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1 4T h e a u t h o r i s i n d e b t e df o rt e c h n i c a l a s s i s t a n c ei n t h e p r e p a r a t i o no f t h e s e m a t i n g s

t o W. G e n c a r e l l a .1 6No d e f i n i t ec a s e o fa d o u b l ec r o s s o v e ri n t h e y s p lr e g i o nw a s o b s e r v e d .1 6 T w o v e r i f i e dc r o s s o v e r sb e t w e e ny a n d s c w e r e d e t e c t e da s y 2s c w s p ip h e n o t y p e s .1 7M o r g a n ,T . H . , B r i d g e s ,C . B . ,a n d S c h u l t z ,J . , Ye a r b o o kC a r n e g i eI n s t . ,3 0 , 4 0 8 -

4 1 5 ( 1 9 3 1 ) .1 8P a n s h i n ,I . B . ,C o m p t .r e n d .a c a d .s c i . U . R . S . S . ,3 0 , 5 7 - 6 0 1 9 4 1 .1 9 B r i d g e s ,C . B . ,J . H e r e d i t y ,2 9 ,1 1 - 1 3( 1 9 3 8 ) .1 0O t h e rs t u d i e sc i t e db y B r i d g e sa n d B r e h m e , 8i n d i c a t i n gt h a t t h i sb r e a k l i e sb e t w e e n

3 C 2 a n d 3 C 3w o u l dn o t a l t e rt h e p r e s e n ta r g u m e n t .2 1H o r o w i t z ,N . H . , t h e s e P R O C E E D I N G S ,3 1 ,1 5 3 - 1 5 7( 1 9 4 5 ) .

ON THE NUMBER OF BOUND S TAT E S IN A CENTRAL FIELDOF FORC E

By V . BARGMANN

PRINCETON U N I V E R S I T Y

C o m m u n i c a t e db y E . P . W i g n e r ,S e p t e m b e r1 8 ,1 9 5 2

1 . T h e p r e s e n t n o t e c o n t a i n s some f a i r l ye l e m e n t a r yr e m a r k sc o n -c e r n i n gt h e n u m b e ro f b o u n ds t a t e s o l u t i o n so f t h e S c h r 6 d i n g e re q u a t i o n

V 2 + E = V ( r ) * ,f o ra c e n t r a lf i e l do f f o r c e ,more s p e c i f i c a l l y,t h e n u m b e rn l o fb o u n ds t a t es o l u t i o n so f t h e r a d i a lw a v e e q u a t i o n

I - 1 ( 1+ 1 ) r - 2 q+ Eb = V ( r ) + o ( 1 )f o ra n g u l a rm o m e n t u m 1 . We a s s u m et h e i n t e g r a l

I = J ; X r IV ( r ) ld r ( 2 )t o b e f i n i t e ,a n d w e w i s ht o e s t i m a t en Ii n t e r m so fI . ( I nt h e u n i t sc h o s e nV h a s t h e d i m e n s i o n( l e n g t h )2 , s o t h a t I i s d i m e n s i o n l e s s . )R . J o s t a n dA . P a i s ( r e f .1 , p . 8 4 4 )h a v e s h o w nt h a t n o b o u n ds t a t e s o c c u r i f I < 1 .O u ra i m i s t o d e r i v et h e more g e n e r a li n e q u a l i t y

2 l+ 1)n1 1 / 2( I- 1 ) . Th e e s t i m a t e ( 3 )i s b e s t p o s s i b l ei n t h e s e n s e t h a t f o ra g i v e n I p o t e n t i a l sm a y b e c o n s t r u c t e dw h i c hh a v e a p r e s c r i b e dn u m b e rn 1o fb o u n ds t a t e sf o rt h a t a n g u l a rm o m e n t u m a n d f o rw h i c hI a p p r o a c h e s

