUzUi^l; Ui= wo+P*«i+p""2+p"w3+p''w4, the general F = k having been reduced as in § 2 to this form by a linear transformation (equivalent to an affine transformation), then F^.= {l/Uo+P'lU\+p''/U',+ p''IV3 + p''/U\)F'; (i = 0, ...4). (1) Hence the equation of the tangent surface to F = 1 at the point (u'o, • • • . «!) is 'Papers, Vol. IX., p. 79. ON THE CTCLOTOMIG QUINARY QUINTIC. 77 uo{ilu;-{'iiu[+i/u; + i!u; + iiu;,o, 0,0,0) l/U;, 1/ U-,) + u, {1/ U'o,!/ U;,ll U;,lf U\, II IT,) + u, (1/ f/'o, 1/ U;, 1/ U'„ II U\, 1/ U\) -5 = 0; (2) the ( ) being the usual notation for complex 5-numbers; F' = \, since the point ( w'o , • • • , k'i ) is on the surface F = \. The equation may be shortened to Uc to + Wi t)i -f W2 1)2 + W3 fS + W4 f 4 — 5 = , where the V's are the quantities in ( ) of (1). In virtue of the fact that points and surfaces of the same kind as the tangent surface are dual elements in Ss, the c's will be referred to as the coordinates of the tangent surface. Nearly all the properties of F = 1 that follow are from a direct reading of these two equations. The surface whose coordinates are I'o , • • • , »4 will be denoted by (»o, ti, V2, V), Vi) . Multiplying the c's by 1, 1, 1, 1, 1 and adding, it is seen that the point (1,1,1,1,1) will lie on the tangent surface (2) if U'q=1, that is, if the point of tangency lies on the surface ( 1 , 1 , 1 , 1 , 1 ) ; or, the tangent surface to the surface F = 1 at any point of the intersection of F and the surface (1, 1, 1, 1, 1), passes through the point (1, 1, 1, 1, 1); or again, the surfaces which pass through the point (1,1,1,1,1) and touch F, meet the latter in the points common to Fo = Ui Uo h\ ir, - 1 = and F . By analogy, Fo is called the "polaf 4-surface " with regard to the point (1,1,1,1,1), and the surface (1,1,1,1,1) the " polar 1-surface " of, etc. The " 4-surface," " 1-surface," are so named to distinguish two things having similar properties. As in the foregoing, similarly for 1 , p , p'j p'j p^ , multiplying in turn by these and adding, it is seen that the 1-surfaces enveloping the point (1 , p, p-, p', p^) and touching F , meet the surface (F) in the points common to F and the polar 4-surface Fi = C7o C/j Uz Ui-l = 0; and so on for each of the remaining 3 special points ( 1 , p' , p* , p , p' ) , ( 1 , p' , p , p* , p^ ) , ( 1 , p* , p' , p^ , p ) whose polar 4-surfaces are respectively F^UeUiUiUi- 1 = 0; F3^UoUiU2Ui-l = 0, and F4S UoUiUiUi- 1 = 0. These 5 special points are among the 120 which lie at zero distance from the origin on the 5-oid nt + l4+y|+^'l+^'l-5=^ 0; 78 ON THE CYCLOTOMIC QUINARY QUINTIC. distance being defined as the square root of the usual quadratic function, in accordance with one usage^ not as the fifth root of a quintic function, which in some respects seems better here. On this 5-oid there are at once 3,125 ( = 5* ) special points obtained by assigning all 5 possible values of p* to each of the w's in the general point ( j/q , Mi , ^2 , «3 , «4 ) . The 5-oid is transformed into itself by any substitution which merely permutes the suffixes, hence it is invariant under the symmetric group of degree 5. Also, if a^, • • • , gj be any of the 5 numbers p', then Sj^ fay bj Cj dj ffy transforms the 5-oid into itself; and the Sj generate an abelian group of order 3,125; hence the special 5-oid is invariant under the abelian group of order 5^, If Tij denotes the result of putting all the elements in Sj except that one in the tth row = 0, then the group is generated by T.-y, (i = 1, • • • , 5); that is by 5 operators of order 5, Ti, • ■ • , Ts, each of which generates a cyclic 5- group, etc., so that the entire group may be represented by { Ti , ^2 , Ts , 2^4 , Ts} . Returning to the 5 tangent 1-surfaces already considered, it is easily seen that each passes through five special points not on the 5-oid (nor on F). Also it appears presently that each of the 1-surfaces gives rise to four others, so that associated with any of the 1-surfaces there are 25 special fundamental points lying on the associated 5-oid, and these 25 points correspond to the powers of Ti, • • • , Tb. Again, the mean-center of each of the sets of five points lies on the corresponding l-surface, and the five mean centers lie on a 5-oid of 5~^ times the linear dimensions of the original 5-oid. Here in passing, the remark made concerning the analogy of the 5-oid to the director sphere or circle may be partly justified; the working out in detail is reserved. The theorems quoted regarding circles, etc., all follow at once from the parametric equation of a circle, and the resulting sin (a =*= 6) formula that such repre- sentation implies. The analogy here is possibly due at bottom to the fact that the sine is singly periodic of a single argument, the functions used here to represent a point on F = I are by § 2, 4-fold periodic on 4 arguments; the addition theorem is algebraic, and geometrically can be made to interpret the properties of mean centers in the same way that the trigonometric functions give rise to the following. The tangents at the vertices of the parabolas touching each three of 4 lines joining 4 concyclic points and having those ON THE CTCLOTOMIC QUINARY QUINTIC. 