EXCHANGE UNIVERSITY OF PENNSYL\'AXIA THE RESOLVENTS OF KONIG AND OTHER TYPES OF SYM- METRIC FUNCTIONS BY STANLEY P. SHUGERT A THESIS PRESENTED ;rO THE FACULTY OF THE GRADUATE SCHOOL IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY PRESS OP THE NEW ERA PRINTING COMPANY LANCASTER, PA. UNIVERSITY OF PENNSYLVANIA THE RESOLVENTS OF KONIG AND OTHER TYPES OF SYM- METRIC FUNCTIONS BY STANLEY P. SHUGERT A THESIS PRESENTED TO THE FACULTY OF THE GRADUATE SCHOOL IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY PRESS OP THE NEW ERA PRINTING COMPANY LANCASTER, PA. The author wishes to acknowledge his indebtedness to Professor 0. E. Glenn at whose suggestion this paper was undertaken for his helpful supervision and encouragement. 44432G THE RESOLVENTS OF KONIG AND OTHER TYPES OF SYMMETRIC FUNCTIONS. By Stanley P. Shugert. INTRODUCTION. Symmetric functions of the differences of the roots of an equa- tion are an important part of many theories, particularly of binary invariant theory, and much attention has been given to the computation of such functions; while the functions which are not symmetric in differences have received very little atten- tion, but in some instances have an important bearing in general theories. As a particular instance, there are the resolvents of degree ('") of an equation, fix) = a;*" + Aix"^-^ + AiX^-^ + \- A„, = 0, which were studied by J. Konig in the Mathematische Annalen, Vol. 15 (1879). In brief, these are constructed as follows: the roots of F(z) = f(z -\-f,) = z^ + B,z^-' -{-■-■-{■ Bm=0 are (ai — fx) {i = 1, 2, • • • m}, where the ai are the roots of f{x) = 0. The equation whose roots are the ("') quantities of the type, (- ma, - /x)(a2 - m) • • • («r - m) (r = 1, 2, . . . m), is rational in Bi and is a resolvent of f{x) = 0. In volume 15 (1914) of the Transactions of the American Mathe- matical Society, Prof. O. E. Glenn has published a paper in which is given the expansion of the binary form of order m = fxv, fix) = aox{^ + aia-r-^a + • • • + amX2"' as a power series in which the argument is the binary form of order v, ^x" = ^oa^i" + ^iXi'-'Xi 4- . . . + ^^a:2^ In § 2 of this paper, we obtain an independent proof of his result 1 THE RESOLVENTS OP KONIG. for J' = 2 by a method depending upon the solution of a system of linear equations and a preliminary lemma on determinants, § 1. The remainder of the present paper is devoted to a succes- sion of applications, introducing theories, which depend upon this expansion for the case v = 2, and finally introducing in § 6 a method for the determination of the Konig Resolvents with the aid of the tables given in § 7. In § 3 a minimum set of seminvariant conditions that a given quadratic form may divide a given form / is obtained and § 4 is devoted to a problem in rational fractions. There are obtained in § 5, methods for the computation of symmetric functions of the sums and of the products of the roots of an equation, taken in pairs. §1. A LEMMA ON DETERMINANTS. We first prove the following lemma concerning determinants: Lemma: The determinant {ckrs] in which every constitiient