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( 2 1+ 1 ) n la r b i t r a r i l yc l o s e l y( s e es e c t i o n5 b e l o w ) . T he w h o l eq u e s t i o ni s t a k e n u p b e c a u s et h e f i n i t e n e s so f I p l a y s a s i g n i f i c a n tr o l ei n s e v e r a lr e c e n t i n v e s t i g a t i o n so n s c a t t e r i n gt h e o r y . 1 - 4 V m a y h a v e a n y s i n g u -l a r i t i e sc o n s i s t e n tw i t h a f i n i t ev a l u e o fI . )

2 . A s i s w e l lk n o w n ,n I i s t h e n u m b e ro f z e r o s( n o tc o u n t i n gr = 0 )o f t h a t s o l u t i o n+ ( r )o f t h e e q u a t i o n

' - - 1 ( 1 + 1 ) r - 2 4 b= V ( r ) + 5 ( 4 )

( E = 0 ) w h i c hv a n i s h e sa t t h e o r i g i n .S p e c i a lc a r e m u s t b e t a k e n w i t ha p o s s i b l eb o u n ds t a t eE = 0 . S i n c eI i s f i n i t ea n y s o l u t i o no f ( 4 ) h a st h e f o l l o w i n ga s y m p t o t i cb e h a v i o ra t i n f i n i t y .T h e e x p r e s s i o nr ( l + l ) 0 ( r )a l w a y sa p p r o a c h e sa f i n i t el i m i t ,s a y X ,a s r - * c. I f X 0 0 ,+ 5 ( r )i n c r e a s e si n d e f i n i t e l y .I f X = 0 t h e e x p r e s s i o nr l 4 ( r )a p p r o a c h e s af i n i t el i m i tu ,a n d , . 0 0 . I n t h e l a t t e rc a s e + ( r )i s s q u a r e i n t e g r a b l ei f I > 0 , a n da c c o r d i n g l yE = 0 i s a b o u n ds t a t e . 5F o r 1 = 0 , E = 0 i s n e v e r a b o u n ds t a t ei f I i s f i n i t e . F o r t h e p u r p o s eo f o u r d i s c u s s i o n ,h o w e v e r ,w e s h a l lc o u n tr = c oa s a z e r o o f4 ( r ) - e v e ni fI = 0 - w h e n e v e rl i m r . . . .r - ( l + l ) O ( r )=0 , a n d i n t e r p r e tt h e i n e q u a l i t y( 3 )a c c o r d i n g l y . 6

R e p l a c ei n e q u a t i o n( 1 )V ( r )b y a p o t e n t i a lV 1 ( r )s u c ht h a t V I ( r )< V ( r )

f o ra l l r , a n d d e n o t eb y n 1 t h e n u m b e ro fb o u n ds t a t e sf o rt h e ne w p o t e n -t i a l . T h e n n l > n . W e s h a l lc h o o s e V I ( r )=- W r ) , w h e r e W ( r ) = V ( r )| , a n d s t u d y t h e e q u a t i o n

- 1 ( 1 + 1 ) r 2 4 0= - W ( r ) 4 W ( r ) = V ( r ) | . ( 5 )

D e n o t eb y V i ,P 2 ,. . . , v P ( n = n , ) t h e s u c c e s s i v ez e r o so f g ( r )( O< V I< 2< . < P n ) ,a n d s e t P o= 0 . W e s h a l lp r o v e

J f , r W ( r ) d r > 2 1 +1 ; a = V k _ l , 8= P k ,k > 1 . ( 6 )

T h e i n e q u a l i t y( 3 ) i s o b t a i n e db y a d d i n g t h e n i n e q u a l i t i e s( 6 ) ,f o rw ef i n dt h e n

I = r W ( r )d r > f o r W ( r )d r > n l ' ( 2 1+ 1 ) 2 n i ( 2 1+ 1 ) .3 . P r e l i m i n a r yR e m a r k so n + ( r ) . - T h e s o l u t i o no f e q u a t i o n( 5 )w h i c h

v a n i s h e sa t t h e o r i g i ni s u n i q u e l yd e t e r m i n e du p t o a c o n s t a n t f a c t o r .As r - - O , r - ( l + l ) + ( r )a p p r o a c h e sa f i n i t en o n - v a n i s h i n gl i m i tK . C h o o s i n gK = 1 , w e f i n df r o m ( 5 )