79 points as foci pass through the center of the rectangular hj-perbola circum- scribing the 4-point, and if 5 points be taken, the corresponding tangents at the vertices pass by 4's through 5 points lying on a circle of one-half the linear dimensions of the original circle; and if a sixth point be taken, the six cor- responding circles are concurrent; the seven such points obtained by taking a seventh point on the circle, are concyclic, and so on indefinitely, all the circles being equal to one half the original circle. Again, ]\Ii^uel's theorem states that the intersections of the 5 circles circumscribing the triangles formed by producing the sides of a 5-side are concyclic, (the intersections being the foci of 4-line parabolas; and it may be shown that the 5-point circle is the director of the conic touching the 5 lines; taking a sixth line, a seventh, • • •, there is a chain of theorems as above. But the center of the rectangular hyperbola is midway between the mean center of the 4 concylic points and the center of the circle. In the same way there is a surface in Sj corresponding to the hyper- bola, and so on; there is a chain of theorems. But again, as indicated in § 2, the theorems may be extended to surfaces of lower degree in other spaces; e. g., to quadrics in 4 space, the existence of a generalization of these circle- properties depending on that of an algebraic theory of composition for the degree-order considered. For ordinary space, Eisenstein's form quoted in § 1 could be used to find results precisely analogous to the present for Ss . Returning; for brevity denote by 0;, 1,-, 2,, 3,-, 4,- respectively the points all of whose coordinates except the first, . . . fifth, are zero, the non-zero coordinate being p'. Also by {tq, • ■ • , lu; Ic} the 1-surface lio I'o + • • • -\- UiVi = 5k , (?\ p'\ p«;5pM;also 1, P' and by {0, i, 2i, 3j, 4f, k } the 1-surface by [i,j,k, I; c] the 4-surface Ui Uj U, Ui = c; and hyl/s {0 ,{,2i,3i ,Ai) the point whose coordinates are 1 / S , p* / S , p^'/ 5 , P^'/S, .p*'/S, then: 1-Sur- Contains face Points (Ii) {0,4,3,3,1 (III) {0,4,3,2,1 (IIIi){0,4,3,2,l (IVr) {0,4,3,2,1 04,lo,2i,32,43 (4, 0, 1, 2, 3)/o [0,1,2,3 Oo,l2,24,3i,43 (0, 2, 4, 1, 3)/o [0,1,2,4 (Vi) {0,4,3,2,1 ih) {0,3,1,4,2 (II2) {0,3,1,4,2 (III2) {0,3,1,4,2 (IV2) {0,3,1,4,2 (V2) {0,3,1,4,2 WTiose Mean And Polar 4-Surf. Has for Center is Corrcsp. Pole 0} ; Oo, 1:, 2., 83, 4, (0, 1, 2, 3, 4)/5 [0, 1, 2, 3 1) Oi, 12,23, 34, 4o (1, 2, .3, 4, 0)/5 [0,1,2,3 2} 02,13, 24, 3o,4i (2,3,4,0,l)/5 [0,1,2,3 03,l4,2o,3i,42 (3,4,0,l,2)/5 [0,1,2,3 Oi, I3, 2o, 3;, 44 (1, 3, 0, 2, 4)/5 [0, 1, 2, 4 02,14, 2,, 33, 4o (2, 4, 1, 3, 0)/5 [0,1,2,4 03,lo,22,34,4i (.3,0,2,4,l)/5 [0,1,2,4 04,li,23,.3o.42 (4, 1, 3, 0, 2)/5 [0,1,2,4 0] (0,1,2,3,4) 1] (1,2,-3,4,0) 2] (2, 3, 4, 0, 1) 3] (3,4,0,1,2) 4] (4,0,1,2,3) 0] (0,2,4,1,3) 1] (1,3,0,2,4) 2] (2,4,1,3,0) 3] (3,0,2,4,1) 4] (4,1,3,0,2) Q 80 ON TUE CTCLOTOMIC QUINARY QUINTIC. (13) {0, 2, 4, 1, 3; 0} Oo, I3, 2i, 84, 42 (0, 3, 1, 4, 2)/5 [0, 1, 3, 4; 0] (0, 3, 1, 4, 2) (113) {0,2,4,1,3;!} Oi, l4,22, 3o,43 (l,4,2,0,3)/o [0,1,3,4;1] (1,4,2,0,3) (Ills)! 0,2, 4, 1,3; 2} O2, lo,23, 3i,44 (2,0, 3, 1, 4)/5 [0,1, 3, 4; 2] (2,0,3,1,4) (IV)3 {0,2,4,1,3:3} 03,li,24,32,4o (3, 1, 4, 2, 0)/5 [0,1, 3, 4; 3] (3,1,4,2,0) (V3) {0, 2, 4, 1, 3; 4| O4, I2, 2o, 3,, 4i (4, 2, 0, 3, l)/5 [0, 1, 3, 4; 4] (4, 2, 0, 3, 1) (14) {0, 1, 2, 3, 4; 0} Oo, I4, 23, 32, 4i (0, 4, 3, 2, l)/5 [0, 2, 3, 4; 0] (0, 4, 3, 2, 1) (114) {0, 1, 2, 3, 4; 1 } Oi, lo, 24, 83, 42 (1, 0, 4, 3, 2)/5 [0, 2, 3, 4; 1] (1, 0, 4, 3, 2) (III4){0,1,2,3,4;2} O2, li,2o,34, 4, (2, 1,0,4, 3)/5 [0,2, 3, 4; 2] (2,1,0,4,3) (IV4) {0, 1, 2, 3, 4; 3} O3, I2, 2i, 3o, 44 (3, 2, 1, 0, 4)/5 [0, 2, 3, 4; 3] (3, 2, 1, 0, 4) (V4) {0, 1, 2, 3, 4; 4) O4, I3, 22, 3i, 4o (4, 3, 2, 1, 0)/5 [0, 2, 3, 4; 4] (4, 3, 2, 1, 0) (lo) {0, 0, 0, 0, 0; 0) Oo, lo, 2o, 3o, 4o (0, 0, 0, 0, 0)/5 [1, 2, 3, 4; 0] (0, 0, 0, 0, 0) (IIo) {0, 0, 0, 0, 0; 1 } Oi, li, 2i, 3i, 4, (1, 1, 1, 1, l)/5 [1, 2, 3, 4; 1] (1, 1, 1, 1, 1) (Vo) {0,0, 0,0,0; 4} 04,14,24,34,44 (4, 4, 4, 4, 4)/5 [1,2, 3, 4; 4] (4,4,4,4,4; The whole configuration may be expressed by the symbol | 1 2 3 4 j; the first set of elements is written down from this, the l-surface by replacing each non- zero symbol by its complement (mod 5), the points are clearly enough indicated, as also is the mean center, the polar 4-surface is written down by changing the last symbol 4 to 0, and the pole comes at once (as throughout) either from the l-surface, the points, or the mear. center. The first row being found, the 4 following come by addition of unity (mod 5) to the variable symbols. This gives the complete set (h) • • • (Vj). The sets (I2) • ■ • (Fo); (I4) • • • (V4); • • • (I3) • • • (V3) come by doubling and reducing each corresponding symbol in the first set. The last set starts with zero as a basis, and then is written down as was the first set. There are here 25 of the special points, and 25 of the special 1-surfaces; each l-surface contains 5 of the special points, and through each point pass 5 of the 1-surfaces. In order to place the mean points, some further definitions regarding a 5-oid are required. The special 5-oid was wg+ |-w»= 5, which by making the equation homogeneous may be written the point (0, 0, 0, 0, 0) will be called the center, and the constant Ws the radius of the 5-oid. The tangent l-surface at (ro, • ■ • , Ps) is Wo rj + «l V\ + U2Tt+ «3 li +Uiii-2o = 0. It is now seen at once that the 25 mean centers all lie on a 5-oid whose center ON THE CTCLOTOMIC QUINARY QUINTIC. 