Science Abstracts. Section A, Physics Abstracts. Vol. 48, No. 569

FEB 4 194S

PHYSICS ABSTRACTS

S E C T I O N A

o f

SCIENCE ABSTRACTS

S E C T IO N A , P H Y S IC S

S E C T IO N B , E L E C T R I C A L E N G I N E E R I N G . . .

THE INSTITUTION OF ELECTRICAL ENGINEERS

T H E P H Y S I C A L S O C I E T Y T H E A M E R I C A N P H Y S I C A L S O C I E T Y

T H E A M E R I C A N

I N S T I T U T E O F E L E C T R I C A L E N G I N E E R S E dited and Issued M o n th ly by

In Association with

A B S T R A C T S 1172-1348

V O L U M E 48

MAY 1945

HiqA O'Aeq.u-encfy J,eAieA

9 G ives a ro u g h in d ic a tio n o f v a c u u m to 0 -0 0 0 1 m m . I ig .

9 Im m e d ia te d e te c tio n o f leak s in glass v a c u u m s y ste m s.

9 S u ita b le fo r c o n tin u o u s use in F a c to ry a n d L a b o ra to ry .

9 In e x p e n siv e so u rce o f h ig h freq u e n c y fo r school ph y sics.

9 O p e rate s fro m A.C. o r D .C . su p p lies.

■ A s lj f g r list T E S . I.

W . E dw ard s & €b;(lMdon):&^.; r - l

I ia n g le y B rid g e R o a d , 'X d w ef S yde.nharo. . L o n d o n , S.E.‘2(>.-' / •_ ■. ■ . . ; ; ■ Telephone: SYDcnham'7026 • ■ •

Telegrams: Edcoliivac, Phone, London Special Note—Ready Shortly 1

‘ HIGH VA CU U M 1 by J. YA RW O O D , M.Sc.

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C O N T E N T S A B S T R A C T S 1172-1348

061 Annual reports

37 Education

51 M athem atics 511 A rithmetic

512 Algebra

513 Geometry

517 Analysis

.5 Functions

.9 Dili, equations 518 Calculation 519.2 Probability

.4 G roups

52 Astronom y

521.0 Astrophysics 521.1/.9 Theoretical 523 Descriptive

.1 Cosmology

.7 Sun

.8 Stars

525 Earth

526 Geodesy

Page Page

129 53 Physics 135

129 530.12 Relativity 135

129 530.145 Q uantum theory 135

129 531 M echanics o f solids 136

129 .7 Measuremets 139

130 532 Mechanics o f fluids 140

131 .13 Viscosity 140

131 532.14 Density 141

131 .5 Hydrodynamics 141

133 .61 Surface tension 143

133 .62 Films, liquid 143

134 .69 Surface activity 143

134 .7 Theory o f liquids 144

134 .72 Diffusion 144

134 .73/.78 Solution 144

134 533 Mechanics o f gases 144

134 533.15 Diffusion 144

134 .6 Aerodynamics 145

134 .7 Kinetic theory 146

135 534 Acoustics 146

135

Abstracts signed " E . R. A .” are supplied by courtesy o f the British Electrical and Allied Industries R esearch.

Association. Abstracts signed “ M.-V. ’ are supplied by courtesy o f the M etropolitan-Vickers Electrical Co. Ltd.

061.1.055.5

512.831

061.1.055.5 1172

Twenty-seventh Annual R ep o rt o f the N ational R esearch Council, 1943-44. A .R . N a t. Res. Cowi.

Can., 31 p p ., 1943-1944.

37 : 511 see A bstr. 1174

51 = 3 1173

G eneralization in m athem atics. Ba b l e r, F.

Schw eiz. B auztg, 124, pp. 215-218, O ct. 21, 1944.—

A n essay o n th e ev o lu tio n a n d d evelopm ent o f m ath em atical co n cep ts, in cluding n u m b er theory (p rim e n u m b er, algebraic n u m b er, hypercom plex n u m b er, tran sfin ite n u m b er, n u m b er field, ideal, etc.).

O th e r subjects discussed a re geom etry, se t theory, to p o lo g y a n d a b strac t algebra. l. s. g.

5 1 1 : 3 7 1174

O n th e ^ te a ch in g o f m a th e m a tic s . Mu r n a g h a n, F . D . Science, 100, pp. 479-486, Dec. 1, 1944.— A n a d d ress in w hich th e a u th o r suggests a ch an g e in th e m eth o d s o f teach in g elem en tary arith m etic and algebra. M echanical form alism sh o u ld be rep laced by m o re em phasis o n th e m ean in g o f th e elem entary o p eratio n s, a n d o n a clear u n d e rstan d in g o f the different types o f n u m b ers, e.g. integers, fractions, o rd ered p airs o f n u m b ers. l. s . g.

511.21 1175

O n m consecutive integers. IV . Pillat, S. S. Bull.

C alcutta M a th . S o c., 36, p p. 99-101, S ep t., 1944.—

A p r o o f , e a s ie r th a n t h o s e g iv e n e a r lie r , is p r e s e n te d fo r t h e th e o r e m th a t w h e n rn > 17 w e c a n fin d m c o n s e c u t iv e in t e g e r s s u c h th a t n o n u m b e r in th e s e t is p r im e t o a ll t h e o th e r m e m b e r s o f th e s e t . l. s. g.

511.213 1176

B e rtra n d ’s postulate. Pil l a i, S. S. Bull. Calcutta M a th . So c., 36, pp. 97-99, S ep t., 1944.— T h e p o stu late states th a t if n > 2 th ere is a t lea st o n e prim e, p , such th a t n < p < 2/i. A p r o o f (sim pler th a n previous proofs) is given a n d this d o es n o t involve th e use o f S tirling’s a p p ro x . fo rm u la fo r « ! l. s. G.

511.32 1177

O n som e form ulae in analytical theory o f num bers. II.

Ba n e r je e, D . P. Bull. Calcutta M a th . S o c., 36, p p.

107-108, S e p t., 1944.— Som e fu rth e r fo rm u lae a re given in c o n tin u a tio n o f a previous p a p e r [A bstr. 8

(1945)]. l. s. g.

511.5 1178

A m e th o d in r a tio n a l d io p h a n tin e a n a ly s is . Be l l, E. T . Proc. N a t. A cad. S ci., W ash., 30, pp. 355-359, N o v., 1944.— By ra tio n al su b stitu tio n s, th e given d io p h an tin e system is red u ced to a m ultiplicative system , w hich is com pletely solvable. I f th e su b stitu ­ tio n s a re b iratio n al, th e com plete ra tio n al solu tio n o f th e orig in al system is o b tain ed , b u t w hen th e su b stitu ­ tions are n o t ratio n ally reversible, a partial so lu tio n o n ly is fo u n d . T h e m eth o d is general a n d elem entary a n d is a n altern ativ e to th e recent ap p licatio n s o f algebraic geom etry to th e a rith m etic o f ra tio n al varieties [A bstr. 1925, 1926, 1928 (1944)]. T w o exam ples a re given. I n th e first, (£ + rj -)- f )3 — d S zyC

= m is solved ra tio n ally fo r ?, J /,f w hen d a n d m are

v o l. x l v u i.— a . —1945. M a y . 129

given ratio n al n u m b ers. In th e second, (a + y ) ( y + z)(z + x ) — d x y z — m is solved for x , y a n d z.

l. s. G.

511.52 1179

O n trinom ial congruences and F e rm a t’s last theorem.

