The Quarterly of Applied Mathematics, Vol. 3, Nr 2

v :

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QUARTERLY

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Vo l u m e

III JULY • 194s

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89

Q U A R T E R L Y OF A P P L I E D M A T H E M A T I C S

V ol. I l l JU L Y , 194S N o . 2

SOM E NUMERICAL M ETH O D S FOR LOCATING ROOTS OF POLYNOMIALS*

BY

T H O R N T O N C. F R Y B ell Telephone Laboratories

1. In tro d u c tio n , I t is th e p u rp o se of th is p ap e r to discuss th e location of th e roots of polynom ials of high degree, w ith p a rtic u la r reference to th e case of com plex roots.

T h is is a problem w ith w hich we a t th e L ab o rato ries h av e been m uch concerned in re c e n t years because of th e fa c t th a t th e problem arises ra th e r freq u en tly in th e design of electrical netw orks. I shall n o t give a n y a tte n tio n to stric tly th eo retical m ethods, such as th e ex a ct solu tio n by elliptic or a u to m o rp h ic fun ctio n s: nor to th e develop­

m e n t of ro o ts in series o r in continued fractions, tho ug h such m eth od s exist an d one a t least— d evelopm ent of the coefficients of a q u a d ra tic fa c to r1— is of g re a t value in im proving th e accu racy of ro o ts once th ey are know n w ith reaso nab le app rox im ation .

In ste a d , we shall deal w ith ju s t tw o categories of so lu tio ns: one, th e solution of th e equ atio n s by a succession of ra tio n a l o p eratio n s, hav in g for th e ir p u rp ose th e dispersion of th e ro o ts; th e o th er, a m ethod d epend ing on C au ch y 's theorem regardin g th e n u m b e r of ro o ts w ith in a closed conto u r.

P A R T I— M A T R I X IT E R A T IO N

2. D u n c an a n d Collar. W e shall tr e a t th e first categ o ry by a m ethod recen tly elaborated by D u n can and" C ollar in tw o p ap ers in th e Philosophical M ag azin e.2 I do n o t know how th o ro u g h ly these w riters a p p re c ia te th e close relation sh ip of th e ir w ork to t h a t of th e o th e r w riters whom I shall m ention in th e course of m y p resen tatio n . T h e fa c t t h a t th eir in te re st w as p rim arily concerned w ith certain b ro ad d ynam ical problem s m ay p erh ap s h av e inhibited th em from tak in g som e of th e step s which I shall ta k e in th e ir nam e. B u t th ey a t least possessed th e essential idea, an d exhibited q u ite sufficient ab ility in th e dev elo p m en t of it to w a rra n t th e assertion t h a t m y p re se n ta tio n only differs from th eirs in d e ta il— som etim es d etails of om ission, som e­

tim es d etails of am plification.

* R eceived D ec. 26, 1944.

1 T h e essence of this m ethod is contained in a section of Legendre's E ssa i su r la théorie des nombres.

It is a lso attrib uted to Bairstow by Frazer and D u n ca n . I t was develop ed ind ep endently, and perhaps som ew h at more fu lly, by the p resent writer; b u t the exten sion s seem so obviou s th a t it has n ot appeared to w arrant separate publication.

2 W . J. D u ncan and A . R. Collar, A method fo r the solution of oscillation problems by m atrices, Phil.

M ag. (7) 17, 865 -9 0 9 (1934) ; M atrices ap p lied to the m otions of dam ped system s, Phil. M ag. (7) 19, 197-219 (1935).

3. T h e fu n d a m e n ta l id en tity . W e begin by n o tin g th a t th e X -d eterm inan t

90 T H O R N T O N C. F R Y [Vol. I l l , N o. 2

- n ( x + x,) (i)

flu + X «21 0-nl

D ( \ ) =

On 022 "f“ X ■ • a n 2

a \ n 02 n a nn ~i" X

is th e ch a ra c te ristic fu n c tio n3 of th e m atrix

M =

|JW| (2)

and its d e te rm in a n t

A = j ais | . (3)

I t is obviously a polynom ial of degree n, which we m ay w rite

■D{\) = X" + M n_1 - M "~2 + • • • ± pn. (4) A ny q u a n tity which satisfies th e eq uation

D ( \) = 0 (5)

and obeys th e asso ciative and co m m u ta tiv e laws of alg eb ra— w h e th e r it be a n u m b er or n o t— m u st also satisfy th e relation

X* = - + p 2\ n~ 2 H + p n ,

and if we m u ltip ly th is by X th ro u g h o u t an d th en elim inate X" we get

„ n - f l , 2 n—1 , . . n— 2

X = (pi + />2)X — ( p i p ! + p l ) \ +

w hich is of th e form

n + 1 ( n + l ) n - 1 ( n + 1 ) n - 2

X = p i X + p i X +

Sim ilarly, by a co n tin u atio n of th e sam e process we m ay g e t a succession of eq u a­

tions, all of th e form

m (m) n—1 (m) n—2 , ,

X = p i X + p-i X + • • • . (6 )

W e call th e ty p ical polynom ial on th e rig h t of (6) / m(X): g raph ically it re p resen ts a curve of degree n — 1 passing thro u g h th e n po in ts —X,-, ( — X,)m. B u t we do n o t wish to em phasize th is geom etric in te rp re ta tio n b u t ra th e r th e form al alg ebraic fa c t th a t our d e riv a tio n has required only th e elem e n ta ry rules of algeb ra and th e relatio n (5), and t h a t w hen these rules are satisfied

X”1 = f m(\). (7)

S uppose, now, th a t we expand th e q u o tie n t f m( K )/D ( \) in p a rtia l fractions. T h e re su lt is

/m®X) ^ , J-rn ( Xjj I D (\) ,_i X + \ j I I ( — Xj + X*)

w

3 T h e u nconven tion al pecularities of sign in (1) and in (4) b elow happen to be con ven ien t for our purposes later on.

1945] M E T H O D S F O R L O C A T IN G R O O T S O F P O L Y N O M IA L S 91

B u t / m( —X,) = ( ~ X ,)m by (7), and D ( \ ) = H (X + X ,) by (1). H ence

j - l X j /

Now (9) is an algebraic id e n tity , an d though w e h av e used th e process of division in se ttin g it up, it does n o t req u ire division b y X as a process of verification. H ence it is again tru e th a t if X is an y q u a n tity w hich obeys th e d istrib u tiv e and associative laws, such for exam ple as a differential o p erato r, and which satisfies (5),

X“ = Z ( - ( i o)

1 w here

a «

k ^ j \ A j /

N o te th a t th e q u a n titie s d enoted b y x^X ) are polynom ials of degree n — 1 in X and are independent of ini.