+ ( r )= r 1+ l - f G ( r , p ) 4 ( p ) W ( p )d p . ( 7 )+ ( r )i s t h e n r e a l . H e r eG ( r ,p ) i s t h e f u n d a m e n t a ls o l u t i o no ft h e e q u a t i o nf - 1 ( 1 + 1 ) r - 2 f= 0 , i . e . ,6 2 G ( r ,p ) / b r 2- 1 ( 1+ 1 ) r - 2 G ( r ,p ) = 0 , G ( r ,r )= 0 ,a n d 6 G ( r ,p ) / b r= 1 f o rr = p . W e h a v e

9 6 2 P R O C .N . A .S .

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G ( r ,p ) = ( 2 1+ 1 )- H ( r , p ) ; H ( r , p ) = r ( r / p )- p ( p / r ) ( 8 )

i f r>0, p>0. C l e a r l y H ( r ,p ) > Oif r> p . S i n c e , f o r r - * . 0 , r - ( ' + ' 1 01 , t h e i n t e g r a li n ( 7 )i s a b s o l u t e l yc o n v e r g e n t .

I n t h e s e q u e lw e s h a l ln e e d t h e i n e q u a l i t y

H ( f t ,p ) H ( p ,a ) < p ( H ( f 3 ,a ) - Y ( f l ,a ) )

Y ( f l ,a ) = 2 ( a 8 ) 1 / 2[ 1 - ( a / 3 ) ' I + ' / 2 ]> 0 ( > a ) . ( 9 )

T o d e r i v ei t c o n s i d e r Z ( p ,( 3 ,a ) = p H ( ( 3 ,a ) - H ( ( 3 ,p ) H ( p ,a ) . Bys t r a i g h tf o r w a r dc o m p u t a t i o n

Z ( p ,( , a ) p ( a ) ' / 2 { ( p 2 / a C , ) l + ' / 2+ ( a 1 / p 2 ) + l / 2- 2 ( a / ) 1 + 1 / 2 }( a f / 1 /{ [ Va ) I 2 _ ( V / a ( / p ) i + / 2 ] 2+ 2 [ 1 ( a / f l ) ' + 1 / 2 ] }

2 p Y ( Q ,a )w h i c he s t a b l i s h e s( 9 ) .

4 . P r o o fo f( 6 ) . - W ed i s t i n g u i s hf o u r c a s e s 7a c c o r d i n ga s a = 0 , a > 0 ;< o = o.( a ) a = 0 , , B = v < c . On t h e o p e n i n t e r v a l( 0 ,( 3 )q 5i s p o s i t i v e ,

a n d h e n c e ,b y ( 7 ) ,4 ) ( r )< r + 1 . S i n c e4 ) ( ( )= 0 w e h a v e f r o m ( 7 )a n d ( 8 )

( 2 1+ 1 ) 3 + 1= J 7 H ( f 3 ,p ) 4 ) ( p ) W ( p ) d p< I f o H ( f l ,p ) p 1 + W ( p )d p( 2 1 1 ) ( 3 + 1< 3 + 1 [j p W ( p )d p f J l( p / 3 ) 2 1 + 1 p W ( p )d p ] .On d i v i d i n gb y 3 + 1w e f i n d( 6 )b e c a u s et h e l a s ti n t e g r a li s p o s i t i v e .