is zero for s > r -\- 1 and not zero fors=r can be reduced to a deter- minant {Ofg} in which every constituent is zero for s > r. Let Ors (A) an, ^21, ai2, ^22, 023, Oil, ai 0, •• 0, •• (ti i+1. .. •• Or-l i+1, • • flr-l r Or t+l, •• Orr Or-i 1, Or-i i, Or-i 3, Or 1 Or 2, Or 3, Multiply the last row by Or-i rjor r and subtract from the pre- ceding row. This will give (B) On, O21, 0.1, 021 0, 022 O23, ail 0, 0, (ar-l(ar)r)l (Or_i(ar)r)2 (ar_i(ar)r)i+l (Or-l(Or)r)r- -, arr arr arr Orr Orl, Or2, ••• (h i+1, ••' Orr-l, ttrr in which (ar_i(ar)r)f+i is a symbolic notation for the second order A LEMMA ON DETERMINANTS. 3 determinant (ar-i i+iO>rr — dr-i rdr i+i). The minor determinant Arr is of the same type as the determinant A and by applying the same process to it and continuing the method for each succeeding minor determinant, we finally obtain: (aiiazitts ' • • {ar)r • • •)3)2)l, 0, • • • azias ' ' • {ar)r ■ • •)3)2 * {aiiasia^ • • • {ar)r • • •)4)3)i Mas • • • (ar)r • • Os) (a3(«4 ••• («r)r •••)4)3 ' («3(«4 ' * * Wr * * ' )4)3 * ai{ai+i • ' ' {ar)r • • ')i+l)l (at(«t+i • • • Mr - • •)i+l)2 {ai+i{ai+2 • • • Mr • • •)i+2)i+l' ai+i(ai+2 • • • Mr ' • ')i+2)i+l' ai{ai+i • •• Mr • ' ')i+i)j Q (ai+l(«i+2 • • • i0,r)r ' " •)i+2)i+l' (ar-iMr)l ar-lMr)2 > "0 arr Orr Orl, Ort, " ' drr (^ = 1, 2, . • . r, i = 1, 2, . . . i). The interlacing determinant notation is merely an extension of the symbolic method explained above for a determinant of the second order. These determinants are expanded by means of a succession of second order determinants and the method as shown for (ar-i(ar)r)i+i = (cfr-i i+i«rr — ar_,>ari+i) is to take the determinant subscripts, those written outside, first in an order symmetric with respect to the center with the product of the constituents and then to subtract the product obtained by re- versing the determinant subscripts. Except in the last reduction, two constituents of the second order determinants will be sym- bolic determinants. Corollary: The determinant {ar8}s>r+i can be written as a symbolic determinant: i. e. {«rsls>r+i = {ai{a2{az{ai • • • Mr ' ' 04)3)2)1. 4 THE RESOLVENTS OF KONIG. §2. EXPANSION OF A POLYNOMIAL AS A POWER SERIES OF A QUADRATIC POLYNOMIAL. Theorem: Any form of degree m can be expressed in terms of a power series of a quadratic form with linear expressions for the coefficients. First: if m is even, m = 2n, we may write, a^"* = aox"^"" + aia:2n-i _|_ a^a^""-^ + h «2n = («i + cax) (1) + (as + a,x)^.' + • • • + (a2r-i + a^rXm^r-'^ where ^^^ = ^o^^^ + ^ix + ^2- Since ao = ^o"* and is determined, the number of a's on the right is equal to the number of a's on the left and since the coeffi- cients of the powers of x on the right are linear in the a's, we have upon equating coefficients 2n = m linear non-homogeneous equa- tions in the 2n quantities ai. For the case 2n = 4, we have iaQaia2azai){xy = (ai + oi2x) + (as + a4x){^QX^ + ^la; + ^2) and upon multiplying out and rearranging, ^oai = ai — 2^o^i, ^otts + ^10:4 = 02 — ^1^ — 2^o^2, «2 + ?i«3 + ^2014 = as — 2^i^2, ai + ^2^3 = 04 — ^2^. Second: If m is odd m = 2n — 1, we can write ax^ = «oa:"^~^ + Oia^^^-^ + 02^:2'*-^ H h a2n-i (la) = (ai 4- Q;2a^) + (aa + aiX)^J^ + • • • + (a2r-l + a2rX)aJ'y-' + ' ' ' + («2n-l + A;^:) (^,2)n-l^ where A; is determined to be ao/^o"~^ and we have upon equating A QUADRATIC POLYNOMIAL. 