[m(^^MU^^ 
 
 o 
 
 2 
 
 < 
 
 < 
 
 z 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1406 
 
 ON THE USE OF THE HARMONIC LINEARIZATION METHOD IN THE 
 AUTOMATIC CONTROL THEORY 
 By E. P. Popov 
 
 Translation 
 
 **K voprosu o primenenii metoda harmonicheskoi linearizatsii v teorii 
 regulirovaniya." Doklady Akademii Nauk (SSSR), vol. 106, no. 2, 1956. 
 
 UNfVERSfTY OF FLORIDA 
 
 DOCUMENTS DEPARTMENT 
 
 1 20 MARSTON SCIENCE LIBRARY 
 
 RO. BOX 117011 
 
 GAINESVILLE, FL 32611-701ll!JS.i\ 
 
 Washington 
 January 1957 
 
NACA TM 1406 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1406 
 
 ON THE USE OF THE HARMONIC LINEARIZATION METHOD IN THE 
 
 AUTOMATIC CONTROL THEORY* 
 
 By E. P. Popov 
 
 The method of harmonic linearization (harmonic balance), first 
 proposed by N. M. Krylov and N. N. Bogolyubov (ref . l) for the approxi- 
 mate investigation of nonlinear vibrations, has been developed and re- 
 eived wide practical application to problems in the theory of automatic 
 control (refs. 3 to 6) . Recently, some doubt has been expressed on the 
 legitimacy of application of the method to these problems, and assertions 
 were made on the absence in them of a small parameter of any kind. Never- 
 theless, the method gives practical, acceptable results and is a simple 
 and powerful means in engineering computatations . Hence, the importance 
 of questions arises as to its justification. The underlying principle of 
 the method is the replacement of the given nonlinear equation by a linear 
 equation. In establishing the method, a small parameter is considered 
 whose presence makes it possible to speak, with some degree of approxi- 
 mation, of the solution of this new equation to the solution of the given 
 nonlinear equation. In an article by the author (ref. 7), certain con- 
 siderations were given on the presence of the small parameter, but this 
 question has not as yet received a final answer. In the present report, 
 a somewhat different approach to the problem is applied that permits: 
 (a) establishing, in the clearest manner, the form of the presence of 
 the small parameter in nonlinear problems of control theory, solvable by 
 the method of harmonic linearization; (b) connecting it with previous 
 intuitive physical concepts (with the "filter property") and extending 
 the class of problems possessing this property; and (c) discussing various 
 generalizations of the method. 
 
 The free motion (transition process and autovibration) for a very 
 wide class of nonlinear systems of automatic control (ref. 7) are 
 described by differential equations of the form 
 
 Q(p)x + R(p)F(x,px) =0 (p = ^) (1) 
 
 * "K voprosu o primenenii metoda harmonicheskoi linearizatsii v teorii 
 regulirovaniya." Doklady Akademii Nauk (SSSR), vol. 106, no. 2, 1956, pp. 
 211-214. 
 
2 WACA TM 1406 
 
 where Q(p) and R(p) are polynomials of any degree, of which some re- 
 quired properties will be established in the following paragraphs, and 
 F(x,px) is a given nonlinear function possible only with respect to as- 
 sumptions of the most general character. However, in problems of the 
 theory of control, no assumptions must be made as to the smallness of 
 the nonlinear function F(x,px) or to its small difference from a linear 
 function. In order to render explicit the form in which it would be 
 possible to write a small parameter in equation (l), we shall proceed as 
 follows . 
 