Va n d i v e r, H . S. Proc. N a t. A cad. S ci., W ash., 30, p p. 368-370, N ov., 1944.— U sin g th e results o f a prev io u s p a p e r [A bstr. 1180 (1945)] criteria are given fo r the so lu tio n o f th e eq u atio n aum - f b i m + 1

= 0 (m o d p ) a n d th e results o b tain e d a re ap p lied to th e e q u atio n x l + y 1 + z ! = 0 . l. s. g .

5 1 1 .5 2 :5 1 9 .4 8 1180

Som e theorem s in finite field theory with applications to F e rm a t’s la s t theorem . Va n d i v e r, H . S. Proc.

N at. A cad. S c i., W ash., 30, pp. 362-367, N o v., 1944.—

T w o th eo rem s a re o b tain e d c o n cern in g finite fields a n d fro m th ese explicit expressions follow fo r the n u m b er o f sets o f so lu tio n s o f c ertain e q u atio n s. A n a p p lic atio n is m ad e , in c o n tin u a tio n o f a previous p a p er [A bstr. 2213 (1942)], to c riteria fo r th e solu tio n o f x l + y l + z l = 0 (F e rm a t’s last theorem ). A n expression is o b tain e d fo r the n u m b er o f distinct integral so lu tio n s (um, vm) o f aum + bvm+ 1 = 0 (m o d p) w here a a n d b a re integers an d p is prim e. L. s . G.

511.92 1181

D istribution o f the types o f the product o f two dcm lo- num bers. Ch a n d r a t r a y a, M . L ., a n d Ka p r e k a r, D . R . J . Univ. B om bay, 13, pp. 4 -6 , N ov., 1944.—

A c o n tin u a tio n o f prev io u s w o rk [A bstr. 516 (1944)].

A tab le is given o f th e v ario u s types o f p ro d u cts.

512.25 1182

T he “ e scalato r” process for the solution o f L a - grangian frequency equations. Mo r r is, J ., a n d He a d, J. W . P hil. M a g ., 35, pp. 735-759, N o v., 1944.— T h e process is described in d etail, the analysis being k ep t o n sim ple a lgebraic lines. T h e m eth o d is based o n the successive in tro d u c tio n o r e lim ination o f eac h o f th e v ariab les involved, by definite self- co n ta in ed stages. It does n o t involve a n y iterativ e p rocess n o r th e ev alu atio n o f d e te rm in a n ts. A special featu re c onsists o f a check a t each stag e to en su re an d , if necessary, a d ju st th e accuracy. T h e m eth o d is p articu larly suitab le w hen th ere is a larg e n u m b e r o f eq u atio n s. T h e can o n ical fo rm o f th e eq u atio n s is discussed a n d th ere is a section d ealin g w ith la te n t-ro o t e q u atio n s arisin g fro m the so lu tio n o f lin ea r sim ul­

tan eo u s e q u atio n s by iterative processes. A n u m erical exam ple o f th e e sc alato r process is given. T h is relates to 6th -o rd e r sym m etrical c an o n ical eq u atio n s, l. s. g.

512.831 1183

A note on the H erm itian m a trix . Gh osh, N . N.

Bull. C alcutta M a th . Soc., 36, p p . 87-90, S e p t., 1944.— A n ew m eth o d is given fo r red u cin g th e general H e rm itian m atrix to th e sum o f sim p le H e rm itian m atrices. M is sim ple if it is o f th e fo rm H H * w here H is a co lu m n v e cto r (o f com plex elem ents) a n d 11*

is th e tran s p o se o f th e c o lu m n v ecto r h av in g as elem ents th e com plex c o n ju g a tes o f th e elem ents

o f H . l. s. g.

512.831 : 530.145.63 see A b str. 1241

512.831 : 621.3.012.8 : 531.25 see A bstr. 1253

512.86

513.813

512.86 1184

O n substitutional equations. Ru t h e r f o r d, D . E.

Proc. R o y. S o c. Edinb. A , 62, 2, pp. 117-126, 1944.—

T h e n ! p erm u tatio n s e, a 2, . . ., <x„i o f n letters fo rm a g ro u p a n d th e su b stitu tio n a l expressions,

L — !\e + h a 2 + . . . + /«! °n!

a re th e elem ents o f a n alg eb ra. I t is sh o w n th a t su b stitu tio n al eq u atio n s o f th e ty p e L X = 0 w here X is a n u n k n o w n expression a re intim ately related to th e th eo ry o f id em p o ten ts (L is id e m p o ten t if L2 = L).

A n eq u atio n L X = 0 is show n to h av e th e sam e so lu tio n as L M = 0 w here M = A L (A being suitab ly chosen) a n d w here th e m in im u m e q u a tio n o f M is xij)(x) = 0, being prim e to x-. T h e expression

>1>(M) is id em p o ten t a n d the general so lu tio n o f L X = 0 is X — <p(M)Y, w here Y is arb itrary . T h e n u m b er o f linearly in d ep en d en t so lu tio n s o f L X — 0 is k n ! w here k is th e coefficient o f e (th e u n it p e rm u ta ­ tion) in ^ (M ). C o rre sp o n d in g results a rc o b tain ed fo r th e eq u atio n L X = R a n d m eth o d s a re given fo r solving sets o f sim u ltan eo u s e q u atio n s o f ea c h type.

l. s . g. 512.864 : 530.145.63 see A bsir. 1242

512.88 : 512.972 1185

O n invariant theory under restricted groups. Lit t l e- w o o d, D". E . Philos. Trans. A , 239, p p . 387-417, N ov. 25, 1944.— In an earlie r p a p e r [A bstr. 773 (1944)] in v a ria n t th eo ry w as stu d ied u n d e r th e full lin ear g ro u p a n d th e m eth o d s a re h ere extended to restricted groups, especially th e o rth o g o n al g ro u p (O ) a n d th e sym plectic g ro u p (S). It is show n th a t a know ledge o f th e ch ara cte rs o f th e g ro u p is a n essential p relim in ary to th e stu d y o f th e g ro u p in v arian ts, a n d th e ch ara cte rs o f O a n d S (previously o b ta in e d by tran scen d en tal m eth o d s) a r e n o w o b ta in e d by entirely algebraic m eth o d s. A fu n d a m e n ta l th eo re m proved, c o n cern in g g ro u p s w ith a system o f fu n d a m en tal ten so rs, is th a t every co n co m ita n t is o b tain a b le by m ultip licatio n a n d c o n tra ctio n o f g ro u n d fo rm ten so rs, ten s o r v ariables, fu n d a m e n ta l ten so rs a n d the a lte rn a tin g ten so r. A n analysis is developed e n abling th e n u m b ers a n d types o f th e c o n co m ita n ts o f any given degree in a n y system o f g ro u n d form s to be p red icted . T h e d ete rm in a tio n o f a ctu al c o n co m ita n ts is also discussed. F o r O , th e th eo ry is ap p lied to th e q u a d ratic , th e tern a ry cu b ic a n d th e q u a te rn ary q u a d ra tic com plex, a n d fo r th e te rn a ry S, to the q u ad ratic, th e lin e a r com plex a n d th e q u a d ra tic com plex. A p p licatio n s a re also m a d e to in tran sitiv e a n d im prim itive gro u p s. l. s.-g.