4. M a tric e s. W e n ex t observe th a t, th o u g h m a trix m u ltip licatio n is n o t in general c o m m u ta tiv e , it is so if we re s tric t ourselves to c e rta in groups. In p a rtic u la r, if we begin w ith th e u n it m a trix I , a n y o th e r m atrix M , an d all sca la r q u a n titie s (i.e., num bers), th en all m atrices w hich can be form ed from th ese by a finite n u m b er of a d d itio n s or m u ltip licatio n s are co m m u tativ e. F o r obviously M is c o m m u ta tiv e w ith itself an d its pow ers, an d w ith I, an d w ith scalars, which o b serv atio n s to g eth er w ith th e associative law are sufficient to w a rra n t th e general s ta te m e n t.

F u rth e rm o re , we know from th e H a m ilto n -C ay ley T h eo rem4 t h a t D{M) = 0,

w here D ( \ ) represen ts, as in §3, th e c h a ra c te ristic fu nction of M . In o th e r w ords, M satisfies all th e req u irem en ts im posed upon X in deriving th e id e n tity (10), w hence we conclude th a t

(12) w here x / ( M ) is a m a trix independent of m.

A s a final step , we m u ltip ly th is eq u a tio n th ro u g h o u t by an a r b itra ry m atrix K -—which need n o t be c o m m u ta tiv e w ith th e rest, since we shall perform no fu rth e r o p eratio n s— th u s o b tain in g

~ M '"K « ; ( f f v j ) , . I S ) (1 3 )

w here — x / ( M ) K is again in d ep en d e n t of m.

This is the fu n dam ental identity upon which Duncan and Collar rely for their method.

I t is eq u iv ale n t to n 2 eq u atio n s of sim ilar form conn ecting corresponding elem ents in th e vario u s m atrices. F o r exam ple, if a m is w ritte n for th e elem en t in th e f-th row and j - t h colum n of M mK , and i j for th e correspon dingly placed elem ent in irf.M ), it m u st

be tru e th a t

am = e f — Xi) m + en(— X2) m + • • • + e „ ( — \ n) m■ (14)

* M . B o ch er, Intro d u ctio n to higher algebra, M a c M illa n , N ew Y o rk , 1929, p. 296.

92 T H O R N T O N C. F R Y [Vol. I l l , N o. 2

W e again recall th a t —X,- is a ro o t of th e ch a rac te ristic eq uation of M — th a t is, of th e polynom ial Z3(X)—~and hence is a num ber. T h e set —X; are, in fact, ju s t th e roots which we wish to obtain. Similarly, t h e e / s are nu m b ers independent of m. B u t a m is a num erical function of m.

5. T h e ro o ts. Suppose now, t h a t one of th e ro o ts w hich we will call —Xi, is larger in ab so lu te v alu e th a n all th e rest. T h en if we select corresponding elem ents a m and a m+1 from tw o consecutive orders of M mK we will h ave

as — Xl

1+ — ( ^ ) +

e\ \A i/ C i\X [/

K — ( —) k r — ( — ') + e\ \ i / e\ \X i7 1 +

an d hence obviously

lim = _ X]. (15)

In o th er words: i f an arbitrary matrix K is multiplied repeatedly by M , and i f its characteristic equation has a largest root, then the ratio of corresponding elements in two consecutive products approaches this largest root as a limit as m —r <x>.

Similarly, we readily find t h a t CLm-\-l &m

CLftt— 1

a

w hence if |X2 and X2 are g re a te r th a n all o th er X’s | , (w h ether th e y are th e m ­ selves equal or n o t), we again have

^m-fl CCm OLm Ctm— 1 lim

CLm CL m — 1

CL m—1 CLm~ 2

— ( — X i ) ( — Xi). (17)

In th e sam e w ay it can be sho w n 5 t h a t provided X j|, • • • , X,| are all g re a te r th a n X,-+i * X„

lim

in-f-1

77i -f-1—2 ’ ' OCm

ttm+l CL fti * 1+2

O t m + i — 2 * CL m C L m -\-i— 3 * & m — 1

CLni OLm - l ■ OLm— i - f l

= ( - X J ( - Xi) ■ ■ • ( - X,). (18)

5 A . C. A itken, Proc. R oyal S oc. Edinburgh 46, 289-305 (1926), o b tain s form ulae eq u ivalen t to these in a discussion of Bernoulli’s m ethod.

1945] METHODS FOR LOCATING ROOTS OF POLYNOMIALS ⁹³ T hese eq u atio n s are sufficient to d eterm in e all th e ro o ts in th e p a rtic u la r case w here

Xl > X; > X3 | • • • > X„ | .

6. E xam ple. As a sim ple exam ple we m ay tak e

in w hich case

3 1 1

M = K =

2 2 0

3 11 43

M K =

2 M - K =

10 M*K =

42

171 683

M*K = , M SK =

170 682

T a k in g th e ratios of th e first elem ents of consecutive m atrices we g e t as th e successive ap p ro x im atio n s to - X |,

11/3 = 3.667, 43/11 = 3.909, 171/43 = 3.977, 683/171 = 3.992.

S im ilarly we find th a t 171 43

43 11 43 11

11 3

= 4 and

683 171 171 43 171

43 43 11

= 4,

which should be th e p ro d u c t of X2 an d X2.

T h e ch a ra c te ristic eq u atio n in th is case is, how ever, Z)(X) = X + 3 1

2 X + 2 = X5 + 5X + 4, and its ro o ts are —1 an d —4. T h e appro x im atio n is obvious.

7. C om plex ro o ts. So fa r we have considered o nly real ro o ts: for obviously, since com plex ro o ts occur in c o n ju g ate p airs (th e coefficients being assum ed to be real) th e re can be no largest one. S uppose, th en , th a t X* = X and th a t all o th e r roots are sm aller in ab so lu te value. T h e n b y (17),

Xi X2 = lim

Ct m -r 1 CCm dm OCtn—1 OLm Ctm—1

Oim—i OL m—2

(19)

T h is gives us th e ab so lu te v alu e of th e roots. I t does n o t, how ever, d eterm in e th e angles. T o g et this, we can b est re tu rn to equ ation (14) an d w rite (retain in g only th e leading term s)

«m = « i ( - XO”1 + et ( - X2) ’\

94 T H O R N T O N C. F R Y [Vol. I l l , N o. 2

W ritin g th e sim ilar eq u atio n s for m — 1 and m +1, and elim in ating ei and e2, we get XiX2a:m_i + (Xi + X2)o;m + a m+i = 0.