 Let equation (l) have a periodic solution or a solution approximately 
 periodic differing slightly from the sinusoidal. We write this solution 
 in the form 
 
 X = X* + ey(t) X = a sin oot (2) 
 
 where e denotes a small parameter, and y(t) denotes an unknown bounded 
 function of time. In the case of the existence of a periodic solution, 
 we write 
 
 00 
 
 ey/t) = e^ a^ sin(koDt + <!>^) (3) 
 
 k=2 
 
 We represent the given nonlinear function F(x,px) in the form 
 
 F(x,p) = F(x*,px*) + [F(x* + £y, px* + epy) -F(x*,px*)] (4) 
 
 Expanding the two components separately in a Fourier series, we obtain 
 
 F(x,px) = q^ + (A + ^ p) sin oot + 2_. F, + &Y2 
 \ / k=2 k=0 
 
 \ (5) 
 
 k 
 
 where q^. A, and B are the coefficients of the initial term, the sine, 
 
 and cosine terms, rexpectively , the cosine being replaced by — p sin oot 
 
 of the expansion of the function F(x*,px*) in a Fourier series. ZF-^ 
 denotes all the higher harmonics of the expansion of F(x*,px*) in a 
 Fourier series (they must not be considered small, since the nonlinearity 
 is not small) where we write 
 
 F^ = bj^ sin (koit +^^) (k = 2, 3, . . ., -) (6) 
 
 e2§, denotes all the terms of the expansion in a Fourier series of all 
 the expressions shown in brackets in formula (4) . This entire expression 
 
NACA TM 1406 
 
 is written with a small parameter which, according to equation (4), is 
 small if the derivatives Sf/Sx and SF/Spx are finite. This expression 
 is also computed as small in the case of certain discontinuous nonlinear 
 characteristics (e.g., the Raleigh type where the preceding derivatives 
 at the points of discontinuity are delta functions) . We may write 
 
 e* 
 
 ^ = ecj^ sin(tojat + ^^) (k = 0, 1, 2, 
 
 ,-) 
 
 (7) 
 
 We substitute equations (2) and equation (4) in the given equation (l) , 
 so that 
 
 Q(p)x* + Q(p)ey + 
 
 I \ OP OP 
 
 R(p)qO + R(P)U + ^ p) sin oDt + R(p)e^ F^ + R(p)£^<i>j^ = 
 
 (8) 
 
 k=2 
 
 k=0 
 
 Since the equation must be satisfied identically, we separately 
 equate to zero all coefficients with the same order harmonic. We note 
 that formula (8) , in the case of the existence of a periodic solution, 
 is exact . 
 
 From the equating of the zeroth harmonics of equation (8), there is 
 obtained with an accuracy up to e the relation 
 
 (9) 
 
 which is a certain general requirement for F(x,px) . 
 
 From the equating of the first harmoics of equation (8) , taking 
 account of equations (2) and equation (7), we have 
 
 a sin oot = 
 
 - A^ 
 
 A'^ + B^ 
 
 R(iao) 
 
 Q(icjo) 
 
 sin (out + y + p) - 
 
 ECn 
 
 R(icu) 
 
 Q(ia)) 
 
 sm 
 
 (oot 
 
 + ^3_ + 
 
 P) 
 
 (10) 
 
 where x ^^^ P s-^e arguments of the expressions A + iB and 
 R(icD)/Q(iaL)) . On the basis of the exact equation (lO), we obtain the 
 following approximation (with an accuracy up to e): 
 
 -/- 
 
 A^ + B^ 
 
 R(iGD) 
 
 Q(ico) 
 
 r + P 
 
 (11) 
 
NACA TM 1406 
 
 It is here assumed that the polynomial Q(p) in equation (l) does not 
 have purely imaginary roots . 
 
 From the equating of the higher harmonics of equation (8), consider- 
 ing equations (3), (s), and (7), we have 
 
 ■.a^ sin(ka3t + 'Pv) 
 
 R(ika), 
 
 Q(ikjD 
 
 sin(ka)t + \ + P^^)-ecj^ 
 
 Rfikoj) 
 
 Q(ika)) 
 
 sin(koDt + *i^ + Pi^) (12) 
 
 where ^-^ denotes the argument of the expression R(ikJo)/Q(il2Jo) . 
 