512.94 1186

Q uaternion centenary' celebration. Proc. R . Irish A c a d , 50 A , 6, p p . 69-121, Feb., 1945.— In clu d ed are th e follow ing p ap ers: T h e sequence o f ideas in the discovery o f q u a te rn io n s, E. T . W h ittak e r; Q u a ter­

n io n s a n d m atrices, A . W . C o n w ay [A bstr. 1241 (1945)]; A m o d ern p resen tatio n o f q u atern io n s, F . D . M u m a g h an [A bstr. 1187 (1945)]; T h e icosian calculus, J. R iversdale C o lth u rst. T h is calculus refers to the system g enerated by 3 elem ents i, j , k such th a t i2 — j* — k 5 = 1, k i — ij a n d ij =4= ji. l. s. g.

512.94 1187

A m o d e m p r e se n ta tio n o f q u a te rn io n s. Mu r n a- g h a n, F . D . Proc. R . Irish A c a d , 50 A , 6, pp. 104-

112, Feb., 1945.— W h en S is a skew -sym m etric m atrix , M = es is o rth o g o n al. W h en th e o rd e r (/i) o f th e m atrices is 2 th is result becom es de M oivrc’s th eo rem ’ since

v - i

= ( ! D

i

K =

H e re i is “ realized” by m eans o f th e second-order m atrix.' T h e case n — 4 is th e n stu d ied a n d it is show n th at S can only satisfy a q u a d ratic equation S 2 -1- 02E 4 = 0 w hen S is o f th e form

/ 0 — c b a \ / 0 — c 6 a \

( ' c 0 — a b \ I c 0 a — b I I — b a 0 c ) o r I — b —a 0 — c l

\ — a — b — c 0 z \ — a b c 0 / E ach o f th ese m atrices leads to a “ realization” o f the q u a te rn io n u n its, th e first by m eans o f th e m atrices

/ 0 0 0 1\ / 0 0 1 0\

0 0 - 1 0 ) ( 0 0 0 1 )

o i o o / ’ J ~ I - 1 0 0 0

- 1 0 0 0 / \ o - 1 o o /

/ 0 - 1 0 0 \ 1 0 0 0 ) 0 0 0 1 )

^ 0 0 - 1 0 /

I t is readily verified th a t these satisfy I 2 = J 2 — K2

= - E 4 a n d J K = — K J = I, K I = - I K = J ,

I J = - J I = K. l. s. G.

512.972 : 512.88 see A bstr. 1185

5 1 2 .9 7 2 :5 3 1 .2 5 9 1188

D yadic analysis o f space rigid fram ework. C h e n , P. P . J. Franklin Inst., 238, pp. 325-334, N ov., 1944.—

It is sh o w n th a t an y space fram ew o rk consisting o f prism atic beam s a n d rigid jo in ts m ay be analysed exactly b y solving a se t o f sim u ltan eo u s linear vector e q u atio n s w ith d y ad ic coefficients. T h e fundam ental v e cto r eq u atio n fo r a beam is fo u n d , assum ing th a t th e ex tern al forces a n d couples a re ap p lied a t the ends o f th e beam . A stu d y is m ad e o f an externally hinged o r slo tted rigid jo in t. E xam ples a re given an d there is a discussion o f th e m eth o d o f iteratio n . l.s.g.

513.468 1189

O pen packing o f spheres. Me l m o r e, S. N ature, L o n d , 154, p . 708, D ec. 2, 1944.

5 1 3 .7 6 6 .5 :5 1 9 .3 1190

T ransversality in h igher space. Dutka, J. J. M ath.

P hys., 23, p p . 126-133, A ug., 1944.—A s h o rt historical review is given o f tran sv ersality relatio n s, w hich first aro se in th e calculus o f v aria tio n s, a n d la te r ap p eared in th e th eo ry o f c o n ta ct tran sfo rm atio n s. A tra n s­

versality in 7i-spacc is defined as a ( 1, 1) corre sp o n d en ce betw een lineal a n d hypersurface elem ents h av in g a co m m o n base p o in t. Such a corresp o n d en c e is stu d ied in d etail, a n d it is sh o w n th a t a necessary a n d sufficient co n d itio n fo r a given corresp o n d en c e to be a tra n s­

versality is th a t a c ertain ind u ced co rre latio n sh ould

be a p o larity . l.s. g .

513.813 1191

O n parallelism in Riem annian space. S en, R . N . Bull. C alcutta M a th . S o c ., 36, pp. 102-107, Sept.,

1944.— A new parallelism is set u p a n d som e o f its p ro p e rties a re d ed u ced . T h e c u rv atu re p ro p erties o f th e space a re stu d ied fro m th e p o in t o f view o f th is parallelism w hich is also u sed to stu d y som e o f th e 130

513.88

517.942.4

fundamental invariants of differential quadratic M cKe r r o w, N . W . P hil. M a g ., 35, p p . 812-818, Dec., 1944.— T h e fu nctions K -Jx), w here v = h, i{

a n d §, arc tab u la te d o v er th e ran g e o f values 0-1 (0-1) 5 -0 for x a n d th e values fo r x = 6 0, 8 -0 an d 10 0 a re also given. T ables a re listed o f K i/^ x ) , A'3/1(.v), a n d K i /^ x ) fo r .v = 0• 1 (0-1) 5-0 an d fo r x — 6 -0 , 8 -0 , a n d 10-0. T h e m eth o d o f calcu latio n is stated , a n d a th eo ry in w hich the fu n ctio n s a p p e a r is given. T h is relates to th e tem ­ p e ratu re field w ithin a cylindrical ro d subjected to a su d d en change in tem p eratu re. l. s. g.

fo rm s. l. s . G .

513.88 : 517.948.35 = 4 see A bstr. 1208-1211

517.38 1192

O n the integral theorem s o f G auss and Stokes.

We s t b e r g, R . K. fysio g r. Srillsk. Lund, F orh., 13, 15, 10 p p ., 1944.—T w o general th eo rem s are given by m eans o f w hich m an y tran sfo rm a tio n s o f integral expressions m et w ith in th eo retical physics m ay be sim plified. T h e 2- a n d 3-dim ensional cases a re co n ­ sid ered a n d exam ples a re tak en fro m th e th eo ries o f electricity a n d h y d ro d y n am ics. l. s. g.

5 1 7 .4 3 2 :5 3 0 .1 4 5 .6 1193

Eigcn-values and eigen-functions for the operator d 2/ d x 2 - \x\. Be l l, R , P. Phil. M a g ., 35, pp. 5 8 2 - 588, S ep t., 1944.—T h e cigen-values a n d eigen­

fu nctions o f th e eq u atio n , d.2<!>\dx2 — \x\<j> = — Aiji, a re fo u n d in a fo rm involving B essel fu n c tio n s o f o rd e r A tab le o f n um erical v alu es is also given fo r th e eigen-values a n d these a re co m p ared w ith the values fo r th e o p e ra to r d 2/d x 2 — ¡jc'7 | w here q is 2, 4, o r CO [for the case q = 4 se e B e ll [A bstr. 1534(1945)].

Som e possible physical a p p licatio n s a rc n o ted , l. s. g.