S u b stitu tin g th e value of XiX2 as given by (19) we get finally

— ,(Xi + X2) = lim

&m ^{+ l} &m— ¹ Olm CLm—2

Otm « m - 1

^ f?i— 1 & m —2

( 20)

T h is, to g e th e r w ith (19) is sufficient to d eterm in e th e p air of roots.

As w ritte n , th e form ula applies even if th e ro o ts are re a l.6 W hen th e y are com plex it is b est to w rite —X i= — T h en obviously we need only replace th e XiX2 of (19) by p 2, and the — (Xi+X2) of (20) by 2p cos p.

Sim ilar, b u t m ore com plicated, form ulae can be o b tain ed w hen m ore th a n tw o roots hav e th e sam e absolute value.

8. T h e m eth o d of D a n iel B ernoulli. W e now note th a t an y polynom ial in X, which we ta k e in th e form

± Pn (4⁾

D { \ ) = X' + M ’,_1 - M " - 2 + • ^{-■ Pn} as before, can be w ritte n as

X 1 0 0 0

0 X i 0 0

D { \ ) = ^{0 0} X 0 0

0 0 0 • • • X 1

pn Pn—1 P n -2 ' ’ ‘ p i p i B u t this is the characteristic function of t h e m atrix

0 1 0 0 0

0 0 1 0 0

M = ^{0 0 0 0 0}

0 0 0 0 1

pn pn—1 p n-1 ’ ‘ ‘ p i p i

( 2 1 )

(2 2)

H ence if we choose for K a n y m atrix w hatever, we m a y solve for t h e largest roots by a n y of the equations of §§4 and 6.

I t is p artic ula rly convenient to t a k e K in th e form

• I t is n ot even necessary th a t they be equal in ab solu te value, though unless th ey are equal (or nearly equal) (15) will ob viou sly be a m ore con venien t form ula.

1945! M E T H O D S F O R L O C A T IN G R O O TS OF P O L Y N O M IA L S 95

T h en we h av e

0 0 0 • «0 0 0 0 • «i K = 0 0 0 • a. 2

0 0 0 • a „ _ i

M K =

0 0 0 • • a i 0 0 0 • ■ a 2 0 0 0 • ■ a 3

0 0 0 ■

n—1 ' jLj p n - i 01]

j'-O

(23)

which is again of th e sam e form as K . If we d en o te b y w e also have

M ?K =

0 0 0 ■ a 2

0 0 0 • ot.

0 0 0 ■

0 0 0

n—1

2 ^ P i—la n—i j - 0

And in general

0 0 0 • • a m 0 0 0 • • a m-f-l

M mK = 0 0 0 • • Otm+ 2

0 0 0 • * 1

w here

¿—0

k > n — 2, ^{(2 4 )}

T h is e n tire se t of m atrices, how ever, is ch aracterized b y a sim ple sequence of a s , of w hich th e defining eq u a tio n is (24). O bviously, it is also tru e t h a t an y set of fo u r consecutive a s in this sequence also constitutes a set of corresponding elements from fo u r cotisecutive matrices of the set M mK . H ence, th e use of th e sym bol a in th is connection is co n sisten t w ith its use in §§3-6. B u t (24) is th e recursion form ula used in B er­

noulli's m eth o d of solution as developed b y E u ler, L agran g e and A itk cn. H ence th is p a rtic u la r special case of th e re su lts of D u n can and C ollar is id entical w ith B er­

n o u lli’s m ethod.

C oncerning th is m eth o d W h itta k e r an d R ob in so n7 say : “T h ou gh h a rd ly now of first-rate im p o rtan ce, it is in tere stin g an d w o rth y of m e n tio n .” O ur te sts a t th e

7 E . T . W hittaker and G. R obinson, Calculus of observations, 2nd ed., B lack ie & Son , London, 1929.

96 T H O R N T O N C. F R Y [Vol. I l l , N o. 2

L ab o rato ries, how ever, h av e show n it as good as an y o th e r m etho d in th e case of com plex roots. Such inferio rity as it m ay h av e com pared to th e ro o t-sq u a rin g m etho d as regards speed is q u ite com pensated b y th e fa c t th a t it is self-correcting: t h a t is, an error a t a n y stag e of th e process m erely prolongs th e calcu lation s, b u t does n o t in v a lid a te it.

9. T h e m eth o d of R. L. D ietzold. A n o th er form into w hich th e general resu lts of D u n c an and C ollar can be throw n is ob tain ed by using th e c o n ju g ate form of (21) to g eth er w ith th e sam e m atrix for K as before. D en o tin g th e c o n ju g a te of M by M ' , we h av e from (22) an d (23)

0 0 0 • ■ • pn < X n - l

0 0 0 ao + p,,-\ctn-.i

M ' K = 0 0 0 ■ ■ ■ a i + i

0 0 0 aB_2 + pia n- i If, th e n , we define

a o p n d n —1, Oij —j- p n—jCCn—l, (25)

M ' K becom es identical w ith (23), except t h a t all th e a ’s are prim ed. In general, if w e set

(m) (m-1) (*•— 1)

ay = a y - i + pn-,-an- i , (26)

a n d u n d e rsta n d th a t v& l is zero for all m, we h ave

M K

0 0 0 ■ • a o

0 0 0 •

(m)

• a i

0 0

\ 0 •

(m)'

• a 2

0 0 0 •

(m) a „ _ i

(27)

In th is case, as in all o thers, th e index m is th e one which is to b e varied in using form ulae such as (16)-(20).

T h is v a ria n t of th e general schem e of D u ncan and C ollar w as developed by M r.

R . L. D ietzold of th e Bell T elep hone L ab o rato ries, b u t has n o t been p ublished.

As com pared w ith B ernoulli’s, it has th e m erit of using a large n u m b er of sim ple op eratio n s instead of a sm all n u m b er of com plicated ones. I t is ap p ro x im a te ly as fast, and like all schem es based on D un can and C ollar's results, it is self-correcting.