 It is thus seen that if bv_ is not small, the magnitude 
 I R(ika))/Q(ikco)| should be of the order of e. The last component in 
 equation (12) will then be of the order e.^ . From the exact equation 
 (12) , we obtain the following approximate equation (with an accuracy up 
 to e) : 
 
 e a^ = b^ 
 
 R(ika)) 
 
 Q(ik£ja) 
 
 \ + Pk ' '^ 
 
 (13) 
 
 Comparing formulas (l3) and (ll) we see that, for example, the wish 
 to have in the solution (see eqs. (2) and (3)) 
 
 (eaj'« 
 
 (14) 
 
 leads to the need of satisfying, in the given equation (l), the following 
 requirement: 
 
 oo 
 
 L 
 
 k=2 
 
 R(ikcD) 
 Q(ikGD) 
 
 ■< (a2 + b2) 
 
 R(ia)) 
 
 Q(icD) 
 
 (15) 
 
 where its satisfying in the concrete system can be checked after cd is 
 obtained. The degree of Q(p) should, in any case, be higher than that 
 of R(p) . A particular case of the general expression (l5) is intuitively 
 the earlier introduced "filter property" of the linear part. 
 
 Thus, condition (l5) has been obtained and must be satisfied by the 
 coefficients of the given differential equation (l) in order that a pe- 
 riodic solution, if it exists, may be approximately determined in the 
 form of sinusoids in the presence of a "strong" nonlinearity of F(x,px) . 
 
NACA TM 1406 
 
 The equation for its approximate determination according to equation (s) 
 with the substitution of sin oot = x'Va, assumes the form 
 
 [q(p) + R(p) (q + 1^ p)] X* = (16) 
 
 where 
 
 '2jt 
 ^ ~ a ~ T^ I -^(^ ^^^ ^> ^^ ^°^ u)sin u du 
 
 F(a sin u, aoo cos u)cos u du (l7) 
 
 The replacement of equation (l) by equation (16) with its subsequent 
 investigation by the linear methods is called harmonic linearization. 
 This is formally equivalent to following the mode of writing the initial 
 equation (l) with a small parameter, so that 
 
 Q(p)x + R(p) fq + _i d X + ef (x,p) = (18) 
 
 where, according to equation (8), we may write 
 
 ef(x,p) = R(p) 
 
 22 ^k + ^ S\ - (<i + of p) ^y 
 
 k=2 k=0 ^ ^ J 
 
 where the first terra of the three terms shown in brackets, taken sepa- 
 rately, is not small. The smallness of the function ef(x,p) is acquired 
 only with the factor R(p) owing to the property of equation (15) . 
 
 There has thus been found the form of the presence of the small 
 parameter e in the equations of nonlinear systems of automatic control 
 required for the application of the method of harmonic linearization 
 (this also refers to the first approximation of the method of small pa- 
 rameter which, according to reference 7, coincides with the given method) . 
 The writing of equation (l) with small parameter in the form of equation 
 (18) is valid in the region of the existence of equation (2). 
 
 For definiteness we shall assume that the polynomials R(p) and 
 Q(p) are such that the characteristic equation (eq. (16)) has all positive 
 coefficients, and that the quotient of the division of the entire left 
 side of equation (16) by p^ + oo^ satisfies the criterion of Hurwitz. 
 It may then be said that the given nonlinear system (eq. (l) ) for the 
 existence of the periodic solution (eq. (2)) is close to the equivalent 
 linear system which is on the boundary of stability, except for the 
 
NACA TM 1406 
 
 circumstance that the linear system is different for various periodic 
 solutions (because of a change in the coefficients q and q, from one 
 periodic solution to another^ since q and q-, depend on the amplitude 
 a and the frequency oo of the required solution). Thus, evidently, the 
 equation of the first approximation (eq. (l6)) is nonlinear if one speaks 
 of the combination of all possible periodic solutions for different values 
 of the coefficients of the polynomials R and Q (i.e., for different 
 parameters of the system) , and is linear only for each given periodic 
 solution (along the given solution) . Such is the property of the method 
 of harmonic linearization in its application to systems with "strong" 
 nonlinearity . 
 
 By writing the given equation (l) in the form of equation (8) all 
 the variants of the method of harmonic balance in control theory and all 
 generalizations of the method also become easily understandable. 
 