517.432.1 1194

T he extension o f the H eaviside expansion theorem to the equations o f engineering and physics in curvilinear orthogonal co-ordinates. Sm it h, J. J. J. Franklin In st., 238, p p . 245-272, O ct., 1944.— P ro b le m s such as th e p ro p a g atio n o f electrom agnetic o r clastic waves, th e c o n d u ctio n o f h e a t th ro u g h solids, th e p ro p a g atio n o f aco u stic w aves in a ir a n d th e d istrib u tio n o f flux in th e airg ap o f electrical m achinery m ay be solved by th e o p e ratio n a l m eth o d s (sim ilar to th a t o f H eaviside) developed in th is paper. O rth o g o n a l c u rv ilin ear c o -o rd in ates a re used . F o rm u lae a re derived fo r the field d u e to a p o in t source in a given space fo r the follow ing cases: (i) L ap lace e q u atio n w ith stead y state co n d itio n s, (ii) e q u atio n o f h e a t co n d u ctio n , (iii) M ax ­ well’s w ave e q u atio n , (iv) th e w ave e q u atio n fo r a d issipative m edium . I t is show n th at by restricting th e ty p e o f p ro b lem to a c ertain fo rm (G re e n ’s fu nction) basic so lu tio n s o f th e e q u atio n s m ay be expressed in se p a ra ted variables. l. s. g.

517.531 1195

A note on th e m axim um modulus of the derivative of an integral function. Sh a h, S. M . J . Univ. Bom bay, 13, pp. 1-3, N o v., 1944.— L et f { z ) be an integral fu n ctio n a n d w rite

log {rM '(r)IM (r)}

[z|=>r log r

Som e ineq u alities relatin g to $(/■) a re n o te d an d 3 new th eo rem s a re given (w ith proofs). T h e first o f these states th at, if

f(z)

limi»(r) = p

r—> 0 0 L . S . G .

5 1 7 .5 4 :6 2 1 .3 1 9 .7 :6 2 1 .3 .0 1 3 1196 T he field between equal sem i-infinite rectangular electrodes o r m agnetic pole-pieces. Da v y, N . Phil.

M a g ., 35, p p . 819-840, D ec., 1944.— [A bstr. 1063 B (1945)].

517.562 : 518.2 see A bstr. 1214

517.564.3 1197

T he tabulation o f som e Bessel functions

KJz)

K{{z)

517.912.2 1198

O n the method o f collocation. Sa ib e l, E . J.

Franklin Inst., 238, pp. 107-110, A ug., 1944.— T he p ro b lem tre a te d is th e a p p ro x . d e te rm in atio n o f the ch ara cteristic n u m b ers A,- associated w ith a differential eq u a tio n E c o n ta in in g a p a ra m e te r A. T h e m eth o d o f co llo catio n a n d G a lerk in ’s m eth o d a re explained an d G ra m m e l’s converse to G a lerk in ’s m eth o d is m entio n ed . In th is w e s ta rt w ith th e hom ogeneous integral eq u atio n I asso ciated w ith th e problem , in stead o f w ith E . A new m eth o d , w hich in m an y cases is a n im pro v em en t, is pro p o sed . I t consists o f a co llo catio n o f I instead o f E. A n exam ple, illu stra t­

ing th e m eth o d , is th a t o f finding a n ap p ro x . value fo r th e low est n a tu ra l frequency o f v ib ratio n o f a u n ifo rm c antilever beam . T h is is trea te d in detail a n d th e result is c o m p a red w ith th a t o b tain e d th ro u g h co llo catio n o f the differential e q u atio n . l. s. g.

517.942.4 = 4 1199

F uchs’s theorem and linear equations w ith periodic coefficients. Pa t r y, J. A rch. Sci. P hys. N a t., 24, J u ly -A u g . (S u ppl. N o. 2, C .R . Soc. Phys. H ist. N a t.

Genève, 59, pp. 118-122), 1942.— T h e eq u a tio n w ith perio d ic coefficients is tak en to be

n d tniL

2 f o n + f m e ~ i x + = 0

mm 0 dx

an d , by w riting z — eLx, th is m ay be w ritten in the form

II d mU

S ( Fm + E mz + = 0

m —0 d zm

F u c h s’s th eo rem m ay be applied to th is eq u atio n u n d e r certain c o n d itio n s w hich a re n o ted , an d the Solution is w ritten in the fo rm

« = S D k é ^ P N lx k= 0

w here th e Dk a rc given by a re cu rre n t system o f difference e q u atio n s. I t is n o ted th a t th e co n d itio n th a t th e series for u converge is th a t th e sin g u lar p o in ts o f th e differential e q u a tio n (ro o ts o f th e eq u atio n f n + e„z + SnZ2 = 0) sh o u ld b o th lie ou tsid e o r inside the u n it circle (in the z-plane). l. s. g.

517.942.4 = 4 1200

A num erical m ethod fo r solving linear equations with periodic coefficients. Pa t r y, J . A rch. S ci. P hys.

N a t., 24, J u ly -A u g . {Suppl. N o . 2, C .R . S o c. Phys.

H ist. N a t. Genève, 59, p p. 122-126), 1942.— A m eth o d is p resen ted fo r use w h en th e e q u a tio n c a n n o t be solved by a n a p p lic atio n o f F u c h s’s th eo rem [A bstr.

1199 (1945)]. T h e so lu tio n is d eveloped in to a F o u rie r series th e coefficients o f w hich a re fo u n d fro m a n infinite se t o f linear e q u atio n s. T h ese c a n n o t be solved directly, b u t a m eth o d is given fo r deriving the 131

517.942.4

517.948.35

so lu tio n a p p ro x im ately (using th e “ e ch o ” principle)..

C o n d itio n s a re fo u n d in o rd e r th a t th e series so lu tio n

converge. l. s. g.

517.942.4 = 4 1201

O n the solution o f linear differential equations with periodic coefficients. Patry, J. Arch. Sci. Phys.

N a t., 24, N ov.-D ec. (S u ppl. N o. 3, C .R . Soc. Phys.

H ist. N a t. Genève, 59, pp. 225-229), 1942.— A c o n ­ tin u a tio n o f a previous p a p e r [A bstr. 1200 (1945)].

A fu rth e r ex am in atio n is m ad e o f th e infinite set o f ' e q u atio n s giving the coefficients in th e F o u rie r dev elo p ­

m en t, a n d th e c o n d itio n s fo r convergence a re m ad e

m o re precise. l. s. g.

517.942.9 1202

The num erical solution o f L ap lace ’s and P oisson’s equations. Mo s k o v i t z, D . Q uart. A ppl. M a th ., 2, p p . 148-163, Ju ly, 1944.— T h e e q u atio n s a re w ritten in finite-difference fo rm a cco rd in g to th e L ieb m an n p ro c ed u re [A bstr. 2703 (1938)] a n d th e system o f difference eq u atio n s th u s o b tain e d is solved sym ­ bolically by usin g th e inverse o f a c ertain o p e rato r.

P ro ceed in g fro m its definition, th e p ro p e rties o f this o p e ra to r a re established, a n d th e so lu tio n o f the orig in al eq u atio n is given as a series involving th e b o u n d a ry values o f th e function in q uestion. N u m erical tab les o f th e coefficients in th e series a re given to facilitate ap p licatio n o f the m eth o d to p articu la r cases.

T h ere is a discussion o f th e p ro c ed u re to b e follow ed w h en th e b o u n d a ry is n o n -rec tan g u lar. l.s.g.

517.946.9 : 539.214 1203

A boundary value problem in plane plasticity for the Coulom b yield condition. Coburn, N . J . M a th . P hys., 23, pp. 117-125, A u g ., 1944.— T h e m eth o d o f a p receding p a p er [A bstr. 63 (1945)] is m odified for use in th e p resen t paper, w here th e n o rm al stresses a x , ay a n d th e s h e a r stress axy a re su p p o sed know n.