10. T h e m eth o d of G raeffe, T h e re is also a close connection betw een D u n can an d C ollar’s processes and th e root-sq u arin g m etho d. T h is m ethod , w hich is u sually a ttrib u te d to Graeffe, seem s ac tu a lly to h av e been developed first b y D and elin , and has had th e a tte n tio n of a long list of m ath em atician s, including L obachevski, E nck e, B ro d etsk y an d Sm ead, and H utch in so n .

T h is connection can best be estab lish ed8 by recalling t h a t th e roots —X, of the

8 M . Bocher, Introduction to higher algebra, M acM illan, N ew Y ork, 1929, p. 283, Theorem 3.

1945] M E T H O D S F O R L O C A T IN G R O O T S O F P O L Y N O M IA L S 97

m atrix \ I - \ - M are in v a ria n t un d er tran sfo rm atio n s of the ty p e T ~ l [ \ I - \ - M ] T . F u rth e rm o re , it is possible to find a tran sfo rm atio n of th is s o rt w hich will th ro w M in to th e form

M* = T ~ lM T =

At 0 • • 0 0 A2 • ■ 0

0 0 • ■ A„

and hence A /+ ik f in to th e form \ I - \ - M * , since T ~ lI T is obviously I.

T h is sam e tran sfo rm atio n , how ever, carries M m in to M * m, as we readily see from th e id e n tity

T - ' M ”'T = T ~ X( M T T ~ ' M T ■ ■ ■ T~lM ) T

= {T~XM T ) ( T ~ ' M T ) • • • (T-'-MT)

H ence th e c h a rac te ristic equ atio n s of M n and M * m m u st also h av e id entical roots.

B ut, obviously,

M

A x • 0

0 m

A2 ^{• 0}

0 0

til

■ A„

so t h a t th e roo ts of its c h a rac te ristic eq uation , and th erefore also th o se of th e c h a r­

ac te ristic eq u atio n of M m, m u st be — A"1.

B u t if w e ta k e K — I in D uncan and C o llar’s process of m atrix itera tio n , th e su c­

cessive m atrices o b tain ed are M m. H ence th e w hole process m ay be regarded as one which sets up a sequence of c h a rac te ristic eq u atio n s w ith ro o ts — A,-, —A' • • • and in general — AJ*.

In th e root-sq u arin g process as originally developed only th e pow ers —<4, —A*,

— A®, w ere o b tain ed , w hich corresponds in m a trix term s to g ettin g first th e p ro d u c t of M by M , which is M 2; th en th e p ro d u c t of M2 by M 2 w hich is M i , an d so on. T h u s high pow ers are reached w ith a sm aller n u m b er of m atrix o p eratio ns, which is th eo re tic ally desirable. P rac tic ally , how ever, th e su p erio rity is n o t so a p p a re n t.

F o r th e zeros of (22) are ra p id ly replaced by n u m b ers in form ing pow ers of M , so th a t a m u ltip licatio n such as M8 i¥ 8 involves m an y m ore a rith m e tic a l o peration s th a n a m u ltip licatio n of th e form M - M * . F u rth erm o re , an erro r a t an y p o in t of th e root-sq u arin g m ethod p e rp e tu a te s itself, w hereas in th e o th e r m etho d an error a t a n y stag e is m erely eq u iv ale n t to s ta rtin g over again w ith a new v alu e of K ,

O ur experience leads us to believe th a t th e m etho ds of §§8 and 9 are generally to be preferred, a t least w hen co m p u ta tio n s are to be perform ed by a clerical staff of com puters.

l b T h e m eth o d of B ernoulli a s developed by L ag ran g e. T h ere is also a v ery close- connection betw een th e ite ra te d m atrix M m and a d ev elop m en t of L ag ran g e s which ho ch aracterizes as based upon th a t of D aniel B ernoulli. In it, he n o tes th a t

98 T H O R N T O N C. F R Y [Vol. I l l , N o. 2

£>'(X)

D(X) X +X,- X X2 X3 X4 (28)

w here

= ( - Xi)" + ( - X * )" + . . . + ( _ x , ) - ; ^{( 2 Q )} and it is th e q u o tie n t sm/ s OT_i w hich L ag ran ge uses. O bviously, th ese are ju s t th e sum s of th e elem ents in th e principal diagonals of M * m. B u t L a g ran g e’s m eth od of o b tain in g them b y dividing J9(X) in to its d eriv a tiv e is preferable. Besides, in s p ite of w h a t m ig h t a t first be assum ed, it is self-correcting.

I t is of historical in te re st to n o te t h a t a v ery sim ilar d ev elo p m en t w as w orked o u t by L egendre9 in d ep en d e n tly of L agrange, and a t a b o u t th e sam e tim e. B oth of th ese writers, how ever, knew of earlier work b y E uler, who h ad carried o u t a sim ilar d e­

v elo p m en t using instead of D ' (X) an a r b itra ry polynom ial of deg ree n — 1, which

0 - PLANE

d ~

\

m erely has th e effect of replacing th e Jm’s in th e rig h t-h an d m em ber of (28) by th e a m’s defined by (14). In o th e r w ords, th e m ethod of E u le r w as ex actly eq u iv ale n t to th a t of D uncan and Collar, except th a t in th e form er th e re was no obvious criterion for th e choice of a co n v en ien t form of n u m e ra to r, w hereas it is easy to choose m atrices K w hich will lead to a sim ple succession of o p eratio ns, as we h av e illu s tra te d in S ections 8 and 9.

P A R T II— C O N F O R M A L M A P P IN G

12. T h e m eth o d of R o u th . T h e second g rou p of m eth o d s to w hich I wish to refer are all founded upon a well-know n theorem of C auch y. If we rep resen t th e com plex v ariab le X by one plane, and th e com plex v aria b le D b y a n o th er, th en th e eq u atio n

D { \ ) = X" + M " - 1 - M H~ 2 + • • •

p

m a y be looked upon as a transform ation by m eans of which th e X-plane is m apped upon th e D-plane. T h e correspondence between X an d D, however, is n o t 1 11 b u t in general n: 1; and hence a simple closed curve C in th e X-plane (Fig. 1) passes into a much more complicated curve C' in the D-plane. In regard to the curve C' the t h e ­ orem in question says t h a t the n u m b er of times it loops a round th e origin is exactly equal to the n u m b e r of roots of D ( k ) = 0 which lie inside C.