( b ) a > 0 , ( < c o . S i n c eq 5 ( a )= 0 , t h e d e r i v a t i v e4 ) ' ( a )d o e s n o tv a n i s h .I f w e r e p l a c e4 ) ( r )b y x ( r )= 4 ) ( r ) / q ' ( a ) ,t h e n X ( a )= 0 ,x ( a )= 1 ,a n d h e n c e

x ( r )= G ( r , a )- f a rG ( r ,p ) x ( p ) W ( p )d p ( 1 0 )On t h e i n t e r v a la < r q b ( r )> 0 f o r- a l l p o s i t i v er . By a s s u m p t i o n ,r ( r )= ( 2 1+ 1 ) r - ( L + 1 ) q ( r )a p p r o a c h e s0 a s r - c. B y ( 7 ) ,

2 1+ 1 - T ( r ) J r K ( r ,p ) q b ( p ) W ( p )d p = 0

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w h e r e

K ( r ,p ) = r - ( Q + l ) H ( r ,p )= p - ' ( 1 ( p / r ) 2 1 + l )< p - 1H e n c e

J f op W ( p )d p 2 J f o rp 1 +K ( r ,p ) W ( p )d p 22 1+ 1 T ( r )+ f o r K ( r ,p ) [ p + l _ - ( p ) ] W ( p )d p ( 1 1 )

S i n c e t h e i n t e g r a li n ( 1 1 )i s n o n - n e g a t i v ea n d T ( r )- O 0 , w e f i n da t oncet h a t f oX p W ( p )d p 2 2 1+ 1 . T o e x c l u d ee q u a l i t yw e o b s e r v et h a t t h e r em u s t e x i s tt w o a d j a c e n t i n t e r v a l s[ ,a ] a n d [ q ,r ] ( t < v < r < c o) s u c ht h a t 1 p W ( p )d p a n dJ 7 Ip W ( p )d p b o t h e x c e e d1 / 4 , s a y .I f r > 1 ,t h e n ,b y ( 7 ) ,r 1 + - + ( r ) 2 . f tG ( r ,p ) q ( p )W ( p )d p a n d t h r o u g h o u tt h ei n t e r v a l[ N ,r ] ( p l + l- 4 ( p ) ) p ( 1 + 1 )> c , w h e r e c i s s o m e p o s i t i v ec o n s t a n t . F o rr >t we f i n df r o m ( 1 1 )

f o r p W ( p )d p > 2 1+ 1 r ( r )+ c f r p ' K ( r ,p ) p W ( p )d pa n d i n t h el i m i tr - - c o

J o p W ( p ) d p o2 1 + 1 +c f oW p )d p >2 1 + 1 , q . e . d .( d ) a > 0 , , B= i = x . W e s t a r t ,a s i n ( b ) , f r o m e q u a t i o n 1 0 ,s o

t h a t G ( r ,a ) 2 x ( r )> 0 f o r r > a . By a s s u m p t i o n ,( 2 1+ 1 ) - 10 ( r )=x ( r ) / G ( r ,a ) a p p r o a c h e s0 a s r X . From ( 1 0 ). w e f i n d

2 1+ 1 O ( r ) 1 r H . p x ( p ) W ( p )d p = 0 .J G ( r ,a )

H e n c e ,b y ( 9 ) ,

p W ( p )d p 2 f G H ( ra ) G ( p ,a )W ( p )d p 2 2 1+ 1 - 0 ( r )+f TH ( r ,p )

G ( r ,a ) [ G ( o ,a ) x ( p ) ] W ( p )d p ( 1 2 )

W e p r o c e e da s i n c a s e ( c )a b o v e . T h ei n e q u a l i t yf . a p W (p )d p . 2 2 1+ 1i s a n i m m e d i a t ec o n s e q u e n c eo f ( 1 2 ) . T o e x c l u d ee q u a l i t y t h ei n t e r v a l s[ t ,a ] a n d [ X ,t ] a r e c h o s e na s b e f o r e( t 2 a ) ,s o t h a t ( 2 1 1 ) ( G ( p ,a ) x ( p ) ) P( ' 1 + )> c > 0 i f I < p < . F o r r > r ( 1 2 )i m p l i e s

p W ( p )d p 1 + 1 - 0 ( r )+ c ' , H ( r Np W ( p )d p, P H ( r ,a )

a n ds i n c el i m , , ( H ( r ,p ) / H ( r ,a ) ) = ( a / p ) w e f i n d f o rr - * c o Y j@ p W ( p ) d p> 2 1+ 1 + c a J ; p W ( p )d p > 2 1+ 1 . T h i s c o n c l u d e s t h e p r o o fo f ( 6 ) .

