stone roundness of moraines connected with taku glacier

stone roundness of moraines connected with taku glacier

STONE ROUNDNESS OF MORAINES CONNECTED WITH
TAKU GLACIER, SOUTHEASTERN ALASKA
MATTI SEPPÄLÄ
SEPPÄLÄ, MATTI 1976: Stone roundness of moraines connected
with Taku Glacier, southeastern Alaska. Bull. Geol. Soc. Finland 48,
87—94.
The roundness and flatness (Cailleux's indices) of granodioritic stones
from three different moraines in contact with Taku Glacier were
measured. Mean roundness of talus material was 60, of late Wisconsin moraine material 240 and of push moraine material 370. Flatness of stones decreased in the same order.
Results were statistically evaluated and the differences were
identified also by analysis of variance.
By comparing the stone size and roundness it was concluded that
the original forms of stones determine their final forms. Flat and
small stones cannot be rounded as easely as cubical and bigger blocks
by glacial and glaciofluvial processes.
Matti Seppälä,
SF-20500 Turku
Department
o/
50, Finland.
Introduction
T h e aim of this p a p e r is to describe t h e i n f l u e n c e of glacial a n d glaciofluvial processes
on t h e r o u n d n e s s of stones in t h e a r e a of a n
active valley glacier.
A p r o b l e m h e r e is t h e p r o c e d u r e by w h i c h
t h e stones w e r e t r a n s p o r t e d a n d r o u n d e d
b e f o r e r e a c h i n g t h e i r p r e s e n t location. Since
t h e r e a r e f e w e r possibilities in t h e case of
a valley glacier, it is easier to s t u d y t h e
p r o b l e m in these conditions t h a n in o t h e r
cases e.g. m o r a i n e s of t h e c o n t i n e n t a l ice
sheets.
T h e T a k u Glacier in s o u t h e a s t e r n A l a s k a
w a s chosen as a s u i t a b l e r e s e a r c h a r e a (Fig. 1).
Geography,
University
of
Turku,
S a m p l e s w e r e t a k e n f r o m : 1) a g r a n o d i o r i t e
m o u n d of stones (talus moraine) s i t u a t e d on
t h e slope of a n u n a t a k in t h e m i d d l e p a r t
of t h e glacier, 2) a m o r a i n e of l a t e Wisconsin
d a t e lying n e a r t h e p r e v i o u s site a n d s i t u a t e d
on t h e slope of Icy Basin, a small t r i b u t a r y
glacier of t h e T a k u Glacier, w h e r e t h e till
is s u b j e c t e d to q u i t e active solifluction, a n d
3) a p u s h m o r a i n e at t h e foot of t h e T a k u
Glacier on t h e seashore, w h e r e t h e m a t e r i a l
is w a s h e d glaciofluvial sand, silt a n d g r a v e l .
Methods
T h e m e t h o d of m e a s u r i n g r o u n d n e s s used
in this s t u d y w a s t h a t of C a i l l e u x (e.g. B l e n k
88
Matti
Seppälä
1960, p. 203). S t o n e s w i t h a d i a m e t e r of m o r e
t h a n 20 m m a n d less t h a n 140 m m (cf. K i n g
and B u c k l e y 1968, p. 202) w e r e t a k e n a n d
t h e i r m a x i m u m l e n g t h (L), t h e m a x i m u m
w i d t h (1) at r i g h t angles to t h e l e n g t h a n d t h e
thickness (E) at r i g h t angles to b o t h of these
m e a s u r e d . In a d d i t i o n t h e m i n i m u m r a d i u s
(r) of c u r v a t u r e in t h e p r i n c i p a l p l a n e L I
at its s h a r p e s t point w a s t a k e n . W i t h t h e
aid of t h e s e m e a s u r e m e n t s an i n d e x of
r o u n d n e s s 2r/L X 1000 a n d an i n d e x of
f l a t n e s s (L + 1)/2E X 100 w a s calculated for
the stones. A n i n c r e a s e in t h e i n d e x of
r o u n d n e s s indicates a h i g h e r d e g r e e of
r o u n d n e s s in t h e stone. A decrease in t h e
v a l u e of E m e a n s a h i g h e r i n d e x of f l a t n e s s
a n d t h e h i g h e r t h e value, t h e f l a t t e r is t h e
stone. W h e n t h e v a l u e f o r t h e i n d e x of
f l a t n e s s a p p r o a c h e s 100, it m e a n s t h a t t h e
values for L, 1 a n d E a r e t h e s a m e as in a
c u b e or a sphere.
roundness higher than the m a x i m u m value
for s a m p l e 1 (Fig. 4). In s a m p l e 2 t h e m e a n
i n d e x of r o u n d n e s s w a s 240 a n d t h e s t a n d a r d
T h e n u m b e r of stones m e a s u r e d at each
s a m p l i n g location v a r i e d b e t w e e n 20 a n d 91.
O n l y g r a n o d i o r i t i c stones w e r e selected for
measurement.