8 Legendre's d evelop m en t w as in term s of the reciprocal powers o f the roots, instead of their direct pow ers. O therw ise th e tw o were identical.

1945] M E T H O D S F O R L O C A T IN G R O O T S O F P O L Y N O M IA L S 99

T h is rule a ppe ars first to h a v e been applied by R o u th to th e problem of d e te r m in ­ ing th e n u m b e r of roots with positive real p arts , a problem which interested him be­

cause of its relation to the s tab ility of linear dynam ical system s. F or this purpose he used as t h e c o n to u r in t h e X-piane th e im aginary axis closed by a semicircle of infinite radius, th u s enclosing th e entire right half of t h e plane. F o r this p a rtic u la r c o ntour he explained in g r e a t detail how from the sequence of intersections of C' with the real and im aginary axes t h e n u m b e r of roots could be found w ith o u t more definite inform ation as to the shape of C '. H e also developed a sequence of functions, similar to S tu rm functions, by m eans of which the n u m b er of roots could be determ ined from t h e polynomial directly w ith o u t even knowing th e real an d im aginary in te r­

cepts of C . H e did n o t extend either of these studies to th e point of locating the roots m ore exactly, b u t both are capable of such extension and h av e actually been used.

13. T h e m eth o d of G. R. Stibitz. T h e second m eth o d — the one using functions similar to t h e S tu rm functions— was developed fu rth e r b y G. R. S tibitz of t h e Bell T elephone Laboratories. H e observes, first, t h a t t h e m ethod can also be used to find th e n u m b e r of roots with real p a r ts g re a te r th a n Xq. T o do this, it is merely necessary to replace X b y X—Xo in th e polynomial (4), and th en proceed as outlined by R o u th . By carrying o u t this process for enough values of Xo, t h e roots can be segregated within strips parallel to t h e im aginary axis. T h e n by a definite routine (resembling in its essentials th e W eierstrass subdivision process in point-set theory) t h e real values of th e roots can be found to a n y desired degree of approxim ation. W he n this has been accomplished, t h e im aginary p a r ts are d eterm ined a t once as a ratio of two of the S turm -like functions.

S tibitz has developed com plete schedules for th e c o m p u ta tio n s required in solving polynomials b y this m ethod, for all values of n up to 10. T h e m ethod has been tried, an d works reasonably well, th ough p erh ap s no t as rapidly as those explained in Sections 8 and 9. I suspect t h a t th e decision in this case, however, m u st remain a conditional one; for th e c o m p u ta tio n a l ro u tin e of S tib itz ’ m ethod is com plicated (i.e., varied) as com pared with th e extrem ely simple (i.e., repetitive) routines of Sections 8 and 9. F o r this reason, it is n o t as well a d a p te d to use in an industrial co m puting laboratory. In th e h a n d s of a m a th e m a tic ia n who thoroughly understood its theoretical origin it m ight show up much better.

14. T h e m eth o d of A. J . K e m p n e r. K e m p n e r ’s m e th o d s 10 resemble more n early the o th er portion of R o u t h ’s work. H e chooses as his c o n to u r C a circle of radius r a b o u t th e origin as center. T h e n X = r e {\ an d (4) becomes

D (\) = [rn cos n6 + p\.rn~l cos (n — 1)0 — p ^ r cos (n — 2)0 + ■ • • ] Hr- ^{i [ r n} sin ^{n d} -j- p \rn~l sin (» — 1)0 — p%rn~2 sin ^{( n} — 2)0 + • • • ] .

T h u s th e real and im aginary p a r ts of D are trigonom etric sums, which, as K e m p n e r rem arks, could be calculated by m eans of a h arm onic synthesizer, such for example as t h e Michelson “a n a ly z e r.” T h u s two curves would be obtained, one giving th e real p a r t of D, an d th e o th e r its im aginary p art, b oth as functions of 6. From this p o in t on, K e m p n e r suggests two possible routines. F irst, to regard these curves as p a ra m e tric representations of D, and from them c o n stru c t th e curve C' itself. Second, to keep them as sep a ra te curves and n o t b o th e r fu rth e r a b o u t C'. In b o th cases, he

11 U n iversity of Colorado Stu dies 16, 75 (1928); B ulletin of the Am er. M ath. S ociety 41, 809 (1935).

100 T H O R N T O N C. F R Y [Vol. I l l , No. 2

develops rules very sim ilar to R o u th ’s for finding th e n u m b er of roots direc tly from th e sequence of in te rc e p ts w ith th e axes.

H e uses th is ro u tin e to segregate th e ro ots in a n n u la r rings, an d th en tra c k s down th e ir ab so lu te values by a su ita b ly chosen succession of in te rm e d ia te circles. T h e angle of an y ro o t is, of course, a u to m a tic a lly determ in ed as th e value of 6 a t which th e real an d im aginary p a rts of (30) vanish w hen r is given th e p a rtic u la r v alu e a p p ro p ria te to t h a t root.

K em p n e r also m entions th e possibility of ap plying th e m eth od to sectorial in ­ stead of a n n u la r regions, b u t does n o t develop th is idea to a significant degree.

15. T h e isograph. K e m p n e r’s m ethod was also developed in dependently, b u t som ew hat later (1934) a t Bell Telephone Laboratories, an d led to th e construction of a machine, called th e isograph, which draw s th e curve C corresponding to a circle of a n y radius r.

Since the indep en d e n t variable in plottin g th e curves is an angle, w h a t is required for th e isograph is a ro ta tin g u n it t h a t provides two linear m otions— one p roportional to t h e sine an d th e o th e r to th e cosine of the angle. T h e re would h av e to be ten of these un its to provide for th e ten variable term s of a te n th degree equation, and while th e first u n it moves th ro u g h an angle 0, th e second u n it m u s t move through an angle 26, t h e th ird u n it th ro u g h an angle 36, and so on. T h e n b y providing a m eans of sum m ing th e sine an d cosine m otions separately, an d allowing these sum s to control two perpendicular m otions of a pencil an d drawing board, a closed curve will be de­

scribed as 6 increased from 0 to 360 degrees.

T o secure m otions proportional to th e sine and cosine of the angle of rotation, th e isograph utilizes t h e “pin and s lo t” m echanism illustrated in Fig. 2. H e re an arm ro ta tin g a b o u t a fixed p oint carries a pin arranged to slide, by m eans of a re ctan g u lar block, in re c ta n g u la r slots c u t in two slide-bars, each of which is free to move bac k and forth in one direction only— the two m otions being a t right angles to each other.