81 is at ( , • • • , ) , and whose radius is 1 / 5^'* . Also the 25 poles lie on the 5-oid whose center is (0, ••• . 0), and radius 1/5^'^; that is, the "mean 5-oid" is 5~* times ths linear dimensions of the " polar 5-oid." The tangent 1-surface at ( 7)o , • • • , ^4 ) to the mean 5-oid being «o «4 + • • • + M4 cj — 5"-* = 0, at the point (0, 1, 2, 3, 4)/ 5 this is «0 + P*Ui+ p' Ui + p^U3+ pUi— 1 = 0, which is parallel to the 1-surface corresponding to this mean center, viz., (Ij). In the same way, it may be shown that for each of the 25 mean centers, the tangent 1-surface to the mean 5-oid at a given mean center is parallel to the 1-surface containing the corresponding pole, or what is the same thing, is parallel to the 1-surface containing the 5 points of which the point of contact is the center of mean position. As usual, defining the distance between two points u, u' of (Sg to be { (wo — y'oY + • • • + ("4 — ■w'*)" V'^, the distances from, the common center of the two 5-oids to the special points ij are equal in sets of 5 to 1 , p , p^ , p' , p^ . Hence these 25 points lie by 5's on the 2-surfaces, Wo+«?+J/2 +«!+«!- P"'= (i = 0, 1, 2, 3, 4); and by Cayley's theory (1. c.) these 2-surfaces are ruled quadrics in S^. These special 25 points are in fact on 25 special surfaces connected rvitli the configuration |01234|, a 1, 2, 3, 4, 5-surface respectively, an t-surface being one of the ith degree; the equations being «^+ W+W2+ W3 + W4- P'= 0; (i,y = l,2,---,5). Each of these surfaces contains 5 of the special points; all the surfaces of the same kind contain all 25 points, and through each point pass 5 surfaces, one of each kind. The entire ( 1 2 3 4 | configuration is clearly only part of a more general configuration connected with the surface in S^; which will now be examined. The figure is more easily kept in hand if a special symbol 1, 2, 4, 3 2, 4, 3, 1 3, 1, 9 4 4, 3, 1, 2 is introduced; the only defining properties of S to be (at present). (1°) That any number occurring in S is positive, < 5 , and hence only one or more of the numbers 0,1,2,3,4; (2°) An interchange of any two rows leaves S unchanged ; (3°) An interchange of two columns is not permissible. The last restriction 82 ON THE CYCLOTOMIC QUIN'ARY QUINTIC. may be removed later, as will be seen, once the properties and manner of operating with S have been established. Considering now the difTerent possible values of 5 when the same (permissible, i. e., > 5 ) constant is added to each member of a row, \eta,b,c,dhe the numbers 1 , 2 , 3 , 4 in any order, but no two of the letters are to have the same value; put S'= 1 + a. 2 + a, 4 + a, 3 + a 2 + b, 4+6, 3 + 6, 1 + 6 3 + c, 1 + c, 2 + c, 4 + c 4 + cf, 3 + d, 1 + d, 2 + d then S' has 24 distinct values, for by definition an interchange of columns changes the value of S' for any system of \'alues for ( a , 6 , c , cf ) , so if two values of 5' become identical, this can happen only in virtue of any two rows becoming identical. Say the first two rows of two values for S become identical (if the two identical rows are not the first, the definition permits them to become so), say 1 + a, 2 + a, 4+0, 3+a = 1 + a', 2 + a', 4 + a', 3 + a', in this order (necessary) ; this requires that a = a' (mod 5) , and since a , a' < 5 , must a =^ a' . But the values oi a,h , c, d must all be different in a given S, therefore S takes 4 • 3 ■ 2 • 1 = 24 difTerent values in all. Hence, if the sum of two symbols S , 5' of the above kind be defined as the symbol of the same kind resulting from the addition of corresponding numbers in cor- responding columns of S and S' , it follows that with respect to addition the foregoing 24 values belong to the sj-mmetric group of degree 4. Arranging the 24 values according to the number of zero values that may occur in any column, consider first the case of 4 zeros in a column; the system (4, 3 , 2 , 1 ) of values for {a,b rC,d) reduces the first column to all zeros, and therefore the systems of values obtained from (4, 3, 2, 1) by doubling and reducing, reduce the second, third and fourth columns respectively to all zeros, for these columns in S are obtainable from the first by the same process. Or again, the com- plementary S to a given S being for a given column that in which all the columns are zeros except the column whose elements are complements modulo 5 of the given column, the initial S is reduced to the form in which its first, . . ., fourth column is all zeros by adding to S the complementary S for respectively the first . . . fourth columns. There is no form of S in which three elements of any cohimn are zeros, unless the fourth element in the column is also zero, for a , 6 , c , 3 4 10 J 4 2 1 ) 13 2 J 12 3 4 13 2 2 4 3 1 