5 . E x a m p l e s . - T h ep r o o f si n t h e p r e c e d i n gs e c t i o ns u g g e s t t h e c o n -s t r u c t i o no fp o t e n t i a l sf o rw h i c ht h e i n e q u a l i t i e s( 3 )o r ( 6 )m a y b e a p p r o x i -

9 6 4 P R O C .N . A . S .

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m a t e l y r e p l a c e d. b yt h e c o r r e s p o n d i n ge q u a l i t i e s .I n 4 ( a ) ,f o r e x a m p l e ,t h e f i r s ti n e q u a l i t y w i l ln e a r l y r e d u c et o a n e q u a l i t y . i fa t t h o s e r w h e r eW ( r ) i s a p p r e c i a b l e+ ( r )n e a r l y e q u a l s r l + l ,i . e . ,t h e f i e l df r e es o l u t i o n .T h i s l e a d s( f o . -n , = 1 )t o t h e c h o i c eo fa p o t e n t i a l V ( r )= - W ( r ) W > 0 )w h i c hv a n i s h e se v e r y w h e r ew i t h t h e e x c e p t i o no f a s m a l l i n t e r v a la < r b (( 2 1+ 1 ) c l= ( I + 1 ) + ( b )- b 4 / ( b ) ; ( 1 4 )( 2 1+ 1 ) C 2= - I + ( b ) - b 4 ( b ) (

I f c 2> 0 , + b ( r )- o o a s r - - c o , s o t h a t + / ( r )v a n i s h e sa t a p o i n t/ 8g i v e nb y ( ( / b ) 2 1 + 1= c 1 / c 2 ,a n d i f c 2= 0 , t h e n , 3= o . O w i n gt o t h e s m a l l n e s so fa t h e r e l a t i v ec h a n g eo f+ ( r )a c r o s s t h e i n t e r v a l[ a ,b ] i s n e g l i g i b l ec o m -p a r e dt o t h e r e l a t i v ec h a n g eo f+ ' ( r ) ,s o t h a t + 6 ( b )- - 4 ( a ) . From t h e c o n -d i t i o nc 2 > 0 w e o b t a i n - + ' ( b ) / + ( b ) > I / b , a n d s i n c e4 / ( a ) / 4 , ( a )=( I+ 1 ) / a ,t h i samounts t o

+ 0 ' ( a ) / + ( a ) + ( b ) / + ( b )- ' ( + ( a ) + ( b ) ) / + ( a )> ( 2 1+ 1 ) / a ( 1 5 )T h u s t h e r e q u i r e di n c r e m e n to f t h e l o g a r i t h m i cd e r i v a t i v ei s t h e s m a l l e rt h e l a r g e ra i sc h o s e n - o ra p o t e n t i a lo fg i v e ns t r e n g t hi s t h e more e f f e c t i v ei n p r o d u c i n gb o u n ds t a t e s t h e f a r t h e ri t i s r e m o v e d f r o mt h e o r i g i n( w h i c hi s t h e r e a s o n f o r t h e w e i g h t f a c t o rr i n t h e i n t e g r a lI ) . W i t h o u ty e ts p e c i f y i n gW ( r ) ,w e s e e f r o m ( 5 )t h a t + ( b )- + ' ( a ) a p p r o x i n d a t e l ye q u a l s- W & t ( a )w h e r e W i s a s u i t a b l ea v e r a g eo f W , p r o v i d e dt h e c e n t r i f u g a lt e r m 1 ( 1+ 1 ) r - 2i s n e g l i g i b l ec o m p a r e dt o W . I f t h e i n c r e m e n ti s a ss m a l l a s

p o s s i b l ew e

f i n df r o m

( 1 5 )t h a t WSa - - '