T h es e m otions are equal to th e length R of th e arm tim es th e sine an d cosine of th e angle of rotation.

1945] M E T H O D S F O R L O C A T IN G R O O T S O F P O L Y N O M IA L S 101

T h e ten u n its provided are geared to a com m on d riv in g m otor, b u t th e gearing is designed so t h a t w hen th e arm of th e first u n it m oves th ro u g h an angle 6, t h a t of th e second u n it will m ove th ro u g h an angle 26, th a t of th e th ird th ro u g h 36, and so on.

T o p rov ide for sum m ing up all th e sine term s and all th e cosine term s, th e ends of all th e slide-bars ca rry pulleys so th a t a single w ire m ay be carried aro un d all th e sine pulleys an d a n o th e r around all th e cosine pulleys as in dicated in Fig. 3. S ta tio n -

a ry pulleys are m ounted betw een th e m ovable ones so as to keep th e d irection of pull on th e wires in line w ith th e m otion of th e slide-bars. T h ese w ires con tro l th e re la tiv e m otions of a pencil and draw ing bo ard to p lo t a curv e as th e an gle is varied from zero to th re e h u n d re d an d six ty degrees.

T h e co n stru c tio n of th e ro ta tin g elem ents is show n in Fig. 4. T h e d riv e sh aft passes th ro u g h th e bed p la te an d is fastened to th e c e n te r of a steel b ar th a t a c ts as th e arm of Fig. 2. T h is b a r is grooved to receive th e pin of th e “pin an d s lo t” m echa­

nism . In o rd er th a t th e pin m ay b e a d ju ste d for different cran k lengths, corresponding to th e coefficients p tfn -k of th e various term s in th e eq u a tio n , a ra ck is c u t along one edge of th e groove so th a t a pinion a tta c h e d to th e pin m ay m ove it along th e bar.

A fte r a d ju s tm e n t th e pin is secured in place by a set-screw .

T h e to p of th e b a r carries a carefully g ra d u a te d scale to w hich th e ce n te r of th e pin m u st be set ac cu ra te ly . T h e scale is m ade visible a t th e c e n te r of th e pin b y con-

102 T H O R N T O N C. F R Y [Vol. I l l , N o. 2

stru c tin g th e la tte r as a hollow cylinder. A v ern ie r scale w ith in th e cy lind er enables th e effective arm length to be a d ju ste d v ery exactly to th e desired v alu e on eith er side of th e c e n te r— one side for positive coefficients an d th e o th e r for n eg ativ e. T h e to ta l rang e of a d ju s tm e n t is th re e inches.

T h e hollow pin tu rn s in a re c ta n g u la r bronze block which fits th e slo ts of two slide bars, one for th e sine m otion and one for th e cosine m otion. T h e slide b ars are identical steel p lates ru n n in g in bronze ways set a c cu ra te ly a t rig h t angles to each

Fi g. 4 .

other. A t th e end opposite to th e slot each p late carries a pulley around which is passed t h e wire t h a t sum s up the sine or cosine m otions of th e ten elements. One end of each wire is fixed. T h e o th e r end of the cosine wire is led b y pulleys to the draw ing board, which consists of a thin alu m in u m sheet m o u n ted on ball-bearing rollers so t h a t it is free to m ove bac k and forth in only one direction. A counterw eight fastened to t h e o th e r edge of th e board keeps th e wire under c o n s ta n t tension. T h e free end of th e sine wire is led b y pulleys to a counterw eighted pencil carriage, which is m o u n ted with ball bearings in a fixed guide crossing t h e drawing board a t right angles to its direction of m otion. T h u s t h e board is displaced b a c k and forth in proportion to th e sum of the cosine term s, and t h e pencil is displaced b a c k an d forth

1945 j M E T H O D S F O R L O C A T IN G R O O T S O F P O L Y N O M IA L S 103

in a p erp en d icu lar d irection in p ro p o rtio n to th e sum of th e sine term s; and this com ­ bined m otion gives th e desired curve.

In o p eration, th e isograph has given accuracies of one p er c e n t or b e tte r: and of course gives them q u ite rapidly. In fact, th e m ost rapid m ethod we h av e a t p resen t is th a t of using th e isograph to o b tain th is degree of accu racy, and th en im proving it eith er by th e m eth o d s explained in §§8 and 9, o r by successive app rox im atio n to th e q u a d ra tic factors.

16. C onclusion, In conclusion I wish m erely to p o in t o u t th a t in none of th e m eth o d s which I h av e described is c o m p u ta tio n w ith com plex n u m b ers involved.

T h e y are all real m ethods. A t p re sen t th is seem s to be a fu n d a m e n ta l re q u irem en t im posed upon us by com m ercial co m p u tin g m achines, since th e m u ltip licatio n of two com plex nu m b ers on such m achines requires six, an d division eight, se p a ra te o p era­

tions. If th is restrictio n w ere rem oved, o th e r m eth o d s m ig h t con ceiv ably prov e to be m ore rapid.

P a rtly w ith this in m ind, an d p a rtly because w e m u st freq u en tly deal w ith com ­ plex q u a n titie s in o th e r connections, we are a t p re sen t developing a co m p u tin g m achine for com plex q u an titie s. W hen it is com pleted, as we h ope it will be in th e course of th e p re sen t y ear, we shall u n d e rta k e a fu rth e r stu d y of m eth o d s which now are clearly ruled o u t b y m echanical lim itations.*

P O S T S C R IP T B Y R. L. D IE T Z O L D

W hen th e foregoing p ap e r w as w ritte n , it w as in ten d ed for im m e d ia te p u b licatio n . B y coincidence, how ever, several o th e r p ap ers of sim ilar c h a ra c te r a p p e are d a t ju s t a b o u t th a t tim e, an d D r. F ry concluded th a t th e su b je c t w as of too lim ited in te re st to ju s tify pu blish ing a n o th er.

Since th en , th e situ a tio n h as changed in several w ays. F irs t, th e in te re st in m e th ­ ods of num erical c o m p u ta tio n h as g re a tly increased, largely because w ar ac tiv itie s h av e led to m uch w ork of t h a t kind. Second, th e specific problem of root-finding h as becom e a live one because its fu n d a m e n ta l im p o rta n c e in lin ear d y n am ics is m ore w idely recognized. F in ally , a few new m ethod s of ite ra tio n h a v e been evolved and som e new ty p es of co m p u tin g m achines developed . T h e p ap e r therefo re now has a tim eliness w hich i t lacked w hen w ritte n , b u t a few com m en ts a re req u ired to bring it up to d a te . T h e m o st im p o rta n t of th ese a re noted in th e following p arag rap h s.