T h e results w e r e processed
statistically a n d t h e m e a n s (x), s t a n d a r d
d e v i a t i o n s (s), coefficients of v a r i a b i l i t y (v),
a n d s t a n d a r d e r r o r s of t h e m e a n (m) calcul a t e d f o r each of t h e t h r e e sites. T h e sign i f i c a n c e of t h e statistical d i f f e r e n c e s b e t w e e n t h e samples w a s s t u d i e d w i t h t h e aid
of a n a l y s i s of v a r i a n c e (Table 1).
Stone roundness
As m i g h t h a v e been e x p e c t e d t h e least
r o u n d e d stones w e r e f o u n d in t h e t a l u s
m o r a i n e (sample 1, Figs. 1, 2 a n d 4, T a b l e 1).
T h e m e a n i n d e x of r o u n d n e s s f o r stones f r o m
this s a m p l e w a s only 60 a n d t h e m a x i m u m
v a l u e 160 (Table 1). A p p r o x i m a t e l y 68 p e r
cent of t h e stones in s a m p l e 2 a n d over 90 p e r
cent of those in s a m p l e 3 h a d an i n d e x of
i
i
i
Fig. 1. General m a p of the Taku Glacier area
with sampling locations (1—3). C 10 is the main
camp on the J u n e a u Icefield. The hatched area
is Taku Inlet.
Stone roundness of moraines connected with Taku Glacier,
Table
1. Roundness and flatness distributions of studied samples.
Roundness -
Sample
n
Range
1
2
3
20
25
91
X
Flatness
L
3 — 160
80 — 520
70 — 700
60
240
370
40
120
150
F = 45.106
dfl = 2
p < 0.001
89
Explanation of symbols in text.
L
1000
s
southeastern Alaska
1
+
•
2 E
i1nn
00
V
m
Range
X
s
V
m
60.33
49.49
40.82
8.93
2.35
15.6
142 — 929
106 — 344
106 — 215
236
185
147
168
50
21
71.26
26.77
13.94
37.6
9.92
2.14
df2 = 134
deviation 120 or, in o t h e r w o r d s , c o n s i d e r a b l y
l a r g e r t h a n in s a m p l e 1 b u t s o m e w h a t s m a l l e r
t h a n in s a m p l e 3 (s = 150) (Table 1). T h e
s t a n d a r d e r r o r of t h e m e a n i n d e x of r o u n d ness w a s l a r g e s t in s a m p l e 3 (Table 1).
More t h a n 16 p e r cent of t h e stones f r o m
the push moraine were more rounded than
a n y single stone in t h e o t h e r samples. T h e
stones in this s a m p l e w e r e also on a v e r a g e
m o r e r o u n d e d t h a n o t h e r s (x = 370).
The
r a n g e (70—700) w a s by f a r t h e g r e a t e s t
(Table 1). T h e c u m u l a t i v e c u r v e for t h e i n dices of r o u n d n e s s in s a m p l e 3 w a s cons e q u e n t l y g e n t l e r t h a n for t h e o t h e r s a m p l e s
(Fig. 4).
W h e n testing statistical d i f f e r e n c e s in t h e
indices of r o u n d n e s s of t h e s a m p l e s by m e a n s
of analysis of v a r i a n c e , a v a r i a n c e ratio
(F = 45.106) w a s obtained, w h i c h is significant
b e l o w t h e 0.1 °/o risk level (p < 0.001)
(Table 1).
14.805
dfl = 2
p < 0.001
df2
134
also decreased in t h e same w a y (Table 1).
This same r e g u l a r i t y w a s also followed by
t h e coefficient of v a r i a b i l i t y a n d t h e s t a n d a r d
e r r o r of t h e m e a n (Table 1).
In t h e c u m u l a t i v e c u r v e s d r a w n for the
indices of f l a t n e s s t h e i n f l u e n c e of t h e
r o u n d i n g of t h e stones can also be seen
(Fig. 4). T h e steepest c u r v e is t h a t for the
most r o u n d e d p u s h m o r a i n e m a t e r i a l and
t h e f l a t t e s t t h a t of t h e slightly or not at all
rounded talus material.
A v a r i a n c e analysis w a s also c a r r i e d out
in t h e case of t h e t h r e e samples on t h e basis
of flatness. This g a v e a similar result to t h e
one obtained f r o m t h e analysis of t h e indices of r o u n d n e s s . T h e v a r i a n c e r a t i o for
t h e t h r e e s a m p l e s (F = 14.805) is significant
Flatness
T h e h i g h e s t v a l u e s for t h e i n d e x of f l a t n e s s
w e r e o b t a i n e d f o r stones f r o m t h e t a l u s
m a t e r i a l (x = 236). In s a m p l e 2 stones w e r e
less f l a t (x = 185). T h e stones w i t h t h e most
even p r o p o r t i o n s w e r e f o u n d in t h e p u s h
m o r a i n e s a m p l e (x = 147) (Table 1 a n d Fig. 4).