 A similar reasoning may also be developed for the application of the 
 method of N. N. Bogolyubov (ref . 2) to the investigation of transient 
 vibrational processes in certain nonlinear systems of control. 
 
 REFERENCES 
 
 1. Krylov, N. M., and Bogolyubov, N. N.: New Methods of Nonlinear Mechan- 
 
 ics. Kiev, 1934; Introduction to Nonlinear Mechanics. Kiev, 1937. 
 
 2. Bogolyubov, N. N. : Sborn. tr. Inst, stroiteln. mekh, AN UkrSSR, 10, 
 
 Kiev, 1949. 
 
 3. Goldfarb, L. S. : Avtomat i telemekh, no. 5, 1947; Method of Investi- 
 
 gation of Nonlinear Systems of Control Based on the Principle of 
 Harmonic Balance. Ch. XXXIII of Fundamentals of Automatic Control, 
 V. V. Solodovnikova, ed., 1954. 
 
 4. Popov, E. P.: Avtomat. i telemekh., no. 6, 1953; Izv. AN SSSR, OTN, 
 
 no. 5, 1954; The Dynamics of Systems of Automatic Control, 1954, 
 pp. 593-663; 704-710; 716-721. 
 
 5. Popov, E. P. 
 
 6. Popov, E. P. 
 
 7. Popov, E. P. 
 
 DAN, 98, no. 3, 1954. 
 DAN, 98, no. 4, 1954. 
 Izv. AN SSSR, OTN, no. 2, 1955, 
 
 Translated by S. Reiss 
 National Advisory Committee 
 for Aeronautics 
 
 NACA - Langley Fifld, V^. 
 
rt r-l 
 
 o 
 
 rt CO 
 O TL ; 
 
 < 
 
 o 
 
 _2 d) 
 
 
 o: S 
 
 Is 
 
 o -• 
 o < 
 
 OS '^ 
 So 
 
 .2 ^?3 
 ^^2 
 
 ^ria 
 
 o 2! 
 
 2S 
 
 Is 
 
 s 
 
 o W 
 
 So 
 
 K 2 
 ?" to . 
 
 Hon, 
 tJ o ■ 
 
 O c W 
 BJ o 
 
 O 5 >> 
 u-S g 
 
 !- O S 
 
 rtS > 
 
 5 c 
 
 o OJ 
 
 CO 
 
 o .2 
 
 2 > w 
 
 '^ TJ CO 
 
 ■*** O Eh 
 
 < rt Z 
 ZiZO 
 
 
 S .-, bD 
 g g '^ 
 
 < CD t; 
 
 H o. > 
 
 Z 0:3 
 "am 
 
 o 2 2 
 
 Sz 
 
 < rt 
 z :a 
 
 ■ C ^ 
 
 ; o 
 
 , CO O 
 
 > rt * 
 
 ! S "1 
 
 ' TO .rt 
 
 ■a 
 
 ,0. rt 
 
 ^2 
 
 if> ^ rt 
 01 ij — 
 
 -M 3 CM 
 
 3 B to 
 
 S oS 
 
 CO tj o> 
 
 i-s C; -< 
 
 V. G 
 
 85 S 2 g 
 
 co" ciJ 73 
 
 •2-° m 
 
 13 c n 
 
 bt , S 
 
 .5 rt Q) 
 
 t< 4) JS 
 
 <1) C '-' 
 
 Sec 
 
 bc ° 2 
 
 <U rt ;- 
 
 CJ ^ *- 
 
 ?> n c 
 
 £ g .S tfl 
 
 o- .2 -S ■■* 
 
 en -g rt „ 
 
 •" O 0) c 
 
 0) "> I. rt 
 
 3 « en f- 
 
 2 •" a 2 
 o o 
 
 o- e ? !3 
 
 ^ 5 OJ cj 
 
 ^ o > a 
 
 13 ^ 
 o 
 
 3 
 < 
 
 e 
 o 
 U 
 
 rt on 
 
 3" i 
 
 Oi CO 
 
 f- .a 
 
 j= 13 
 
 CJ e 
 
 0) <5 
 
 O '^ 
 be — 
 
 §•< 
 
 a< z 
 
 ■icO-S 
 SON 
 
 Z^ ^ 
 > ^ 
 
 <« 
 
 CM 
 
 s-g- 
 
 T3 ^H CO 
 
 -^ 2 o> 
 
 JqS 
 
 »Ba 
 
 Z 
 
 2 
 
 1^ 
 
 K §..: 
 woo. 
 