3 4 2 4 10 3 2 14 14 2 2 13 3 12 4 4 2 3 3 4 1 10 3 4 2 14 3 2 1 4 3 12 4 2 3 after which it recurs. The effect is to reduce the number of zeros by one. Since each of the X^ contains 4 zeros, and no three are in a column, and since no 4 elements in a column are identical, no one of the X^ may be reduced by addition of Si to a form containing less than or more than, and therefore exactly 3- zeros. Hence the 20 values of P ( = ( 1/0 > ~ X^ ) , are in two sets, the first of which contains 4 S's such that their ( 1 / X) (or X^) parts contain 4 zeros; the second set contains 16 S's each of whose X^ parts has only three zeros. (By X^ part is meant what remains in an J? when the first row and column are deleted). Furthermore these 20 values are all distinct values for the Z^; for no one of these F can contain more than 4 elements each equal to the same integer , • • • , 4; and each of them contains exactly 4 equal respectively to 1 , 2 , 3 , or 4 , except the first set of 5 which has 4 zeros, these numbers 1 , • • • , 4 only occurring (in the 4-way distribution) in the places of the original zeros. Clearly then, no two of the sixteen can be identical. Considering now a typical X', say 2 3 4 2 1 4 3 4 1 3 2 1 it is seen in the same way as for the X^, that each has exactly 3 zeros by the addition of Si, and as each has at first not more than 3 identical elements of any kind except zeros, no addition of S,- to any of the X' can produce a form in which there are more than 3 zeros; also, since there are exactly 3 identical elements at each stage, each of the P , except the initial 4, have 3 and only 3 zeros. It is seen at once that the sixteen corresponding values of the P (each with 3 zeros), are distinct from the sixteen P, found above; for in no two of the l^ , P initially is the distribution of zeros similar; hence in no two of the derived P , P can the 4 identical elements be similarly arranged. By precisely similar reasoning, it is easy to see that no two of the similarly derived P , P , P CMi be identical among themselves, or to any of the I, P, P. Hence the distinctness of the whole 120 l,t-,P,P,P,Pis proved, also the separation of them into classes according to the distribution of their zeros in one case, identical elements in others. Summing up; there are of all the Z", 4 which 86 ON THE CYCLOTOMIC QUINARY QUINTIC. contain in their (X) part 4 zeros in the same column; 4 with 4 zeros, 2 zeros in each of two columns; and 16 with 4 zeros, 2 zeros in one column and one in each of two others; that is, 24 with any distribution of 4 zeros in all; and 6 sets of 16 each derived from the above by additions of multiples of Si, con- taining each 3 zeros, each set of 16 being derived from an initial X, X^ • • • , X' by the addition of 5i , 2Si , 3Si , 4Si . Of these last, just 8 have their 3 zeros in the same column; and so on. To each of the 120 symbols so derived corresponds a configuration of one point ard a definite surface such that the tangent 1-surfaces to the original surface F at its points of intersection with the definite surface, all pass through the fixed point. This 120 configuration is not studied in detail here, nor is the total configuration of 3,125 fixed points of which the 120 is only a part. In order to investigate the figure completely, it will be necessary to consider along vdxh F , its " reflected " surfaces; the idea of a reflected curve in a plane, e. g., of a circle in the axis of reals by change of sign of the ?/-co6rdinate, etc., being carried over into 5-space (ideal), where the F-surface has " reflections " in several planes, or about several axes. The extension is natural and con- venient, for reflection as ordinarily defined consists in a multiplication of a coordinate by — 1 , that is by one of the units of the natural numbers; in the ideal 5-space here considered, " reflections " are accomplished by multipli- cations of one (or more) coordinates by units. As there is an infinity of units, so each point has an infinity of reflections about oo lines, super-lines, etc., but all may be referred to the fundamental system =tp'(i=0, 1,2,3, 4). This extension is necessary, if a complete discussion of the geometry of the " configTiration 3,125 " is to be given; algebraically it is a discussion of certain associated forms F {p"'ao, p" ai , p*" ao , p' aj , p' Ui) , which present themselves naturally in considering F {ao, ■ ■ ■ . Ui) . The totality of distinct forms F obtained in this way, constitute in the ideal Ss, all the " images " of F ob- tained by reflections. (Two forms are not distinct if F' = eF , where e is a complex unit). Thus, the more general configuration contains 3,125 points, an equal number of polar surfaces, and the whole configuration is invariant under the group which leaves F and its fundamental images invariant. The special surfaces considered are separated into species according to the form of the 5 (and other symbols constructed in the same way), corresponding to them. For clearly, each 5 corresponds to one and only one form of the equa- tion of condition that the tangent surface to F shall pass through a given point (p° p'' p' p"* p"); the symbols S correspond to a , b , c , d, g = 1 , 2 , 3 , i , 5; no two letters having the same value. The rest of the configuration is derivable on the hypothesis that at least two of the a , ■•• , g have the same value. Again, all these points lie on the special 5-oid. The 120 point-surface con- figuration has been selected out of the total because its properties may be ON THE CYCLOTOMIC QUINARY QUINTIC. 