T h e p ro b lem solved is th a t o f d eterm ining th e stresses in th e in te rio r o f the half-space a: > 0. T h e stresses ax , ay , axy a n d th e fu nctions sin 2y , cos 2y a re expanded in to p ow er series o f th e friction coefficient a n d these, u p o n su b stitu tio n in to L evy’s eq u atio n s, yield a n infinite set o f e q u atio n s fo r th e v ario u s a p p ro x im atio n s to th e stress co m p o n en ts. I t is sh o w n th a t th e latte r m ay be determ in ed by a step-by-step m eth o d o f

c o m p u ta tio n . l. s. g.

5 1 7 .9 4 7 .4 2 :5 3 4 .1 2 1 .2 1204

R elaxation m ethods applied to engineering problems.

X I. Problem s governed by the ‘quasi-plane-potential equation.’ Al l e n, D . N . d e G ., So u t h w e l l,R . V., a n d Vaisey, G . Proc. R o y. S o c. A , 183, pp. 258-283, Feb., 1945.— T h e q u asi-p la n e-p o ten tial eq u atio n

b / b xF \ b

+m x v ) + z = = 0 is discussed in relatio n to p ro b lem s w here Z a n d X a re k n o w n func­

tio n s o f .v a n d y . I t governs, inter alia, th e sm all transverse displacem ent o f a m em b ran e in w hich the ten sio n T a y, a n d its finite-differcnce a p p ro x im atio n governs th e sm all transverse d isplacem ents o f n o d a l p o in ts o f a net in w hich, sim ilarly, th e strin g tension T varies fro m n o d e to n ode. (E q u ilib riu m in th e d irec­

tio n s o f -v an d y can be m ain tain ed , b o th in the m em b ran e a n d in th e net, by forces actin g in th o se d irections a n d accordingly h aving n o effect o n the tran sv erse equilibrium .) T h e rela x atio n al treatm en t, b ased o n th is m echanical (net) analogue, reduces,

w hen X is c o n st., to th e tre a tm e n t developed for p ro b lem s governed by th e p lan e -h arm o n ic (Poisson) eq u atio n [A bstr. 323 (1944)].

517.947.44 : 518.5 1205

N um erical solution o f initial-value problem s by m eans o f punchcd-card m achines. Ko r m e s, J. P ., a n d Ko r m e s, M . R ev. Sci. lu strum ., 16, p p . 7-9, Jan., 1945.—T h e m e th o d is illu stra ted o n the eq u atio n : b h t/b t2 — cHR-ufbx1 = 0 w ith given initial conditions u(Q, x ) a n d u,{0 , x ) as well as b o u n d a ry conditions u{t, xq) = f i i t ) a n d u{t, x„) = f i l t ) . T h e plane co n ­ tin u u m is rep laced by a re ctan g u lar net a n d the differential eq u atio n by a finite-difference e quation.

A value o f the fu n ctio n u(t, x ) in a given p o in t o f the n et is expressed by its values o n tw o low er row s o f th e n et [see A b str. 2662 (1943)].

517.948 - 1206

O n the integral equations o f continuous dynamical system s. Ingram, W . H . Proc. N a t. Acad. Sci., W ash., 30, p p. 370-376, N ov., 1944.—T h e system s a re th o se con sid ered previously [A bstr. 2330 (1940)].

T h e kernels a re rep resen ted in a c ertain form w hich is k n o w n to b e valid in th e case o f the v ibrating string, th e electrical tran sm issio n line, etc. T h e p a p er is d ev o te d to a d e te rm in a tio n o f th e m odal vectors and n u m b ers o f su c h kernels. l. s. g.

517.948.3 : 532.532 see A bstr. 1293

517.948.3 = 5 1207

O n a functional equation. Ascoli, G . Portugaliae M athem atica, 4, 4, p p. 145-157, 1944.— It is observed th a t the logarithm ic derivative, xF (z ), o f the gam m a function, F (z ), satisfies th e functional e q u atio n

= /¡{Y (z) - log »}

T his result is generalized. T h e function f i x , y)

= vF ( x + iy) — lo g y satisfies (A ) /

+ /

x + n - \

>' = «/(-v, ny) an d a stu d y is m ad e o f som e p ro p e rties o f the general so lu tio n o f (A). It is sh o w n th a t a necessary and sufficient c o n d itio n fo r a fu n ctio n f i x , y ) to satisfy (A) is th a t it p e rm it a F o u rie r series dev elo p m en t o f the fo rm

f i x , y ) ~ c0 - f 2 / «(/¡y)e2~‘"x

flea —00

w here th e d a s h (') d en o tes o m ission o f th e term , l

n — 0. H e re a (/) = f l i , t ) e ~ 2ni^d£ fo r t g reater UQ

th a n a certain value. A n asy m p to tic e x p an sio n o f f i x , y) is given in term s o f th e fu n ctio n s f f x , y)

= B s{x)ly* w here th e B f x ) a re th e B e m o u llian polynom ials. S om e exam ples a re given. l. s. g. ‘

5 1 7 .9 4 8 .3 5 :5 1 3 .8 8 = 4 1208

T he spectra] decom position o f H e rm itian operators.

W a v r e, R . A rch. Sci. P hys. N a t., 24, Ju ly -A u g . {Suppl. N o. 2, C .R . Soc. Phys. H ist. N a t. G enève, 59, pp. 112-115), 1942.— A process is given fo r d e co m ­ po sin g a n ‘ elem ent / o f a space w hich is e ith e r a 132

517.948.35

519.281.2

fu n ctio n al space o r is iso m o rp h ic to a H ilb ert space.

T h is proceeds acco rd in g to th e p ro p e r elem ents f a o b tain e d w ith th e a id o f th e o p e ra to r A 2 w here A is a H e rm itian o p e ra to r [A bstr. 783 (1943)]. A sequence o f elem ents xg, x h X2, . . . is o b tain e d fro m a n o r­

m alized elem ent JcoCI lA‘o [ I = 1) by ite ratio n a cco rd ­ ing to /¡Xj = A(x,-—jX i = 1, 2, .;. .) a n d we have W rite £0 g= j ■ j . ’ '.

k < k < • • ., / = lim /,.

i ->=o

I f <5 A O , th e o p e ra to r A is said to b e re g u la r a n d in th is case th e elem ents X2r converge stro n g ly tow ard s a lim iting elem ent x * satisfying th e e q u atio n l2x * = A 2x . I f cû = ‘0, th e X2r converge w eakly to zero. Several th eo rem s a re sta te d a n d th ese a re used to d ed u ce th e m ain resu lt th a t, w hen A is regular, th e elem en t / m ay b e d eco m p o se d in the fo rm / = S c o t / o ') / “ + h w here A (h) = 0 a n d / a is the

a

lim it o f th e elem ents o b tain e d by a n iteratio n involving th e o p e ra to r A 2 actin g u p o n /q. H e re we have f o - = " ( / o ' ) / “ + . t f + 1 - L.'S. ° .

5 1 7 .9 4 8 .3 5 :5 1 3 .8 8 = 4 1209

O n linear equations involving a H erm itian operator.