( 2 1+ 1 )w h i c hi s

e q u i v a -l e n tt o f b r W ( r )d r = J f 7r W ( r )d r 2 1+ 1 .T o h a v e a d e f i n i t ee x a m p l ec o n s i d e rW ( r ) = 1 + 1 ( 1+ 1 ) r - 2 ,a n d

a = ( 2 1+ 1 ) ( 1+ 5 ) / S . T h e n , i n [ a , b ] ,+ 5 ( r )= c o s ( r - a ) + ( ( I+ l ) / a )s i n( r - a ) s o t h a t + ( b )= c o s a + ( ( I+ 1 ) / a )s i n5 , + ( b )= - s i n S +( ( 1+ 1 ) / a )c o s 6 , a n d o n e v e r i f i e se a s i l yt h a t c 2> 0 f o r s m a l l 5 ( e . g . ,a < 1 / 4 ) . T h e z e r o ,f 3 i s d e t e r m i n e db y ( W / b ) 2 1 + 1= c l / c 2 ,a n d a p p r o x i -m a t e l y C 1 / C 2a - ' 5 - 1s o t h a t l , B a . 5 1 / ( 2 L + l ) .F i n a l l y , I= f b r W ( r )d r =5 ( a+ ' / 2 6 )+ 1 ( 1+ 1 )l o g( 1+ 5 / a ) . As 5 - - 0 O ,I 2 1+ 1 . A l t e r a t e l y ,i n s t e a d o fv a r y i n ga a n d k e e p i n gt h e s t r e n g t ho f W f i x e d ,o n e m i g h tk e e pa f i x e da n d v a r y t h e s t r e n g t ho f t h e p o t e n t i a l .

I n a s i m i l a rw a y o n e m a y c o n s t r u c tp o t e n t i a l sw i t h tw o o r more b o u n ds t a t e s o l u t i o n ss u c h t h a t I i s a r b i t r a r i l yc l o s et o ( 2 1+ 1 ) n j .One s i m p l yh a s t o a d d o t h e r t r o u g h si n s u i t a b l yp l a c e di n t e r v a l s[ a ' ,a + 6 ' ] ,e t c . ,i ns u c h a w a y , h o w e v e r ,t h a t tw o s u c c e s s i v ei n t e r v a l sa r e s u f f i c i e n t l yf a r

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M AT H E M AT I C S : F . D . MURNAGHAN

f r o m o n e a n o t h e ra n d f r o m t h e z e r oo f + ( r )b e t w e e nt h e m . N o t e t h a tt h e s ep o t e n t i a l sa r e a d j u s t e d o n l y t o o n e f i x e dv a l u e o f t h e a n g u l a r m o -m e n t u m .

1 J o s t ,R . ,a n d P a i s ,A . ,P h y s .R e v. ,8 2 ,8 4 0( 1 9 5 1 ) .2 L e v i n s o n ,N . ,K g l .D a n s k eV i d e n s k a b .S e l s k a b .M a t . - f y s .M e d d . ,2 5 ,N o . 9 ( 1 9 4 9 ) . J o s t ,R . ,a n d K o h n ,W . , P h y s .R e v . ,8 7 ,9 7 7( 1 9 5 2 ) .4 B a r g m a n n ,V . ,R e v .M o d .P h y s . ,2 1 ,4 8 8 ( 1 9 4 9 ) .6 F o r I = 0 t h e s e s t a t e m e n t sa r e p r o v e di n t h e a p p e n d i xo f r e f .4 . F o r h i g h e r I a

s i m i l a rp r o o fm a y b e g i v e n .6 E v e n f o r I = 0 t h i sc a s e i s s i g n i f i c a n ta l t h o u g hn o b o u n ds t a t ei s p r e s e n t .I n

p a r t i c u l a ri n t h i sc a s e

I s i nv ( 0 ) l= 1 w h e r eq ( k )i s t h e s c a t t e r i n gp h a s es h i f t ,a n d h e n c e

t h e c r o s ss e c t i o ns i n 2 ( k ) / k 2b e c o m e si n f i n i t ea s k - 0 . ( S e e r e f . 4 , e q u a t i o n ( 1 . 9 ) .I n t h i sc a s ef ( O )= 0 . )

7 F o r I = 0 o n e n e e d n o t d i s t i n g u i s ha = 0 a n d a > 0 , b e c a u s eG ( r ,a ) = r - am a y b e u s e d i n b o t h c a s e s .