 ^?, 
 
 u 
 
 n. 
 
 tD 
 
 
 
 0. 
 
 c« 
 
 0. 
 
 m 
 
 .2 o 
 
 Q, (U 
 
 c a o 
 
 > •- rt 
 
 2 « 3. 
 
 at! (u 
 
 S 1^ ""3 
 
 ^ -3 £ § ■ 
 
 o "- ^ ' 
 
 S.°o 
 ^ il -^ . 
 
 - °2 g 
 J: 0) c Q) 
 5 tc o js 
 ■a s*z! ■S 
 "' 13 S 
 
 < 
 
 < 
 z 
 
 to T3 
 
 s5 
 
 ( r- c: ^ 
 
 c 
 
 a 
 
 
 
 g 
 
 Q) 
 
 ,j 
 
 < 
 
 
 
 
 
 
 
 g 
 
 
 
 
 (1) 
 
 S 
 
 
 « 
 
 s 
 
 < 
 
 T, 
 
 n 
 
 U 
 
 r 1 
 
 X 
 
 >. 
 
 H 
 
 u 
 
 U. 
 
 to 
 
 
 
 
 
 > 
 
 W 
 
 •o 
 
 I/l 
 
 < 
 
 W ID . 
 
 o-sw 
 
 K o 
 o ^ >> 
 
 U-S g 
 
 K 2 o 
 
 S ._ bO 
 
 N c^ 
 a o 
 
 a. a 
 
 CO , 
 
 u ^ 
 
 4) -^j -Q "^ ;::3 
 
 CO e^ (ti 
 .2-0 to 
 
 " 2 -5 
 
 ♦^ O 01 c 
 dj ■" t, cij 
 
 s 
 
 < CD Z 
 
 zzo 
 
 3 C ;3 
 
 W.S S 
 
 H a > 
 z o- 
 "am 
 Q §13 
 O 2 2 
 K a C 
 
 CO -— • 
 
 o OS 
 
 < cd 
 
 uz 
 z :a 
 
 <u 
 a c3 
 
 t- ^ ' 
 
 Oi 2 ■^ 
 1-t ^ a 
 
 a-3 
 
 s s 
 
 OJ .H 
 
 m m 
 a _>. 
 
 a> cij 
 
 (U i! 
 
 ■o 
 m 
 
 o • o 
 -»- to — • 
 
 Cl! E c« 
 
 " 2 ^ 
 
 -1 -^ 13 
 
 W ? a! 
 be . C 
 
 .5 CJ 4) 
 
 tU C *J 
 
 Sec 
 
 m 3 c S " 
 m -. 3 B & 
 
 a e ? b 1) 
 *- 5 ") c« ^ 
 
 5a°°S, 
 .S a 13 « 
 
 < 
 
 ':5 S u 
 
 
 a ' 
 a i" ' 
 
 si o 
 H 3 
 
 o ^ _ 
 r! -S m 
 
 IS'" 
 
 2 Sfi 
 
 a lu 
 
 CJ 
 
 ^ ^ CJ 
 
 g&5 
 
 > '-' rt 
 
 M J- cr 
 0,3 <u 
 
 ■o 
 
 'm -o 
 *" 1) 
 
 - ^ "■ ' 
 
 0) "^ S 
 
 <D C <U 
 
 "' 13 "• 
 
 S^ 
 
 m S =• .-^ 
 IS m uj - 
 , ^ c m 
 
 QJ O ™ 
 
 £ 
 3 
 < 
 
 e 
 o 
 o 
 
 3 00 
 
 S ■ 
 o ;t 
 
 3 
 
 <7J 
 
 
 
 O" 
 
 
 1 
 
 
 c 
 
 
 m 
 
 
 s: 
 
 
 0. 
 