Wa v r e, R . Arch. Sci. Phys. N a t., 24, Ju ly-A u g . (Suppl. N o. 2, C .R . Soc. Phys. H ist. N a t. Genève, 59, p p. 157-159), 1942.— U sin g th e results o f a previous p a p e r [A bstr. 1208 (1945)] an e q u atio n o f th e type

<j> = f + ~A(<f>) is con sid ered w here / is a given elem ent o f E, 4> an u n k n o w n elem ent, a n d v is a c o n stan t. A n y so lu tio n o f this e q u atio n is also a solu tio n o f <j> = f * + ~2/l 2(4>), w h e re / * = / + ~ A ( f ) a n d conversely. By deco m p o sin g / * in to / * = 2 f a / “ + h w here A (h) = 0 th e so lu tio n o f the

a

o rig in al e q u atio n is o b tain e d in th e fo rm <f> = f * I I f *

+ E ~ ;---7> fa- S om e rem ark s a re m ad e co n cern in g

a V /a

possible relatio n sh ip s betw een v a n d th e n u m b ers Lx a n d th e consequences o f these. I t is show n how the th eo ry o f in teg ral eq u atio n s o f th e F re d h o lm type w ith a sy m m etric k e rn el m ay be ra p id ly o b tain ed (e.g. th e H ilb e rt-S c h m id t th eo rem s) by th e m eth o d s o f ite ratio n involving H e rm itia n o p e rato rs, l. s. g.

517.948.35 : 513.88 = 4 1210

A note on the iteratio n o f H erm itian operators.

Wa v r e, R . A rch. Sci. P hys. N a t., 24, N ov.-D ee.

(Suppl. N o . 3, C .R . S o c. Phys. H ist. N à t. Genève, 59, p p . 229-233), 1942.— A su p p lem en t to previous p a p ers [A bstr. 1208-1209 (1945)]. S om e fu rth e r re m a rk s a re m ad e c o n cern in g the n u m b ers /j, . . . , lr, . . . o b tain ed fro m th e fo rm u la

A rxo = /i . . . lrxr (r = 0 , 1 , 2 , . . . ) a n d a stu d y is m ad e o f c ertain infinite p ro d u c ts th at arise in th e sp ectral deco m p o sitio n o f a n elem en t o f a fu n ctio n al o r H ilb ert space. l. s. g.

5 1 7 .9 4 8 .3 5 :5 1 3 .8 8 = 4 1211

Som e results com plem entary to the theory o f ite ra ­ tion o f operators by M . W avre. Vig i e r, J. P. Arch.

S ci. Phys. N a t., 24, J u ly -A u g . (Suppl. N o . 2, C .R . Soc.

P hys. H ist. N a t. Genève, 59, pp. 159-162), 1942.— A n extension o f th e w o rk o f W avre [A bstr. 1208 Ü945)]

to skew H e rm itian o p e ra to rs, i.e. o p e ra to rs defined by (A x , x ) = — (X, A x ). T h e th eo ry o f L alesco relating to skew sym m etric kernels is show n to follow rap id ly fro m th e p o in t o f view tak e n . In pa rticu lar, a series so lu tio n o f th e e q u a tio n <}> = -A<f> + / is given an d th is ex ten d s th e th eo rem o f H ilb e rt-S c h m id t to skew

H e rm itian o p e rato rs. l. s. g.

517.949 : 531.234 see A bstr. 1250

517.949 : 624.18 = 3 1212

Application o f difference equations in calculating ferro-concrete protection pillars. Ko l l b r u n n e r, C . F ., a n d Du b a s, C. Schw eiz. B autzg, 124, pp. 191—

194, Oct. 7, 1944.— In calcu latin g protective pillars o f g o a lp o st fo rm , f [, to be placed n e a r m achinery' in a pow er sta tio n , th e w hole p illa r is trea ted as a lam in a a n d th e p ro b lem is solved m athem atically,

using A iry ’s function. o . e. a.

518.2 1213

P rep aratio n o f punched-card tables o f logarithm s.

Th o m a s, G . B., a n d Ki n g, G . W . Rev. S ci. Instrum ., 15, p . 350, D ec., 1944.

5 1 8 .2 :5 1 7 .5 6 2 1214

Punched-card tab les o f the exponential function.

K In g , G . W . Rev. Sci. Instrum ., 15, pp. 349-350, D ec., 1944.

518.3 1215

N om ograph fo r equations o f the form y x n = z.

Ba r n e s, J. C. E ngng J ., M ontreal, 27, pp. 543-545, O ct., 1944.

518.4 : 536.2 see A bstr. 1466 518.5 : 517.947.44 see A bstr. 1205 519.1 : 536.77 see A bstr. 1502

519.217 : 532.72 1216

T he diffusion problem and the theory o f connected events. Da r l i n g, D . A . P roc. A m er. Phys. S o c., Pasadena, Cal., D ec. 16, 1944. A bstr. in Phys. R ev., 61, p . 65, Jan. 1 a n d 15, 1945.— A n extension o f th e 1-dim ensional ra n d o m w alk to th e case w here th e p a rticle s possess in ertia leads to a statistical problem in th e th eo ry o f non -in d ep en d en t, o r M ark o ff, p ro ­ cesses. T h e m odification o f th e hom ogeneous ra n d o m process in this case is deduced, a n d th e m eaning to be a tta c h e d to ra n d o m events extended to th e c o n ­ tin u u m is studied. T h e analogue to th e o rd in ary diffusion e q u atio n becom es a n in tegro-differential e q u atio n m t . x ) , b2p (t, x)

b t b x 2 <!>(/, y )y 2dy, w here p (t, A) is th e p ro b ab ility th a t th e p article (possessing inertia) has a displacem ent a- a t a tim e t, a n d 0 ( t , y ) is a so lu tio n to th e o rd in ary diffusion e q u atio n . Solutions o f th e eq u a tio n are d ed u ced by considering th e m o m en ts o f th e p ro b a b ility d istrib u tio n . Som e o f th e eq u atio n s used in tu rb u len ce a n d B row nian- m ovem ent p roblem s a re special cases o f th e integro- differential e q u atio n , w ith a suitab le in te rp re ta tio n o f the dependence fu n c tio n <I>.

519.281.2 1217

A m ethod fo r the solution o f certain non-linear problem s in least squares. Le v e n b e r g, K . Quart.

A ppl. M a th ., 2, p p . 164-168, July, 1944.— A m eth o d is developed w hich is o f use in cases w here th e s ta n d a rd 133

519.283

523.841.37

m eth o d o f least sq u ares fails. L et h (x) be a p p ro x i­

m ate d by H (x ; a ) w here x — (.v, y , z , . . . ) a re the variab les a n d (a) = (a, (1, y , . . .) a re th e u n k n o w n p aram eters. T h e residuals at th e po in ts (x■,)(/ = 1, 2, . . ., n) a re /¡(a ) = H (xp, a ) — h (xf) a n d th e least-

n

sq u ares criterio n requires th a t s{a) = X f c be a 1 — 1 m in im u m . I f the initial so lu tio n is (ag) = (a<), f t , yo,• • •), th e ls t-o rd e r T a y lo r expansions o f th e residuals tak e n a b o u t (a0) yield

() ft t) f;

fX<x) X FAX) = fA « o ) + P + . . . (1) w h ere A a = a — a 0, A P — P — Po, . . . • T h e sta n ­ d a rd m eth o d consists o f m inim izing

S (x ) = S F?

i = 1

b u t th is m ay lead to values o f A a , A f t A y , ■ ■ ■ w hich a re so large th at th e a p p ro x . ( 1) becom es invalid.