5 coefficients 2n — 1 linear non-homogeneous equations to de- termine the 2n — 1 a's. For the case m = 3, we have (aoaia2az){xy = (ai + aix) + {as + kx){^oX^ + ^ix + ^2) and upon multiplying out and rearranging, ^0«3 = «! — ^b «2 + ^l^S = 02 — ^2, «i + ^20:3 = as, if, as before, we take ao = ^o'*~S which we may do without loss of generality. The particular form that these equations take, enable us to express them in general in terms of an operator, when we observe that any power of a form is a covariant of that form and so is given by its end term and a partial derivative operator. Hence the equations : 2)o^^[(».-i)/2]^ = fli - D^f^"'"', . . . -f Z)-^-^f (-•'■)/=^Jcwy+i • • • _l_ D'-2^mr.-2mc^^ _|_ Z)-i^f (— ^)/2]a,„ = a,- - D'^^'^i'\ D%'(X2 + • • • • • • D%'ai + D^o'a2 • • • + Z)-^-^f K-.^/2],^ .^^ . . . (^ = 1, 2, 3, • • • to) and (j = 1, 2, 3, • • • m). (7H — 7 \ TO "~ 1 — ^ — 1 is the integral part of — ^ — , and D is the Aronhold operator Also DW^,^[(—i)/2] = for j > i. 6 THE RESOLVENTS OF KONIG. Solving for am-y+i, we have CXm-j+l = 0, 0, ... ..., ai-Z)^f-/% 0, 0, ... ..., a2-D'^^-"\ 0, 0, • • • 2)-(i+i)|f (^-c^+iM], ai - Di^^^"\ 0, ^'"-^^o", ••• 2)»»-(i-l)fcJ?[(m-y-l))/2] , . . £)m-2tE[(m-2)l2-\ J)Tn-l tE[(m-l)i2] -^A. where A = ^o*". If we expand the numerator of this expression in terms of the column tti - D'^^'^'^\ the minors of the constituents a,- - D'^f^'^^ are determinants of the type (vl) of § 1. So in the symbolic notation, (2) Ort^j+i = . . . y+i^mn*-u+i)my-i^mm-u-i))i2-] This result enables us to make a rapid computation of the ca in particular cases. Example 1. m = 4, j = 3. a2 = [{iD\D\D')%mo)%}iai - D^o') A QUADRATIC POLYNOMIAL. 7 + D%'D%D%{a^ - D%')]* - ^o^ = [HD^o'D^o - D%D%)(a, - D^o') - l'^o'DUa2 - 2)V) + l-^o^o(«3 - 2)V)] - ^0^ = [(^i' - ^0^2) (ai - 2^o^i) - ^o^i(a2 - ^i^ - ^0^2) + ^o'ias - ^1^2)] 4- ^0^ = [(- ^0^1^ + ai^2 - ^o^ia2 + ^o'as) + 2(^o'^i - ^o«i)y ^ ^0^. Example 2. m = 3, j = 3. ai = [(Z)2(Z)3)V)^^o(ai - D^o) - (2)H^^)V)^^o)(a2 - D%) = KD^o-D^o' - D%'D%){ai - D^o) - D%-D^oKa2 - D%) + Do^,-Do^oKa^ - D%)] - ^0^ = [(^1-0 - l-^2)(«i - ^i) - ^o-0(a2 - ^2) + lo-l(a3- 0)1-^0^ = [^o«3 - (- ^1 + aiM ^ ^0^ Without interfering with the generality of the results we may take ^0=1, and then the values for ai and 0:2 can be put in the following form:t a, = am- H K-^^'a^lX (3) where <=0 \i oai oa2 dm Thus, for example, the results in the particular case, m = 4, * (Z)')*^o' = D'~'{o' = D°y is factored out of each term since it enters the result as a constant multipUer. t Cf. Transactions Amer. Math. Soc, Vol. 15 (1914), No. 1, p. 79. t — fi is substituted for x before the differentiation is performed and (i) is the power of the differential coefficient. 