 
 
 0) 
 
 (0 
 
 L. 
 
 
 H 
 
 CJ 
 
 
 
 
 J3 
 
 ■r| bO 
 
 s 
 
 u 
 
 S 
 
 ^ 
 
 H 
 
 tcl 
 
 1) 
 
 > 
 
 *(" 
 
 q; 
 
 ji: 
 
 n 
 
 U 
 
 rn 
 
 
 n 
 
 
 
 n 
 
 <: 
 
 b: S 0. z 
 
 "2 
 en ■= 
 
 ■gco'^ 
 
 z-^ ^ 
 
 1-g^ci. 
 
 ^ -H CO 
 
 ^q2 
 
 o 
 
 
 -CM 
 
 (U ■ 
 3 <J1 
 C*^ — ' 
 
 tu m 
 H 
 
 as 
 
 o 
 
 3 w> 
 
 o 
 
 •§ 
 
 
 (SS 
 
 o < 
 (^ Z 
 
 So 
 
 CO ^ 
 
 ■g CO i-i 
 
 d 
 
 eg 
 
 S^a 
 •S S - 
 
 XJ ^H CO 
 
 :Sd2 
 
 ^oB 
 
 W CSl 
 
 ,aB 
 
 z 
 
 H 
 
 ■ <! 
 
 O N 
 
 .H *-• 
 
 ^! 
 C U 
 
 2S 
 
 •sl 
 
 S3 
 
 eS 
 s 
 
 o" 
 >,^ 
 
 i^ <^ 
 § §0 
 
 2; ■> w 
 
 -^ T3 M 
 
 g<; p 
 
 ^ B PS 
 < c< Z 
 
 zzo 
 
 3h m • 
 
 H d) Q, 
 
 T -e 
 
 O c W 
 
 « o 
 
 &: s„ 
 
 o ^ > 
 
 G 2 o 
 g E 3 
 0:gS? 
 
 5 = :3 
 < d) Jj 
 
 W.§ S 
 33 C - 
 H a > 
 
 °:a 
 
 3 m 
 
 3 § 
 
 
 CO -^ 
 o « 
 ■» m 
 
 H^ 
 
 < cH 
 UZ 
 i< d 
 Z:a 
 
 dJ 
 
 eg' ?! 
 
 m 
 
 , u <u 
 
 en 3 > 
 
 .2 2 c o 
 
 ^ o i I 
 
 d) -M J3 — 
 
 .s s :^ .2 
 
 S == ? ^ 
 
 M o a 
 
 •^ ..^ J3 d 
 
 !3 d 15 
 Sam 
 
 m 
 
 d ' 
 
 -i o 
 d ■" 
 
 >.a ii 
 ■° m a 
 
 c d m 
 11^ 
 
 o c 
 
 <= 2 
 
 2 -3 
 
 O d) 3 
 
 "> t. d 
 
 d) d -f- 
 
 J3 d> ^ 
 
 5 S d) 
 
 •" a a 
 
 o 
 
 _ C t, 
 
 c d) d 
 
 g- 
 
 5^ 
 
 Q §13 
 
 o 2 2 
 a a t. 
 
 H 9 d 
 
 w^ S 
 
 o o S "> 
 
 <" -2 2 I 
 
 ^ ^ a d) 
 
 d) d " c 
 
 5 S .2 ■& 
 
 CO a, J- (u 
 
 ^ j2 t-i _ 
 
 ire "O .-K J^ 
 
 J£ Cm -^ O • CJ 
 
 2 _. oQ -^ CO :j 
 
 :;i o > (1) >» 
 
 a m In bb.5' 
 