In this case the values o f A a , A/?, A y , . . . a re lim ited o r “ d a m p e d ” by m ak in g a m inim um the expression

5 (a ) = « 5 (a ) + a (A a )2 + ¿»(Aft2 + . . . w here a, b, . . . a n d w are w eighting factors. T his leads to “ d a m p e d n o rm al e q u a tio n s” w hich only differ fro m the o rd in ary type in th a t th e coefficients o f th e p rin cip al d iagonal a re altered . T h e 2 system s a p p ro a c h coincidence as w - > oo. T hese n o rm al eq u atio n s are discussed an d several m eth o d s a re given fo r d eterm in in g th e w eight facto rs tv, a, b, c, . . . . A sim ple g eom etric in te rp reta tio n is given o f the d a m p in g im p o sed o n th e p a ra m e te r v ariab les in the

above process. l. s . g .

519.283 1218

A second approxim ation to S o p er’s epidemic curve.

Wil s o n, E . B ., a n d Wo r c e s t e r, J. Proc. N a t. Acad.

Sci., W ash., 30, pp. 37-44, Feb., 1944.— A differential eq u atio n is con sid ered w hich expresses S o p e r’s fo rm u la tio n o f th e co u rse o f a n epidem ic. V arious a p p ro x . m eth o d s a re con sid ered for solving the eq u atio n a n d a n exam ple is given relatin g to an

epidem ic o f m easles. l. s. g.

519.283 1219

T he epidemic curve with no accession o f sus- ccptibles. Wil s o n, E . B „ a n d Wo r c e s t e r, J. Proc.

N a t. A cad. S ci., 1 Vash., 30, p p . 264-269, S e p t., 1944.—

A c o n tin u a tio n o f a prev io u s p a p e r [A bstr. 1218 (1945)], m o re a ccu rate calcu latio n s being n o w given to g eth er w ith som e n u m erical resu lts. l. s. g. 519.3 : 513.766.5 see A bstr. 1190

519.444 1220

Groups involving a sm all num ber o f sets o f conjugate operators. Mil l e r, G . A . Proc. N a t. A cad. S ci., W ash., 30, pp. 359-362, N ov., 1944.—T h e g roups a re classified in th e cases w here the n u m b er o f sets o f conjugate o p e rato rs is 2, 3, 4 a n d 5 respectively, e.g. if a g ro u p involves exactly 4 sets o f co n ju g ate o p e rato rs, it is o n e o f 2 g ro u p s o f o rd e r 4 w hen it is abelian, a n d w h en it is n o n -ab elia n it is e ith e r the d ih ed ral g ro u p o f o rd e r 10 o r th e tetrah ed ral group.

l. s . G.

519.48 : 511.52 see A bstr. 1180

5 1 9 . 5 = 4 * 1221

C haracterization o f the operation o f closure by m eans o f a single axiom . Monteiro, A. Portugaliae M alhem atica, 4, 4, pp. 158-160, 1944.— T h e closure o f a subset, X C 1, is a subset X * C l , such th a t (i) X C X * , (ii) X c Y im plies X * C Y*, (iii) X * * = X * . I t is proved th a t these 3 con d itio n s a re equivalent to th e single c o n d itio n Y + Y *

+ 3T** C ( X + Y)*. L .s.g.

521.031 1222

E nergy liberation in red giant sta rs. Va n Al b a d a, G . B. Physica, 's Grav., 10, p p. 604-612, July, 1943.—

T h e th eo ry o f energy lib eratio n (G am o w an d T eller) [Phys. R ev., 55, p. 791, 1939)] b ased on suggestions by B eth e [A bstr. 1550 (1939)] is criticized a n d it is show n th a t th e ra te o f ev o lu tio n p redicted by the th eo ry is n o t in agreem ent w ith secu lar v ariatio n s in th e p erio d s o f the C epheids. T h e conclusions o f G reenfield [A bstr. 2267 (1941)] a re also show n to be erro n eo u s.

It is suggested th a t, fo r th e ex p la n atio n o f th e g reat energy o u tp u t o f the re d giant stars, it is p ro b a b ly necessary to a d o p t th e h y pothesis o f superdense cores a n d highly inflated envelopes (hom ology tran sfo rm a ­ tio n s being invalid in this case). A m eth o d fo r fu rth e r

research is suggested. l. s. g.

521.14 1223

A note on the minimum radius for degenerate stellar masses. Ko t h a r i, D . S., a n d Au l u c k, F . C . Phil.

M a g ., 35, pp. 783-786, N o v., 1944.— P revious w o rk is discussed [A bstr. 3088 (1930)] a n d it is show n th a t the theory o f degenerate stella r m asses com posed o f relativistic degenerate gas predicts a non -zero ra d iu s fo r an y m ass. T h e ra d iu s a tta in s a m in. value fo r a m ass w hich is a b o u t 10 tim es larger th an the “ critical”

m ass in tro d u ced previously by C h an d rasek h ar.

523.11 : 536.422.15 see A b str. 1475

523.746 1224

Provisional sunspot-num bers for Ju n e to J u ly , 1944.

Br u n n e r, W . Terr. M agn. A tm os. E lect., 49, p . 238, D ec., 1944.

523.746 1225

Provisional sunspot-num bers for August to N ovem ber, 1944. Br u n n e r, W . Terr. M agn. A tm os. E lect., 50, p . 56, M arch, 1945.

523.746 1226

A tab le o f secular variations o f the so lar cycle.

Gleissberg, W . Terr. M agn. A tm o s. E lect., 49, pp. 243-244, D ec., 1944.

523.746 : 621.396.11 1227

Sunspot minimum. Be n n in g t o n, T. W . W ireless World, 51, p p . 81-82, M arch, 1945.— [A bstr. 1279 B (1945)].

523.841.37 : 523.872 1228

T he spectra o f the Cepheid variables. St r u v e, O . O bservatory, 65, p p . 257-273, D ec., 1944.— R e ce n t ad v an ces in stellar sp ectro sco p y a rc applied to th e in te rp reta tio n o f C ep h eid sp ectra. C onflicting evi­

dence as to th e extent o f th e a tm o sp h eres o f su p er­

gian ts th ro w s d o u b t on th e validity o f c o n v en tio n al th eo ries o f ab so rp tio n -lin e fo rm atio n in th ese stars, a n d raises th e q u estio n w h e th er th e v a ria tio n ' in velocity a n d in light a re a t all closely related to any p h e n o m e n a in th e d eep e r in terio rs o f C epheids.

O b servations at M c D o n a ld O b serv ato ry o f 7 C epheids 134

523.872

530.145.6

a re used to d eterm in e w h eth er C epheid sp ectra a t an y stage exactly resem ble su p c rg ian t sp ectra. C lose resem blance is found a t m in. light, b u t anom alies o c cu r to w ard s m ax. w hich c a n n o t be m atc h ed in n o rm al stellar sp ectra a n d suggest th e fo rm a tio n o f a n ex ten d ed a tm o sp h ere. T h is m odel req u ires som e m o dification o f the accepted view o f th e observed ra d ia l velocities, o n w hich is b ased th e p u lsation th eo ry . T h e differential m o tio n s observ ed in certain sp ectral lines are p ro b a b ly re la te d to th e m echanism o f shell fo rm atio n a t m ax. light. It is suggested th a t a close stu d y o f th e sp ectra o f clu ster variables sh o u ld •

n o w be u n d ertak en . a.h u.