8 THE RESOLVENTS OP KONIG. can be written (4) i=0 U; or ai = tti- ^2l24'«if, - ^2'043ai^y, = 04 - (ao^i^ - ai^i + 02)^2 + ao^2^ 0:2 = fi4aifi + ^2fi4^0-fi = (— ao^i^ + ai^i^ — a2^i + as) + (2ao^i - ai)^2. This result agrees with that of example 1 if we make |o = 1. From (3) ^[(w-l)/2]_ tJ+1 ... ti ... There is proved in the paper just cited a general result* which is applied here for the case v = 2; namely, 1 f) {0i2k+l + 0;2fc+2a^) ~ ~ J^qF (<^2A-1 + (X2kX). Hence the assumed expansions (1) and (la) take the form dF 1 d^F (6) \\ iL ^LjL_ (ti\kA !^ tl ^ (t 2\E(ml2) ^ \k d^^^''^^ \E{ml2) a|2^c>"^^^^''^ The coefficients in this expansion are symmetric functions of the roots of ^^ and also of those of a^. Expressed in terms of the coefficients of a^"*, and ^x^, they are linear in a^, ai, • - •, Om, but, in general, not linear in ^0, ^1, ^2. These functions have a seminvariant character as appears in the application which follows. * In Glenn's paper the corresponding expansions of the general form of order i» in the general argument ^a," is given. A QUADRATIC POLYNOMIAL. \i § 3. CONDITION THAT A FORM CONTAIN A GIVEN QUADRATIC FORM AS A FACTOR. Using the expansion (6) obtained in § 2, a minimum set of necessary and sufficient conditions that a given form be divisible by the rth power of a quadratic form is the identical vanishing ^ ^ dF d'F d'-'F _. ^, . ,. 01 F, -rr , wjr-o, ' ' • >.^ ,_, . Smce these expressions are linear in X and do not vanish identically for every value of x, then the identical vanishing of each of these r expressions necessitates the vanishing of two seminvariants, and hence the simultaneous vanishing of 2r seminvariants furnishes the minimum set of necessary and sufficient conditions that ax"" be divisible by the rth power of ^J^. In particular, if ax"* contain ^J^ as a factor, then from (4) and (5) j;(m-l/2) t i+l (7) '- ^ ' ^(m-l/2) t i f=0 [^ and conversely. These two conditions are equivalent to the redundant set of m + 1 conditions furnished by the identical vanishing of the co variant $, given by Clebsch,* which for w = 4 is $ = - 2D{aO^,'ax' + 2^x'{a^')ian^Jax = 0, where D is the discriminant of ^^^ and ^^ = ^J"^ = ^x"^- B. Igelf has developed a method depending upon the vanishing of a series of resultants of every two forms of a certain system. * Theorie der Binaren Formen, p. 94. ■\ Sitzungsherichte der Kon. Akad. der Wissenshaften Mathematische., Bd. LXXXII, p. 943. 10 THE RESOLVENTS OF KONIG. §4. SEPARATION OF a,-/(y)^ INTO PARTIAL FRACTIONS. From (6) § 2, we have dF 1 d'^F Hence dF^ d'^-'F a^ ^ _iL _ ^^2 , (- 1)^^ d|2^ {^.'Y {^.'Y i^.')'-''^'" \h-l ^.^ , (- 1)^ d'F (- 1)^+^ d^+'F ..,. "^ 1^ 'd%'^ [Mil d^s'^+i'^^"^ or, upon writing the numerator in explicit form, E[{m-V)li] y i+1 y i /2) + E To illustrate, let a/ :x^- x^-\-Zx^- X i^.')'-'' F dFld^2 + i^- x-\-iy (x^- x+ 1)2 ' \2{x^ -x+1)' From (5) F= - (^1^ + ^1 + 3)^2 + |2^ + {(- li^ - ^i' - 3^1 - 1) + (2^i+m2}a:, A COMPUTATION OF SYMMETKIC FUNCTIONS. 