 O 
 
 ^ a 
 
 O -3 O 
 
 = ■§3 
 
 cn;3 ^ 
 o-9x; 
 
 en J3 .^ 
 
 -H 3 CM 
 
 3 a <o 
 
 Co"' 
 d £ eji 
 t^ ^ ^^ 
 
 .. ■" 2 
 
 fc. a 2, a 
 &&S>.c 
 
 o-S o g 
 m _ i! > 
 
 £2 §2 
 H 3 o a 
 
 as- 
 
 si c 
 
 a o 
 dj -^ ■ 
 
 o 
 
 ••3bc5 
 Sag 
 
 •H J3 O , 
 « -^ c • 
 O «-■»-, ■ 
 
 §■13 
 
 V-i *^ S ■ 
 
 o o d ■ 
 
 d; ■" a 
 
 d; c d) 
 
 h O J3 
 
 ^§a. 
 a"|- 
 
 o S ^■ 
 m dj -■ 
 _ c m 
 
 ^ ■^^ 
 
 < 
 < 
 
 ZtH 
 
 O « 
 
 SO 
 
 to M SB 
 
 2SS 
 
 « J z 
 
 o Z ,, 
 ' 2 ■> 
 
 5 KO 
 rR w 3 
 
 5S ^ 
 
 J5 f^ 
 So 
 
 g cm' 
 
 a-H • 
 ■-1 o ti o 
 
 oa,i2 c 
 
 o 
 
 a -CD --y 
 
 u -Jo aj 
 
 d sir}. M 
 
 d) ,t! <; d 
 „-3 U Z 
 .-. bO<! d 
 
 « a 
 
 c :3 d) 
 
 d 3 
 
 2 S c 
 
 c a Si 
 
 d) ■*-• -Q a 
 33 _ 3 o 
 
 a "^ d » "■ 2 t^ 
 u h-c g >>a i^ 
 
 2 g j= 2 -o » a 
 
 o C y d g d m 
 H o * Q. eg 2 a 
 
 ' 5r"S — 
 a " 
 
 d) 
 d) « 
 jq 13 
 
 '^ a 
 
 bO 
 
 C J3 
 
 gco , 
 
 2; V W 2 
 '-' T3 M a 
 
 <!• c a s 
 
 < dZg 
 ZZOS 
 
 a > . CM 
 o^St- 
 
 3 (0 OS ."^ ' 
 
 is ^ 
 
 
 3 a 'o 
 
 ba S e« t. e3> 
 
 Sr3 n i; ^ 
 
 d) 2 2 -S en 
 = J a d) :a 
 
 d) d en c 3 „ _ 
 
 °-^s 2 asl 
 
 t, 'S, m a d) 3 
 §.&S-c_§^o 
 
 2.S°£&5.2 
 
 m _ i! > '-' d t. 
 H *j CJ at3 dj Q 
 
 g d) d 
 
 g > d) 
 
 s s 
 ■a 5 
 
 O s-i 
 
 t. o 
 a^ 
 
 ft e« 
 
 ° a 
 
 dj 
 
 d} 3 
 b o 
 bOrl 
 
 S a 
 ^ 3 
 
 a) tT 
 
 ^> 
 
 o S 
 
 « di 
 
 j= = 
 ti d) 
 
 ?5 
 
 o ^ 
 .» o 
 
 •^5 
 
 aS 
 
 QJ O 
 
 j3 
 
 2 = 
 
 3 o 
 
 ;^ ^^ 
 H Si 
 
 < 
 
 < 
 z 
 
o 
 
 o 
 
 q 
 
 cil 
 
 I 
 
 
 o 
 
 
 S » 
 
 
 1 6 
 
 to 
 
 to >> 
 
 3 " 
 
 o 
 
 Si3 
 5 « 
 
 f-t 
 
 s 
 
 bO.S 
 
 < 
 
 
 O 
 
 QJ *< 
 
 < 
 
 s? 
 
 s 
 
 S Id 
 
 < 
 < 
 
 2 
 
 S 
 < 
 
 5 s 
 3 s 
 
 
 < 
 
 < 
 z 
 
 o 
 
 o 
 
 u 
 
 u 
 
 s 
 
 I 
 
 CO 
 
 « S 
 
 "S « 
 
 «> 
 
 < 
 
 o 
 
 I 
 
 e 
 o 
 u 
 
 u 
 
 •■3 
 s 
 
 I 
 
 "S 
 
 n 
 
 s « 
 
 3 £ 
 
 "53 m 
 
 
 -^ 5 
 
 S 
 
 < 
 
 z 
 
 bo .3 
 l§ 
 
 X C 
 
 (U C4 
 
 < 
 
 < 
 Z 
 
 bo.a 
 
 I = 
 ■o o 
 
 c c 
 
 >< S 
 <u at 
 
CM 
 
 2 CO 
 
 -rj «« 
 
 
 B . 
 