87 seen by direct reading, also its points are all those and only those no two of whose coordinates are equal; also, as already noticed, in the sphere-inter- pretation, these 120 are those and all those which reduce to points (as spheres). It is clearly not invariant under the group of all possible simple reflections; this group is that already found of order 5'; hence the substitutions which leave the 120 points unchanged, that is, permute some or all of them among themselves, must be in number 1, 5, 25, 125 or 625. First, looking at a given pole, e. g., that represented by (0, 1,2, 3, 4), the only substitution of the group which leaves this unchanged is the identity, but obviously, it is transformed into another pole of the set by r= fp P p. and therefore also by T-, "P , T^. Similarly, it is seen that { T ] transforms any pole into 5 poles of the set. Also no transformation which contains two dissimilar powers of p can change any one of the set into another of the set. Therefore the poles may be arranged in 24 sets of 5 each, such that all in a given set are derived from any one of them by the transformations { T } . Five such sets have been shown in the table, for the pole sets (Ii) , (h) , ih) , (h) , (lo)', (III), (Ila), (Ills), (IV3), (lo), etc. The configuration of 120 poles is thus composed of 5 sets of 5 five-point figures which have a point in common, — ^that is, from the fundamental points or" vertices " lo, Ho, HIo, IVq, Vo, are " drawn " sets of five-point figures; 5 from each fundamental vertex, and any set of 5 five-point figures having a common vertex is transformed into itself by 1 , T , 'P , P; T^; and the whole is inscribed in the special 5-oid vl + ti\ + t/H- «! + lil 5 = 0. Other properties were given above. The entire set of 120 poles, etc., is char- acterized by the symbol {ah cd g) , all the letters distinct; the remaining 3,005 are given by the symbols in sets (aaedg);{a,a,a,d,g);{a,a,a,a,ff} (^a, a , a , a , a) . A set of values for a, • • • , ^ in a given S was denoted hy (a, b, c, d, g); and it has been seen that from a X , • • • , X^ , other values of S are derived by additions of multiples of Si; and it is clear that the resulting S is the same regarded as an (Z) • • • (/^) if the sjmbols (a, b , c, d, g) are augmented by addition of a constant to each letter, or if a distinct set of symbols (a, ■••,§) is found for each I as it arises; that is, from the set ( 1 , 2, 3, 4) , 0?/ the 120 symbols are derived by successively increasing each number in the symbol by 88 ON THE CYCLOTOMIC QUINARY QUIXTIC. a constant, and then reducing modulo 5 . But the /'s are 5 columned and 5- rowed symbols; hence it is not sufficient to increase the numbers in the (12 3 4), etc.; a means of identification must be given to show to what particular / this 4-row belongs. Clearly this is only a convenience, not a necessity, for no (12 3 4), • • • can belong to more than a single I, and any such symbol being given, it is possible to identify the I to which it appertains, but if a suffix indicating the constant by which every number is to be increased, is subjoined to the symbol, the trouble of identification is avoided. Hence the whole 120 symbols are conveniently represented by (Xy.), (X5,), ••., (Xy.)«; (i = l,2, 3, 4, 0); (y = l, 2,3,4); thus Xi2= (1, 2, 3, 4)2^(3, 4, 0, 1); n.z= (3, 2, 1, 4)3= (1, 0,4, 2); if the j denotes the index of the column in which the given function of X occurs in the " anharmonic table." For the sake of uniformity with preceding parts, the order will be X,,,= (1234),-; X,.,= (2413);-; X3.,= (3, 1, 4, 2),; X,.,= (4,3,2,1),-; where, e. g., Ki^(3+j, l+j, i + j, 2+j) (mod 5). Summing up; the symbols of the form S are written down as follows: the six usual forms of the anharmonic ratio of 4 elements, 1,2,3,4 are written down, and in each of the six classes so defined are put respectively the 4 equally valued anharmonic ratio-symbols. The six classes so formed all have suffix zero, the other classes correspond to suflSxes 1,2,3,4. In the 5-row symbol I corresponding to any of these (abcd)i the first row and first column are composed of the element j repeated in each 5 times, the (X)-part of the / is the iS derived directly from the {ah cd)j. Having written down the six anharmonic ratio signs as a first step, the three remaining sets of 6 each are deduced from this by doubling and reducing modulo 5 every number Ln the sj-mbols. As these symbols represent the polar surfaces, also the corresponding poles, and as the form of a surface is determined by the number and location of the zer^in its symbol, it is convenient to classify the ( ) on this basis. The pole of X.y ^W b' c' d')j is the point denoted by (j, a' ,b' ,c' ,d') , that is, its coordinates are p'', ■■■, p"". This may be taken as a definition, or an extension of the usage already adopted; the X's arose from the properties of the configuration, now conversely the X's may be assumed a priori and the resulting configuration deduced. The ON THE CYCLOTOMIC QUINAKT QUINTIC. 