11 riF ~ = - (^1^ + ^1 + 3) + 2^2 + (2^1 + l)x, Now, ^1 = — 1, ^2 = 1- dF__ _ _ _L^_-, Then a^-:i^+Sx^-x x-2 a; + 1 , 1 (a:2_a;+l)3 (x^-x+lf ' {x'-x+iy ' x^-x-\-l §5. COMPUTATION OF SYMMETRIC FUNCTIONS. The two seminvariants obtained in § 3, whose identical vanish- ing furnished the necessary and sufficient conditions that a form of order m be divisible by a given quadratic form, also furnish a direct method for the computation of certain classes of symmetric functions. Let the m roots of fix) = ax"" = aox"^ + aix"^-^ + a2X^^ + • • • + a^ = be Xiy Xi, • • ' Xm, and denote the totality of the sums of these roots taken in pairs by ^u and the products in pairs by ^ij- [i, j = 1, 2, . • . \m{m - 1).] The number of quadratic forms which will divide aj^ is |w(w — 1) and they are of the form (8) x^ - {Xi + Xj)x + XiXj = a;2 + ^^^x + ^2*. Hence, if from the two conditions (7), i=0 [J^ where ai is of order m — 2 in ^i and E i — ^ — J in ^2, and 012 is of order w — 1 in |i and E I — ^ — ) '^^ ^2, we eliminate ^2, 12 THE RESOLVENTS OF KONIG. we obtain a resolvent of order |m(w — 1), J»i(m— 1) (9) = Z (- iyi^i^"'^^-'^-' = 0, whose roots are the ^i^. The 0^ are values of the elementary symmetric functions of the ^u and are rational functions of the a's. By eliminating ^i, we obtain in the same manner Jm(m— 1) (10) ^ = E (- i)%^2''"^"*-'^-^' = 0, a resolvent whose roots are the ^2y. The \f/j are the values of the elementary symmetric functions of ^2j and are expressible rationally in terms of the a's. §6. THE RESOLVENTS OF KONIG. The ^ = of § 5, is the Konig resolvent for the case v = 2. $ = is a resolvent of a similar nature. Both of these resolvents have been studied by Glenn* in his theory of the rational resol- vents of the factorable ternary forms. The general form of the $ resolvent is, from (7) 12a!^f„ 9i'a':fll • • u'^^+y^-t^T'''"^ 0, • • • • *t = 0, am, ^'al^„ •" 0, 0, ^^(.+3/2)^».[^(-l/2)] = 0. -fl The elimination of ^i from equations (7) does not exhibit the ^t resolvent in a concise form like the determinant above. The determination of the ^m{m — 1) quadratic forms which divide a given form can be obtained by solving either $ = or ^ = 0. If = be solved, then by substituting the ^u in the condition a2 of (3), ^2i can be obtained and the quadratic factor is determined. * Amer. Jour, of Math., Vol. 32 (1910), p. 80. t The order of the determinant is m — 1. I The order of the ^ determinant ism — 1+m— 2=2m — 3. THE RESOLVENTS OF KONIG. 13 These resolvents are given for the case m = 4. From (4'*), ai = — «o^2^ + («o^i^ - ai^i + 02)^2 + «4 = 0, ai = (2ao^i-«i)^2+(-ao^i^ + a^^x - 02^1 + as) = 0. Eliminating ^2, — ao, ao^i^ — aili + ^2, $ = 2ao^i — ai, - ao^i^ + axl\ — o.i^\ + as, 0, 2ao^i — «!, — ao^i^ + o.x^\ — 02^1 + as Likewise, eliminating ^i, = 0. ^ ao^2, — ai^2, — ao^2^ + a2^2 + cii, 0, ao^2, — ai^2, 0, 0, ao^2, ao, ai, 2ao^2 — a2, 0, — ao, «!, — ao^2' + a2^2 + a^ — ai^2, — ao|2^ + a2^2 + a4 =0. — ai^2 + as, 2ao^2 — a2, — ai^2 + as The explicit expanded forms of $ = and ^ = can be ob tained from the tables in § 7. §7. TABLES OF SYMMETRIC FUNCTIONS OF THE SUMS AND OF THE PRODUCTS OF THE ROOTS OF A GIVEN FORM, TAKEN IN PAIRS. In § 5, it was shown that there existed two resolvents $ = and ^ = 0, whose roots were the ^u and the ^2y of (8) respectively, and hence the coefficients of the powers of ^1 in $ = and I2 in ^ = are the values of the elementary symmetric functions of ^u and ^2y, respectively, expressed rationally in terms of the a's of / = fla;"*. These symmetric functions have been computed 14 THE KESOLVENTS OF KONIG. for the cases m = 3, 4, 5, 6 and a table of them in homogeneous form completes this paper. In order to check these results, the following formulas have been derived : (a) (b) k= Z^i«0^'«l'*"- (im\ <-»-(f)=-?