In th e Bell T elep h o n e L a b o ra to rie s th e av ailab le co m p u tin g e q u ip m en t h as been m a te ria lly im proved th ro u g h th e d ev elo p m en t of th e relay c o m p u te r b y S tib itz and th is in ev itab ly re a c ts upon th e re la tiv e convenience of v ario u s m eth o d s of solution.

A lthough the relay co m p u ter is v ery flexible in resp ect to the ty p e of problem it can h andle, it is p a rtic u la rly well suited to ite ra tiv e processes such as B ernoulli's m ethod of ro o t e x tra c tio n ; for once the p ro p er in stru ctio n s have been s e t in to th e control ta p e which governs th e m achine, all successive o p eratio n s are perform ed w ith o u t fu rth e r supervision. T h e sim plicity of B ernoulli's rule, w hich requ ires on ly th a t th e m achine accu m u late n — 1 of th e a ’s, each m ultiplied b y th e a p p ro p ria te coefficient from th e polynom ial, recom m ends it for m echanization . T h e in stru ctio n s are easiiy

* T h is m achine was p laced in serv ice in 1940 and w as dem onstrated a t th e sum m er m ee tin g of th e A m erican M a th e m a tic a l S o c iety in H anover, N ew H am pshire in Septem ber o f th a t y e a r. T h e re la y com ­ puters referred to in Dr. D ie tz o ld ’s p o s ts c rip t are s till more v ersatile devices which h a v e b een d eveloped since t h a t tim e.

104 T H O R N T O N C. F R Y [Vol. I l l , N o. 2

set up, and the m achine is n o t required to recall very m a n y num bers a t a n y stage in the process. Bernoulli’s m e th o d is likewise well a d a p te d to com p u tin g eq u ip m e n t of the punched-card type, provided only t h a t the ac cu m u lato r is designed to recognize algebraic sign.

One of th e routines which m ay be set up in a relay c o m p u ter enables th e algebraic operations to be performed on complex nu m b ers with the ease t h a t th e sam e o p era­

tions are perform ed on real n u m b ers w ith a m echanical co m puting machine. T h e availability of this aid m akes N e w to n ’s m eth o d useful for root im p ro v e m e n t in th e complex dom ain, and some on th e L a b o ra to rie s ’ com puting staff prefer it to Bair- s to w ’s m ethod, a lthough t h e margin of choice is no t great.

B airsto w ’s v aria tio n of N e w to n ’s m ethod avoids c o m p u ta tio n w ith complex q u a n titie s by im proving t h e coefficients of a trial q u a d r a tic factor. T h e trial factor, say Q(X) = X 2+ o X - |-6, is divided twice into th e polynomial, and th e ra te s of change of t h e re m a in d er coefficients found from th e second rem ainder, as in H o r n e r ’s process.

T h e m eth o d has b y now been sufficiently publicized;" nevertheless, it can be given here, since it is sh o rt to state. T h e polynomial being expressed as

Z)(X) = (foX + So) + Q (X )(n X + S i) + Q 2( \ ) ( l B + /iX + • • • ) ,

im proved coefficients for Q are r0

So

r i a = a Sl

ar, — Si r i

'brj Si

V = b +

ari — Si r0

br i s0

ar, — si hi

br i Si

N e w t o n ’s m ethod typifies a class which is deliberately excepted from t r e a t m e n t in F r y ’s p ap e r; m ethods in this class are characterized by the p ro p e rty t h a t only som e­

tim es do th e y lead to a solution. N e w to n ’s m ethod, for example, can never lead to a complex root if t h e ite ra tiv e process is s ta rte d from a real trial value. B airsto w ’s m ethod has a similarly restricted region of convergence and was, quite properly, a d v a n ced b y him only as a m eans for im proving roots already located approxim ately.

M e th o d s which som etimes fail to converge m ay still be v ery useful if, in ap p li­

cation, th e y converge often enough an d fast enough. N e w t o n ’s m eth o d and its v a r ia ­ tions, however, alm ost always fail unless t h e y can be s t a r t e d from values closely corresponding to roots. B u t in 1941, S hih-N ge Lin revealed an alg o rith m 12 re m a rk ab le

11 Bairstow gave the m ethod only in R eports and M em oranda N o. 154, A d visory C om m ittee for Aeronautics, O ct., 1914 (H . M l Station er’s Office), but it was m ade generally availab le by Frazerand D u n ­ can, Proc. R oyal S oc. London 125, 6 8 -8 2 (1929). H itchcock offered the m ethod as A n improvement on the G.C.D . method fo r complex roots, Jour. M ath . P hys, 23, 6 9 -7 4 (1944). H itchcock proposes th a t the roots be improved b y th is m ethod a fter on ly approxim ate location by the G .C .D . m ethod, which he gave in Jour. M ath. P hys. 17, 5 5 -5 8 (1938). T h e G .C .D . m ethod is nearly identical with the m ethod of G. R.

S tib itz, described by F ry. B airstow ’s m ethod was also rediscovered b y Friedm an, whose work is noted in Bull. Am er. M ath. Soc. 49, 8 5 9 -8 6 0 (1943). B airstow ’s form ulae give the leading term s of series d evelop ­ m ents of the coefficients by Fry, w ho concluded, after an investigation of th e convergence, that the ex ­ pansion w as su itab le on ly for root-im provem ent.

u A method o f successive approxim ations of evaluating the real an d complex roots o f cubic a n d higher order equations, Jour. M ath. P h ys. 20, 153 (1941).

1945] M E T H O D S F O R L O C A T IN G R O O T S O F P O L Y N O M IA L S 105

b o th for sim p licity and convergence. By L in ’s m ethod, th e polynom ial is divided only once by a trial q u a d ra tic fa c to r; if13

D (X ) — p 0 -j- p i \ -f- p2^ + ■ ■ ■

— (reA + io) + Q(X)(qo + ¡?A + ‘ im proved coefficients for Q are

Qopi ~ qlpj P*

a — , — — .