89 pole whose coordinates are the above will be said to correspond to X,y a pole is denoted by the corresponding X . If 5 poles X , X' , • • • , X" are so related that they are cyclically permuted among themselves by the transformations { T ] , then it is natural to say that the 5-point X' X" • • • X" has been rotated about that from which the 5 points are equidistant, distance having been already defined. If {ao, bo, Cq, do, go), •••, (at, •••, gi) are 5 fixed points, and (wo, ■ • • ,Vi) any point equidistant from them, then the 4 independent equa- tions (wo - aty +... + (u,-gJ-= (wo - a.+i)- + • • ■ + (w4 - gi+iY (i=0, ••-. 4), defining a line in Sj, determine, if it exists, the" axis of rotation " for the 5-point. The points X.-y give rise to 24 axes of rotation, one for each cyclic 5-point; e. g., the axis of the set Xi, o, Xi, i, Xi,?, Xi, 3, Xi,4, that is of the poles (0 1 234), (1 2 3 40), (2 3 40 1), (3 40 1 2) (4 1 2 3) is («o - py+{u^- p^y + (t/2 - p')= + (W3 - p'y+ (w* - D- = (wo - 1)^+ (wi - py+ (m2 - p^)-+ («3 - p')^ + (W4 - p*y, (uo- p'y+im- p''y+ (th- p'y+(u,-iy+ (u,- py = iuo - iy+{th - py+(t>2 - p'y+ («3 - p'y+iv^ - p^y, (wo + p'y+ (wi - p'y + («2 -iy+(u,-py+ (u, - p=)= = (wo-i)='+(wi-p)^+(«2-p=)-+(w3-p')-+(w4- p'y, (Mo - p'y + («i - 1)^ + (W2 - py+ («3 - P-y + (u^ - p^y = (vo-iy+in,- py+(m- p^y+im- p^y+in,- p^y. By inspection, this axis passes through (0,0,0,0,0) the center of the 5-oid, and in the same way all 24 axes pass through the center. It may be seen that all the axes intersect in the locus constituted by all points all of whose coordi- nates are equal for the above equations obviously pass through (arbitrary) {a, a, a, a, a) when they are written «0 (P - 1) + Wl (p' - p) + "2 (p' - p') + W3 (p' - p') + W4 (1 - P^) = Mo (p'' - 1 ) + wi (p' - P) + W2 (P* - P-) + J/3 (1 - p') + W4 (P - P') = i/O (p' - 1 ) + «1 (P' - P) + «2 (1 - P-) + «3 (p - p') + W4 (P- - P') = Mo (p' - 1 ) + «1 (1 - P) + «5 (P - P') + M3 (P'- P') + «4 (p' - P') = If the equations of the 1-surfaces corresponding to 5 poles Xi , 0, • • • , Xi , < be solved for their corresponding intersection, it is found that the point lies 90 ON THE CYCLOTOMIC QUINARY QUINTIC. wholly at 00; i. e., these 5 1-surfaces may be said to be parallel; in the same way all 24 cyclic sets of 5 are parallel among themselves, so that in this case the 120 poles and corresponding 1-surfaces may be said to constitute a regular body in 5-space. The body from which this was obtained by projection at the start is not regular, but symmetrical in 5 directions; in other words, the pro- jected body corresponds to a regular solid (pentehedron), the unprojected to an irregular star-shaped body; also the 5-oid is the correspondent in Ss of the sphere, whereas the surface from which it was obtained, viz., «g/A-g+wf/A-f+ ••• =0 by a- projective transformation may be considered the analogue of the central quadrics; the degree is 5 instead of 2, but the properties of the 5th degree are in some respects closer analogues than those of the 2d; in all events it is simply a fashion of speaking of algebraical (arithmetical) facts embodied in the form. Since 1'+ (P')'+ (P^')-+ (P")-+ (P^*)'= if i is not divisible by 5, it follows that the 120 poles are at zero distance from the origin, and do not coincide with it, in fact they lie on vl+ hwf-5 = 0, whose center is the origin. As in Ss or Si the geometrical language becomes slightly paradoxical, but the. 5-oid may be regarded as a semi-minimal surface; semi because all the points on it are not at zero-distance from the origin. The set of 120 poles are all those of the entire 3,125 which have this property. A result more consistent with ordinal^' 3-space could have been obtained by defining distance not as a function of squares, but of fifth powers for the ideal (Ssj such a definition (or its analogue) is not usually adopted. The apparent contradiction arises from the ideal nature of the elements considered. The distribution of 1-surfaces corresponding to the 120 poles in relation to the 25 fundamental points Oo, ■ • • , 44 is readily determined. If (01234), that is Xi, be the pole considered, the l-surface corresponding is found by writing down the complementary symbol {04321;0j, similarly from Xi, 3 (say), or (3 4012) is derived {0432 1;3); first by subtracting 3 from every term, and then taking the complement. The sets of fundamental points corresponding to Xi,o and Xi,3 are respectively Oo, li, 2;, Sa, 44, and O3, 14, 2o, 3i, 4; or as already pointed out, the suffi.xes are the same numerals in the same order as those which occur in the pole; hence the pole indicates by position of numerals what fundamental points lie on the cor- responding l-surface; e. g., given pole (2 3 1 4 0), the l-surface will contain O2, I3, 2i, 34, 4o. And generally, all X-symbols having a same numeral in ON THE CYCLOTOMIC QUINARY QUINTIC. 