523.872 : 523.841.37 see A bstr. 1228 525.24 : 537.591.5 : 538.691 see A bstr. 1531 526.918.523 : 771.35 : 535.317.1 see A bstr. 1370

530.12 1229

T he two-body problem in BirkhofT’s and E in stein ’s theories. Be r e n d a, C . W . Phys. R ev., 67, p . 56, Jan., 1945.— E instein a n d o th ers have given a so lu tio n o f the tw o-body p roblem , w hich gives results agreeing closely w ith those o f B irk h o ff’s th eo ry [A bstr. 488 (1945)]. T h e m axim um difference betw een th e tw o th eo ries is w ithin th e experim ental e rro r, a n d m ay arise from differences in th e a p p ro x im atio n m eth o d s

em ployed. a. j. m.

5 3 0 .1 2 :5 3 0 .1 4 5 = 4 1230

A new m ethod fo r the quantization o f fields. I.

St u e c k e l b e r g, E . C. G . A rch. Sci. Phys. N a t., 24, p p. 193-222, S e p t.-O c t., 1942.— P rev io u s w o rk o n the th eo ry o f fields is discussed [A bstr. 3489 (1936), 983 (1939)] a n d , in p articu lar, th e re ce n t w o rk o f Pauli [A bstr. 35 (1941), 22(1942)] w ho sh o w ed th a t p articles o f integral o r half-integral spin m u st o b ey th e F e rm i- D ira c a n d B o se -E in ste in statistics respectively.

P au li’s stu d y how ever is incom plete a n d , in th e p resen t p a p er, th e fo u n d a tio n s a re laid fo r a new p re sen ta tio n o f th e classical a n d q u a n tu m th eo ry o f b o th c h arg e d a n d un ch arg ed fields. C o n sid e rab le use is m ad e o f th e ten s o r a n d sp in o r calculus. T he p ro p e rties o f th e tw o fu n d a m e n ta l so lu tio n s o f th e w ave eq u atio n [A bstr. 1352 (1941), 1475 (1942)] are stu d ied a n d th e classical th eo ry o f n o n -ch arg ed fields is p resen ted . T h e in te rac tio n betw een tw o fields is con sid ered a n d a p a rticu la r exam ple is discussed in

detail. l . s. o .

5 3 0 .1 2 :5 3 0 .1 4 5 = 4 1231

A new m ethod fo r the quantization o f fields. II.

Q uantum theory o f non-charged fields. St u e c k e l­ b e r g, E. C. G . A rch. Sci. Phys. N a t., 24, pp. 261-271, N o v.-D ee., 1942.— A c o n tin u a tio n o f a previous p a p er [A bstr. 1230 (1945)]. T h e sym m etrical q u an tizatio n (B o se-E in stein statistics) a n d th e skew -sym m etric q u a n tiz atio n (F e rm i-D ira c statistics) are discussed a n d a stu d y is m ad e o f th e different possible theories o f n o n -ch arg ed fields. A co n d itio n is found in o rd e r th a t th e q u a n tiz atio n be uniquely d eterm ined. T his relates to th e p ro p e r values o f th e w ave energy, l. s. g.

530.12 : 530.145 = 4 1232

A new m ethod fo r the quantization o f fields. I I I . C lassical and quantum theory o f charged fields.

St u e c k e l b e r g, E. C . G . A rch. Sci. Phys. N a t., 24, p p. 5-34, Jan.-F eb., 1943.— A c o n tin u a tio n o f previous p a p ers [A bstr. 1231 (1945)]. T h e w orld-lincs follow ed b y w ave pa ck ets o f c h arg e d a n d n o n -c h arg ed fields

are stu d ied a n d th ese lea d to th e classical theory.

In th e q u a n tu m th eo ry th ere a re 16 a priori theories, b u t reasons a re given fo r ad m ittin g only 4 o f these.

T h e creatio n o f q u a n tu m p airs is th e n discussed, an d it is finally c o n clu d ed th a t only 2 th eo ries are p er­

m issible. l. s. G.

5 3 0 .1 2 :5 3 1 .5 1 1233

O n M iln e’s theory o f gravitation. Sc h i l d, A.

P hys. Rev., 66, pp. 340-342, D ec., 1944.— It is arg u ed th a t th e e q u atio n s w hich M ilne gives in his k incm aticai relativ ity fo r th e m o tio n o f tw o g rav itatin g particles a re m eaningless so long as n o ru le is given for asso ciatin g each ev en t on th e w orld-line o f one particle w ith a c o rre sp o n d in g event o n th e w orld-line o f the o th e r. M ilne h im self uses his d efinition o f sim ul­

tan e ity to relate th e p airs o f events b u t, since this definition is n o t in v a rian t u n d e r L q ren tz tran s fo rm a ­ tio n s, n e ith e r is th e resu ltin g g ra v itatio n al th eo iy . N evertheless, in v arian t rules c a n be a d o p te d , e.g. the re ta rd ed p o ten tial, b u t this in tro d u ces a n a d hoc non-epistem ological elem en t in to th e th eo ry . A n e xception o ccurs w hen o n e o f th e tw o g rav itatin g particles is a fu n d am en tal p article, in w hich case the eq u atio n o f m o tio n o f the o th e r is L oren tz-in v arian t.

G. c . McV.

530.12 : 539.153 1234

M ass-encrgy relation. Du s h m a n, S . Gen. Elect.

R ev., 4 7 ,pp. 6—13, OcU, 1944.— E xp erim en tal evidence fo r th e tw o relatio n s w hich follow from the principle o f relativity is given. T h e first sta te s th a t th e m ass o f a p article increases w ith velocity a n d becom es infinity fo r speeds th a t a p p ro a c h th a t o f light. T h e seco n d relatio n states th a t energy an d m ass a re equivalent.

530.145 : 530.12 = 4 see A bstr. 1230-1232

530.145.6 1235

U nification o f the theories o f photon and meson.

Bo r n, M . N ature, L ond., 154, pp. 764-765, D ec. 16, 1944.— A sim ple exam ple is given o f th e in teractio n o f 2 fields w hich m ay be stu d ied rigorously by m eans o f sim ple m ath em a tics. T h e L ag ran g ian densities fo r fields having a sp in 0 a n d 1, respectively, a re w ritten dow n a n d th e sim plest possible in te rac tio n is tak en , assu m in g th a t in e ac h field th e m ass term is zero.

T h e field e q u atio n s a rc deduced, a n d a discussion based o n these leads to th e c onclusion th a t particles o f spin 0 a n d 1 w ith o u t re st m ass are tra n sfo rm e d by th e sim plest in te rac tio n in to 2 new types o f p articles, one h aving the rest m ass zero (p h o to n ), th e o th e r a finite rest m ass (m eson). T h u s p h o to n a n d m eson seem to be different p h en o m en a o f th e sam e q u a n tiz ed field.

l. s . G.

530.145.6 1236

Expansion o f positive-energy Coulomb wave functions in powers o f the energy. Be c k e r l e y, J. G . Phys. R ev., 61, p p . 11-14, Ja n ., 1945—T h e non -relativ istic w ave fun ctio n (in sp h erical p o la r co -o rd in ates) fo r a c h arg e d p article in a C o u lo m b field is expressed in a fo rm suitab le for p ro b lem s in w hich th e particle h a s a sm all positive energy. T h is fo rm u la tio n a m o u n ts to e x p an d in g th e ra d ia l p a r t o f th e w ave fu n ctio n in pow ers o f th e energy E a n d is achieved by sim ple algebraic m an ip u latio n o f p o w e r series a n d recu rren ce fo rm u lae. T h e coefficients o f the ex p an sio n are fu n ctio n s o f th e ra d ia l c o -o rd in a te a n d a re identified w ith Bessel fu n ctio n s o f in teg ral o rd e r.

135