ON T H E INVARIANT T H E O RY OF T H E CLASSICAL GROUPS

By F . D . MURNAGHAN

I N S T r T r u T oT E c N o L 6 G I c oDB A E R O N A u T I C A ,S A oJ o s eD o sC A M p o s ,B R A s n LC o m m u n i c a t e dA u g u s t1 8 ,1 9 5 2

I t h a s b e e nr e c o g n i z e df o rs o m et i m e t h a t t h e t h e o r yo f i n v a r i a n t sa n dc o v a r i a n t s ,w i t h r e s p e c tt o a - g i v e ng r o u p ,r e s t so n t h e a n a l y s i so f r 0 r ,w h e r eF a n d r a r e i r r e d u c i b l er e p r e s e n t a t i o n so ft h e g r o u p ,i n t oi t si r r e d u c -i b l ec o m p o n f n t s . T h u si f w e d e n o t eb y IX t h e i r r e d u c i b l er e p r e s e n t a t i o no f t h e n - d i m e n s i o n a ll i n e a rg r o u p w h i c hi s a s s o c i a t e dw i t h t h e p a r t i t i o n( X )= 1 , * * . . .Xn ) ,X l, X 2) . . . X3 >n 0 ,o fa n y n o n - n e g a t i v ei n t e g e r

m i n t on o t m o r et h a n n p a r t st h e c o r e

o ft h e t h e o r yo f i n v a r i a n t sa n d c o -

v a r i a n t s ,u n d e rl i n e a rt r a n s f o r m a t i o n s ,i s t h e a n a l y s i so f { X I { , u }w h e r e( X )a n d ( j u )a r e p a r t i t i o n so f a n y t w o n o n - n e g a t i v ei n t e g e r sm a n d j , r e -s p e c t i v e l y .T h e c a s e s w h e r e( X )i s e i t h e rt h e 1 - e l e m e n tp a r t i t i o n( m ) o rt h e m - e l e m e n tp a r t i t i o n( l ' )a n d ( , u )i s e i t h e rt h e 1 - e l e m e n tp a r t i t i o n( j )o r t h e j - e l e m e n tp a r t i t i o n( 1 ' )a r eo fp a r t i c u l a ri m p o r t a n c ea n d t h e p r o b l e mo f a n a l y z i n g{ ( X0 { , A ,e s p e c i a l l yi n t h e s ec a s e s ,h a s b e e nm u c h s t u d i e d ,f o l l o w i n gt h e i n i t i a li m p e t u sg i v e nb y L i t t l e w o o d , 1d u r i n gt h e p a s t d e c a d e .However t h e m e t h o d su s e d h a v e b e e nl a b o r i o u sw h e nm a n dj a r e g r e a t e rt h a n 2 ; i n t h e s e c a s e s { X I0 { gI c o n t a i n sm a n y c o m p o n e n t s ,e a c h c o r r e -s p o n d i n gt oa p a r t i t i o no fm j ,a n d e a c ho f t h e s eh a s h a dt ob e d e t e r m i n e ds e p a -r a t e l yb y a t e d i o u sc a l c u l a t i o n .W e p r e s e n ti n t h i sn o t e a m e t h o dw h i c hy i e l d s ,i n t h e c a s e so fp a r t i c u l a ri m p o r t a n c er e f e r r e dt o , t h e c o m p o n e n t so f{ XI { , u }i n p l a t o o n s ,r a t h e r t h a n i n d i v i d u a l l y ,e a c hp l a t o o nc o n s i s t i n go ft h o s ep a r e n t h e s e sI . . . I w h i c hc o n t a i nt h e s a m en u m b e ro fn o n - z e r op a r t s .

9 6 6 P R O C .N . A . S .