91 the same position (e. g., in the third place) will be so related that the corre- sponding 1-surfaces all pass through the so-determined point (or vertex). It is easily seen that the point Oo is common to the 24 1-surfaces "•10 > A20> ^30) ^iO't ••■> AlOJ '■'» ^40> Oi is common to the 24, ^11 J X2i» Xsij X41; •••; XjiJ ■■■» X^iJ — j and finally, O4 is common to "14 J A24> A34 , A44; •■■; Aj4 , *••> A44 . Similarly for the remaining 4 sets li, ••• , lo; ••• 44 ••• 43; each of which gives rise to a permutation of the X's as written above. Hence the 120 1-surfaces pass by 24's through the 25 fundamental points. There will likewise be arrangements with respect to 2 , 3 , or 4 of the fundamental points, so that for (say) 2 chosen from among them all, there will be a deter- minate number of 1-surfaces having these two points in common. Looking at the first line of X's just written, it is clear that only 1 / 4 or 6 of them contain a specified point, as li, other than Oo; these are Xj*, Xfo, \il, Xj*, \l, X%. Similarly for any other point pair of any other set, and since there are 25-24/1 • 2 = 300 such pairs, it follows that the 120 1-surfaces are arranged in 300 sets of six each such that each set of six has two points in common. For the same 24 X's, looking at any set of six with two points common, e. g. those with the arrangement (012**) are Xio , Xio . Remembering that , 1 , 2 in any other order in the ( ) gives a distinct set of common points, there are in this way given 25 ■ 24 • 23/ 6 = 2,300 sets of 2 . Hence the 120 1-surfaces are so disposed that 2,300 pairs of them have three points in common.. Similarly if 4 of the 25 points are to be common there is but a single 1-surface satisfying the condition, if the points are the first 4 in (0 1 2 3 *) the 1-surface is Xio. There are in all 12,650 such. Finally, there are in the configuration of 25 fixed points (fundamental), 21 X 12,650/5 = 53,130 absolutely fixed 1-surfaces. From the equation of the 1-surface tangent at (v'q, • • ■ , v\) to the surface F , the conditions that ( wj,' , ■ • • , v'i ) shall lie on this tangent are written down from equation 1 (at beginning of this section); if (wq ) • • • J W4 ) is the symbol of one of the 120 poles, the condition is ( P"°" + P""+' + P"'"+' + P"'"-*-' + P"'"-^' )/U[+{ p""" + p"'"+2 + p"^'+'' + p""+» + P"'"-^' ) / f/i + ( P"°" + P"'"+' + P"'"+' + P'""+- + P"-'""-' )/U\+( P-" + p-'^' -f P--+' -f p-"-f p"+^ *"+=)/ U', = 5; 92 ON THE CYCLOTOMIC QUGfAHY QUINTIC. or dropping accents, say R1/U1+ ■■' + R,/U^= 5, which intersects the surface F in all points at which the tangent 1-surfaces pass through the fixed pole {ii'a , •••,<). This surface (R) contains the fixed line l/'i = 4i?i/5; ?72 = 47?2/5; C7s = 4i?3/5; U, = iRi/5. There are 120 such 4-surfaces (surfaces of 4th degree) intersecting F in points at all of which the tangent 1-surfaces pass through fixed points, — the 120 poles. A set of 24 will be considered in some detail first; putting P" + p"^^ + P^" + p''+' + p"^! = rj/ 5 ^ r; P'+ p'^+ P^' + p'+' +P'^'=rzl5^s, P' + p'^' + P^i + P''^ + p<>^=n/5^t, and for convenience 1/Ui, I/U2, I/C/3, 1/U,^x, y. z, w, where a, • • • , g are the numbers , 1 , 2 , 3 , 4 in any order, the equation px + ry -jr sz + iw =^ 1 represents the 24 1-surfaces corresponding to the poles derived from the symbols S, if a = 0; if a ^ , the letters b, c, d, g= 1, 2, 3, 4. When any of the 24 systems of values for (fc, c, «Z, ^) in the corresponding iS has been assigned, the values for p , r , s , f respectively are found by adding the elements in the first, • • • fourth column of S, to form a new complex number a', b' , c' , d' , g' . It is evident from the properties of S that the coordinates so found will not all be equal; there can be no confusion if the symbol {a' , ■ ■■ , g') is used for this number. Thus, e. g., let (a, b, c, d) ^(2143); then S becomes 3 4 1 3 4 2 2 1 3 2 1 4 ON THE CYCLOTOMIG QUINARY QUINTIC. 93 then p=(-.-2,2.); r= (2, 1, •••. 1); * = (1, 2, • • ■ . 1); <= (1, -1, 1, 1), or WTitten in full p = 2p^+2p'; r-2+p+p^ * = 1 + 2p + p^ t = 1 + p' + p' + p'; in brief, the numerals in S are the indices of the powers of p, if p" is zero, but if a is (say) 2, 1 will have to be added to every numeral in the third place, so in this case, p,r,5,<= (.-3,2, •), (2,1,1, -1), (1.2,1, • 1), (1-2, 1, 1); for a = 4 , 1 is added in 5th place, (•-221), (21--2), (12.