 o • t/l J 
 
 ^ 00 
 
 3 — 
 
 3 05 w 2 
 
 V ■ 
 
 □•^^•T! M C 
 
 o 
 
 3 
 
 < 
 
 chni 
 
 Pau 
 
 06 
 
 auk 
 
 06, 
 
 214 
 
 3 
 < 
 
 C 
 O 
 
 hTe 
 atlcs 
 Egor 
 M 14 
 ilaN 
 , v.l 
 211- 
 
 ^^ 
 
 N 
 
 S S ."^ S.g-0. 
 
 u 
 
 
 rt cu > rf (u S . 
 
 c 
 o 
 o 
 
 1 
 
 Rese 
 Math 
 Popo 
 NAC 
 Akad 
 Dokl 
 1956 
 
 «dB 
 
 CM 2 03 
 
 ► CM 
 
 K'^ 
 
 ,. • s ■ 
 
 <u . 
 
 W J 
 
 .i^ « o - 
 
 3 Ol 
 
 m ° 
 
 S • is — 
 
 ty-"g 
 
 CO <= 
 
 rt ^ 3 
 
 o 
 
 1 S§ 
 
 •S to rt 
 
 < 1 
 
 h Te 
 atlcs 
 Egor 
 M 14 
 
 ilaN 
 , v.l 
 211- 
 
 ? :S 
 
 ^ e -"^ 
 
 e^a 
 
 rt 01 > < 
 
 ii rt . 
 
 
 0) -; ay 
 
 
 c -s 
 
 a Mia 
 
 o 5 
 
 0) <5 o •< 
 
 OS S ei, z 
 
 ^S2 
 
 ^bH 
 
o 
 
 o 
 
 H 
 
 <: 
 u 
 
 2 
 
 o 
 
 h 
 
 a 
 o 
 u 
 
 CI 
 
 I 
 
 e 
 "3 
 
 ii 
 
 V 
 
 CO 
 
 3 
 
 & 
 
 
 
 i 
 
 :§ 
 
 1 
 
 § 
 
 B 
 
 c 
 
 £ 
 
 ■a 
 
 x 
 
 B 
 
 S 
 
 ol 
 
 < 
 
 s 
 
 < 
 
 < 
 
 a 
 o 
 o 
 
 u 
 
 i 
 
 s 
 
 I 
 
 a 
 
 
 at 
 
 01 
 
 c 
 "3 
 "S 
 
 CD 
 
 3 
 
 S' 
 
 1 
 
 13 
 
 0) 
 
 hn 
 
 .s 
 
 r" 
 
 
 t1 
 
 o 
 
 c 
 
 c 
 
 
 < 
 < 
 
 2 
 
 H 
 
 o 
 
 < 
 
 1 
 
 3 
 
 "S 
 
 B 
 S 
 
 CO 
 
 3 
 
 s- 
 
 <U 
 
 S 
 
 5 
 
 g, 
 
 :§ 
 
 1 
 
 g 
 
 e 
 
 c 
 
 5 
 
 T) 
 
 X 
 
 C 
 
 S 
 
 a 
 
 < 
 
 < 
 z 
 
UK'IVERSITY OF FLORIDA 
 
 .'JSNTS DEPARTMENT 
 1 20 MARSTON SCIENCE UBRARY 
 P.O.BOX 117011 
 GAINESVULE. FL 32611-7011 USA 
 
 ■^ J ; 
 
 •^eSKARCH CENTER 
 704-W 
 . y'SSOTA 
 
 -.V. M;ADOfJ U-V 
 
 T. M. 14Q6