?o

If it converges, the process d eterm in es the fa cto r corresponding to th e roo ts of least ab so lu te v alue; th u s a su itab le initial choice for Q is

+ (p i/p i)^ + [po/pi)-

In ap p licatio n , th e process does v ery o ften converge, alth o u g h som etim es slowly.

W hen th e convergence of L in ’s m ethod is slow, B airsto w ’s m ethod offers a v alu able su pplem ent. L in ’s m ethod is used until th e size of the rem ain d er ind icates th a t an ap p ro x im a tio n to a q u a d ra tic fa cto r has been o b ta in e d ; B airsto w ’s process, s ta rte d from a sufficiently close app ro x im atio n , will converge, an d when it converges, it converges rapidly.

T h e co m bination of these two m ethods prov ides useful, an d usually ad e q u ate, eq u ip m en t for th e w ork a-d ay solution of polynom ial equ atio n s. In re c a lc itra n t cases, m echanical aids are p a rtic u la rly helpful. B erno ulli’s m ethod is alw ays av ailable, b u t is q u ite likely to be slow in cases for which L in 's m ethod has alre a d y failed. T h is m akes little difference if the ite ra tiv e process is perform ed a u to m a tic a lly by a relay co m puter, b u t recom m ends devices to accelerate th e convergence if th e c o m p u tatio n m u st be perform ed w ith o u t aid. An efficient: device for accom plishing th is is given by A. C. A itk en in a very full discussion14 of num erical m eth o d s for ev a lu a tin g the la te n t ro o ts of m atrices.

Like m ost of those who use m atrix m ethods, A itken is concerned not solely w ith th e solution of polynom ial eq u a tio n s, b u t ra th e r w ith th e m ore general problem of d eterm in in g th e c h a ra c te ristic roots (and also th e c h a ra c te ristic vectors) of m atrices.

P relim in a ry red uction of the m atrix to th e rational canonical form involves so m any o p e ra tio n s,15 th a t one would com m only s ta r t th e general problem w ith a m atrix M having few v an ish in g elem ents. In this ev en t, we lose one of th e reasons for preferring B ernoulli’s m ethod (i.e., rep eated m u ltip licatio n by M ) to m atrix pow ering by the ro o t-sq u a rin g m ethod, for the la tte r m ethod arriv es a t high powers of M w ith fewer o p eratio n s, th u s providing a n o th e r m eans for h asten in g the convergence. T h e a d ­ v a n ta g e is, how ever, p a rtly illusory except for th e lim ited class of co m p u ters who are so unerrin g th a t th ey can afford to sacrifice th e self-correcting fe a tu re of th e form er procedure.

*5 A departure from Fry's n otation is co n v en ien t here.

14 Proc. R oyal S oc. Edinburgh 57, 172-181 (1937).

16 Harold W ayland, E xpansion of determ inantal equations into polyn om ial form , Q uarterly A ppl.

M ath . 2, 277-3 0 6 (1945).

106

T H E K A R M A N - T S I E N P R E S S U R E - V O L U M E R E L A T I O N I N

THE

T W O - D I M E N S I O N A L S U P E R S O N I C F L O W O F C O M P R E S S I B L E F L U I D S *

BY

N . C O B U R N U niversity of Texas

13 In tro d u c tio n . T . v. K a rm an and H. S. T sien1 h av e tre a te d th e tw o-dim ensional subsonic flow of a perfect, irro tatio n al, com pressible fluid b y replacing th e a d ia b a tic pressure-volum e curve by the ta n g e n t line draw n a t an a rb itra ry p o in t of th is curve.

F irst, we shall discuss the ap p lica b ility of th e K arm an -T sien idea in th e supersonic range. Secondly, we shall show th a t w hen the K arm an -T sien relatio n can be used (fairly uniform com pletely supersonic flow), th e ch a racteristics form a Tschebyscheff n e t (fish n e t) .2 H ow ever, we shall be concerned w ith those regions of th e physical plane w hich can be m apped into a T schebyscheff n et in a unique one-to-one m anner.

H ence, we shall n o t s tu d y the o n set of shock. F u rth e r, we shall show th a t if th e di­

agonal curves of th e n e t of characteristics are draw n so as to correspond to eq u i­

d is ta n t values of th e arc length p a ram eter along th e ch aracteristics, th en these diagonal curves will be th e fam ilies of eq u ip o ten tials an d strea m lines. A nalytically, th is la st re su lt m eans th a t th e d eterm in a tio n of th e stream lines depends upon two a rb itra ry functions of one real variable. I t is shown th a t th e angle betw een th e c h a r­

acteristics and the angle form ed by a ta n g e n t to a stream line an d the x-axis can be determ in ed in term s of these functions. F u rth e r, the m ag n itu d e of th e velocity and the den sity depend upon only the form er angle an d th e M ach n u m b er of th e flow.

In p artic u la r, if a know n stream line coincides w ith th e x-axis, it is shown th a t only one a rb itra ry function e n ters into th e problem of d eterm in in g th e stream lines. Even in this last case w here th e d a ta are of a sim ple D irich let ty p e (sym m etric flow a b o u t the x-axis and a know n extern al b o u n d ary stream line— as in th e je t problem ), th e direc t problem c a n n o t be solved easily. H ence, an an alytical-g eom etrical m ethod is outlined for solving th e problem ind irectly. A p a rtic u la r exam ple is studied. Finally, in an appendix, we furnish a n o th e r proof (analytical) of th e fa ct th a t w hen the K a rm an -T sien re la tio n is applicable, th e ch a racteristics form a T schebyscheff n e t and conversely.

2. E xten sio n of th e K a rm a n -T sie n m eth o d to superson ic flow. In this section, we shall show t h a t t h e K a rm an -T sien m ethod m ay be extended to the supersonic flow of a perfect, irrotational, compressible fluid. If we d enote the pressure by p , the density by p, the ratio of the specific h eats by 7 , the a diaba tic relation is

pp~y = constant. (2.1)

’ R eceived O ct. 16, 1944.

1 T . von K irm ä n , C om pressibility effects in aerodynam ics, Journal of Aeron. Scien ces 8 , 3 3 7 -3 5 6 (1941).

H . S. T sien, T w o-dim ensional subsonic flow s of compressible flu id s, Journal o f Aeron. Sciences 6, 399 -4 0 7 (1939).

1 L. Bianchi, L ezion i d i geometria differentiale, vol. 1, Enrico Spoerri, Pisa, 1922, pp. 153-162.