-2), (1 -112), commas being omitted from ( ) as no number > 9 can occur. Writing down the six sets of four numbers as values for p, r, s,t, corresponding to, and in the same order as the X's in the anharmonic table; find for a as yet arbitrary; p, r, 5, < = { }; (table on next page). In the table p" =# , but is arbitrary; giving a the values 0, 1, 2, 3, 4 in succession, which is equivalent to increasing the digit in the first • • • fifth place respectively by unity, 120 sets of values for p, r , s , t are found, corresponding to the 120 poles whose coordinates are all distinct, i. e., in any pole no two coordinates are equal. Each of these sets gives one surface of the fourth degree, defining a cone (in Ss); viz., the tangent 1-surfaces from the corresponding pole touch the surface F along its intersection with the fourth degree surface. Again, as has been seen in one special case, each of these { p,r, s ,t I has in it, or envelops, a fixed line, that is a skew curve fixed in Si; there are clearly 120 such. There are 30 sets of 4 surfaces each; the surfaces in any set intersect in 4 1-surfaces, or rather have these in common. E. g., caUing X; what X becomes when a is i, the 1-surfaces common P (X) {(-1111 (X^) {(-31 • (X') { (2-11 (X^) {(112- (X^) {(112- (X«) {(• •3- (1) r s t (•1111), (4 ), (-1111)1 (1 • -21), (2-11-), (1 -1 ^2)} (• -22-), (21 • • 1), (•2- -2)} (2-11-), (12- 1.), (. • • 13) ! (•1-3-), (1-1 -2), (21- - 1) J (12-1-), (21- -1), (1- -21)} 94 ON THE CTCLOTOMIC QUIXAET QUINTIC. (2) (X) (X«) (X) (X«) (X) (X') (V) (x"*) (X«) to the set Xo are g: {(• {(1 {(• {(• {(2 P 1111), ( • -21). (• . •22.), (21 1-3-), (1 ■ •11-), (1- r s t 1111), (-1111), (4, . . •3-1), (12- 1 •), (21 • . •)} 1)} {(li-1-), ( •1), (-2. ■2), (21. •2), (.31 13), (112 2), (2.11-)} 1), (112- .)) •), (1- -21)} •), (2-11.)} (3) p r {(4 ), (-1111), (. {(2.11 .), (12.1.), (. {(21. .1), (.2- .2), (2 {(1-1.2), (.31. .), (1 {(12-1.), (21 . .1), (1 s t nil). (-1111)} • -13), (112. .)} • 11 ■), (• •22-)} • -21), (2- 11-)} • -21), ( .3-1)} {(21. .1), (112. .), (.1.3.), (1 .1.2)} (4)- {(•nil), (4- {(1-1-2), (21 •), (• nil), (. 1 1), (112- .), (. 1 {(•2 {(2 1 {(•• {(!• ven bv ■2), (2-11 .), (. •1), (1 . .21), (• m, (112. .), (2 21), (2-11 •), (1 22-), (21 3-1), (12 11-), (12 1 -2), (-3: t n)} 3-)} •1)} 1-)} • )> (lllll)a:+(lllll)2/+ (5 . . . .)2+(lllll)^«=l, (lllll)a:-f(lllll)2/+(lllll)3+(5- . . ■)io=l, (5 • - • •)a;+ (11111)2/+ (11111)3+ (11111)«;= 1, (lllll)a; + (5. . . •)y+(lllll)2+(lllll)M=l.J Since here (111 11) = 0, z — \ 1 5i = X = y = \o . But X, y , z or w = constant is the equation of a 1-surface, hence these 4 l-surfaces are contained in every surface of set Xo. The l-surfaces so found (when they exist), are all of the form x = const., or y, z, w = const.; where a;, y, z, w = 1 are 4 l-surfaces from among the set of 120, so the 30 ON THE CYCLOTOIIIC QUINARY QUIXTIC. 95 sets of 4 surfaces are such that the members of each set intersect in four fixed l-surfaces parallel to four of the 120 set. The above set of 24, (j), r, s, t) is typical of all 5 such sets, for if any 2 values in one set are equal, the cor- responding values in any other set will be also; that is the internal structure of all sets of 24 4-oids is the same. The above may be written in the tj"pical form for all; a=(.31- •); b = (1 • -21); c= (2-11.); d = (1-1 -2); e= (• -3-1); s- = (1 2-1-); ff=(2 1- •1); h = (• • -13); k=r (112. .), l = (• 1-3-), m— (4 • • • ); n = (-1111); r=(. •22-); s = (• 2- -2); then ()x+()y+nz + {)w=l. {)x+{)y+{)z + ()w=l (Xf) a b c d (Xi) h e f g (M) d a b c (XI) g b e f (xi) c d a b (Xt) f g b e (XI) b c d a (Xf) e f g b (X|) c f k k (Xi) d g k I (xt) k c f h (Xt) I d g k (M) h k f (Xf) k I d g (XE) f h k c (Xi) g k I d (Xa) m n n n (Xi) r g s c (X4) n m n n (X!) c r g s (Xi) n n m n (XI) s c r g (X2) n n n m (Xi) g « c T Since the effect of multiplj ing ( • • ) by p is simply to permute the numbers within cyclically, p'^a= (3 1 •••); P'e = (3-i--); pH= (3 ••!•) 1 ph= (3.-.1), hence p'a + p3e + p^l-\- ph — 8; also a + e + A + f= -4, and m = 4, n = — 1; r + 5= — 1, and so in any case the number of coefficients in the 24 set may be greatly reduced, but the symmetry is thus lost. The 5 different sets are distinguished by suffixes, e. g., Tz means that in the above value of r, p' is to be taken as p', so that ra = (• 21 • 2). This arrangement of the 120 4-surfaces shows 96 ON THE CYCLOTOMIC QUINARY QUINTIC. clearly the distribution of the skew-lines in 5-space through which the 4- surfaces pass. Also each set of 4 is circumscribed to a symmetrical figure of 16 1-surfaces, which surfaces are parallel in sets of 4 (reading the table by coIuQins), the 4-surface passing through the lines parallel to one of the funda- mental axes arid common to the respective sets of 4 non-parallel l-surfaces (reading the table by rows). In short, all the descriptive properties of the 120-configuration are summed up in the table, and it becomes a mere matter of reading the table to elicit further inter-connection, in the same way that the whole 3,125-configuratIon and its consequences arose from the reading of the polar-tangent equation. In order to completely exhaust the properties of this simple figure, the properties of" ruled surfaces " in Si have to be considered, and account taken of the other geometric entities, sub-surfaces, . . . super-lines, etc. (according to Cayley). The 120 configuration being in a sense complete in itself, the farther points are not considered in detail in this place. ON THE CYCLOTOMIC QUINABY QUINTIC. 97 VITA^ The undersigned is a native of Scotland, (born 1SS3); entered Stanfo*> University, California, as an advanced student in 1902, graduated thence 1904 (A.B.); entered the University of Washington, 1907, as Teaching Fellow, and received there the A.LI. degree 1908; entered Columbia University, NovembiT, 1912, as a student of mathematics. E. T. Bell. April 2, 191$