^r)-(r)'- -(:)"■ As a further check upon the \f/'s, it may be noticed that by inter- changing ao with Om, di with ttm-i, etc., \J/j is transformed into ^^y. (i = 0, 1, 2, . • . m.) Table la.^ m 3. 4>i — 2aoC5i, 02 = fli^ + ao«2, 03 = 0,1^2 — doas. Table lb. ^,= aoa2, 1^2 = aias, )/'3 = as\ Table Ila. m = 4. <^l 6 Sao^ai 2a,^a2 + 3aoai2 4aoaia2 - 4ao^a4 + aoai«3 + 2ai2a2 — 4aoaia4 + aia2a3 Table lib. ' lAi ^2 ^Pz '/'4 \^B \^6 ao2a2 - ao2a4 + aoOias - 2a^aia^ + ai2a4 + aia3a4 + a2a42 a4« TABLES OF SYMMETRIC FUNCTIONS. 15 Table Ilia, m = 5. <^2 4>3 i «^5 4ao'ai Sao^a2 + QaoW dodz + 4aoai^ + 4ao^aia3 + daoW — 11 floras — 5ao^aia4 + 2ao^a2az + 5ao«i^«3 + 6aoaia2^ + 3ai%2 Table Ilia (continued). ./>6 9 10 — 22ao2a]a5 - ^a^^a^a^ loo^aza^ 400^04^5 - aoW — 2aQ^a2ai — ^a<^aza^ - 4:a,^a^ + 4aoaia3a5 + 2aoaia4«5 - a^W — 16ao«i^a5 - 9aoaia2a5 - ^aottiGi^ + OoChazas — 2a^^a^ + aotta^as - SaoaiazGi — ao«2^a5 - aoGz^ai + Gaocnoaas + 4aMaz + aoai^a^ - ttoas^ - aiW + aoa2' + aiaz^ - OffiL^az^ - ^aMa^ — aiai^as + 2ai%3 - ^ai^a^ — aiai^ai + aiazazai + 3aiW + aiH2a4, + aiW + 1a\a^az + aia^az^ Table Illb. 1^2 lAs ^6 ao''a2 — ao^a4 + tto^aiaz — ao^aiGs — 2ao^«2«4 + ao^az^ + ao^aza^ — 3aoaia2«5 + aQaiazd^ 2ao2a52 — 2aoCiia4a5 — 2ao«2a3a5 Table Illb {continued). «A6 rf^i ^8 ^9 '/'lO aoaitts^ - aaa^a^^ - aitts^ astta^ 05^ - Saoazttia^ — 2axazai^ + 02^4052 + aoa4^ + aia^^a^ - aiW + aa^as^ + aia2a^a6 16 THE KESOLVENTS OF KONIG. ■^ TtHecTticoeecoeeio^^-Hee (NC0t^O5(NQ0(Mi-i00COi-i e eOJTtH 1 1 1 1 1 1 + 1 +++++++ ■4 oo(Me(MeeeeTtHcseo o«» e e e e U <^ c e e O5cco5i:oco-*(N eco e .o "^^ e e e .- „ „ "rt ''^rt '=^rt e G '^t *■ " e e e "^w '^w e Tjt -^ ^ e e TtH e e e e + I + + + I I ++ I I ++ I I I I + + ■ e e e "^c? II e e e e e „« «« .« c? °^^ (^^0(^^T-lOicocoTt^(Nt^^05co e e-* e ecoooeeeecocoecseee^^vo (Ncoi-iiocoi>TjHTtii-iioi>'*i>-'^ e e(N I ++ I ++ I + I I I + I I ++ I I I I I + + + e "-. e ^^ ^^ e =1, ?, 00-^00 !5e e e csco'* (Se e eeiN e"^e e^-^e « i-H00(NCO(NCDQ0Tt t^ e. ^ cT. e «„ "^ ^^ . o. . -. '^ G o o o C3 G CO e e e (M o e e e e ^. (M-^» lO to 1 ++ 1 + 1 + 1 ++ + 1 1 1 ++ -§: ^« If "^ «^ c? 1 1 + + -i: /I 1 + -5^ ^ > TABLES OF SYMMETRIC FUNCTIONS. 19 e if I + ^ =2. ^ 1. ''„ « + I I + + to cs (N e e CO e e e + I + I + I + '- e CO e « e « *ir a.^ 1^ Tf ^*J «i ^ >*^ ^ (Nco(NTt< e(N(M(N e e I ++ I ++ I I + + ■ to ?3 -f t € e cf (N CO e CO CO I + I + I ++ I N to CO e e '2. e e CO I I + + THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETTURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. OCT 13 1933 my 12 1933 lo^rs" ' 207an^'5flt'#» • '5 ■. "■'--. .- 1 ■'•'JO LD21-100m-7,'33 "--; tt '-i TLi ^ t» S'«r UNIVERSITY OF CALIFORNIA LIBRARY