T h e s t a n d a r d d e v i a t i o n (s), w h i c h gives an
indication of v a r i a t i o n s w i t h i n t h e sample,
Fig. 2. Angular stones f r o m the talus material
(Sample 1). The longer side of the compass is
11 cm in length. Photographs by the author.
90
Matti
Seppälä
Fig. 3. Rounded stones f r o m the push moraine (Sample 3). The longer side of the
matchbox is 5 cm in length.. Stereogram.
below t h e 0.1 °/o risk level (p < 0.001) (Table 1).
Relationship between roundness and
flatness
T h e indices of f l a t n e s s b e c o m e s m a l l e r as
the stones become m o r e r o u n d e d (Figs. 4
a n d 5).
1
A
2
x
In Fig. 5 t h e indices of r o u n d n e s s a n d
f l a t n e s s a r e s h o w n t o g e t h e r . T h e stones of
s a m p l e 1 a r e to be f o u n d in t h e l o w e r p a r t
of the g r i d a n d t h e a r e a w i t h i n w h i c h t h e
values lie e x t e n d s to t h e r i g h t . T h e a r e a
r e p r e s e n t i n g s a m p l e 2 lies h i g h e r u p on the
grid a n d lies f a r t h e r to t h e l e f t t h a n t h e f i r s t
e v e n t h o u g h it, too, is r a t h e r e l o n g a t e d b e cause of a single stone. S a m p l e 3 p r o v i d e s
a r o u n d i s h a r e a lying e v e n f a r t h e r to t h e
3
.
Fig. 4. Percentage cumulative curves of the flatness (L + l)/2Exl00 and roundness 2r/Lxl000 of
the studied samples. Key to symbols: 1. talus material, 2. Late Wisconsin moraine and 3. push
moraine stones (see Fig. 1.).
Stone roundness of moraines connected with Taku Glacier, southeastern Alaska
91
— • 1000
Fig. 5. Relationships between roundness and flatness of stones in different moraines.
symbols: see Fig. 4.
l e f t a n d h i g h e r on t h e grid t h a n t h e t w o
p r e v i o u s samples. T h e g r e a t e r t h e degree
of erosion a n d r o u n d n e s s of t h e stones, t h e
f a r t h e r to t h e l e f t a n d t h e h i g h e r u p t h e grid
t h e c l u s t e r s of v a l u e s f o r t h e indices lie.
r o u n d e d . H o w e v e r , no clear statistical correlation was obtained for the whole material.
If Fig. 6 is studied m o r e closely, t h e r e can
uo
i a
2 x
3 •
X
X
130 •
120
Relationship between stone size and
roundness
110
8 *
100 o a a
90
To i l l u s t r a t e t h e size of t h e stones t h e
l e n g t h of t h e l o n g i t u d i n a l axis (L) of stones
w a s chosen (cf. K i n g a n d B u c k l e y 1968).
W h e n this w a s c o m p a r e d w i t h t h e i n d e x of
r o u n d n e s s of t h e stones it w a s noticed (Fig. 6)
t h a t l a r g e stones, w h i c h w e r e most f r e q u e n t
in the t a l u s m a t e r i a l , w e r e g e n e r a l l y less
r o u n d e d t h a n small stones, w h i c h h a d u n d e r gone d i f f e r e n t a b r a s i o n processes. This obs e r v a t i o n is r a t h e r n a t u r a l since t h e size of
t h e stones decreases as t h e y a r e w o r n a n d
Key to
80
70
,'
60
*
a
a
^ X
& a
aa aa
•
X
X
.
50
X
•
40
•
X
>»
w
•
• »
30
20
•
0
100
200
300
•
400
500
600
700
*JL. 1000
Fig. 6. Relationship between stone length (L) and
roundness (2r/Lxl000) of stones in different
moraines. Key of symbols in Fig. 4.
92
Matti
Seppälä
be seen some i n t e r e s t i n g h y p e r b o l i c lines
f o r m e d by t h e points f r o m the samples. F r o m
this t h e guess m i g h t be h a z a r d e d t h a t a stone
of a c e r t a i n s h a p e w h e n it h a s b e e n loosened
f r o m t h e rock u n d e r g o e s a c e r t a i n developm e n t process in t h e course of its becoming
r o u n d e d . In o t h e r words, t h e f i n a l f o r m of
a stone is d e t e r m i n e d p r i m a r i l y by its original
s h a p e w h e n c r u m b l e d a w a y f r o m t h e rock
a n d secondarily by t h e processes t h r o u g h
w h i c h it s u b s e q u e n t l y passes (cf. K i n g and
B u c k l e y 1968). In Fig. 6 it is possible to follow f i v e or six d i f f e r e n t h y p e r b o l i c r o u n d n e s s
g e n e r a t i o n lines. T w o of t h e lines a r e not
v e r y clear b e c a u s e of t h e small m a t e r i a l used
and t h e f a c t t h a t t h e largest stones w e r e not
m e a s u r e d b u t t h e y a p p e a r only in t h e section of t h e f i g u r e i l l u s t r a t i n g m o r e r o u n d e d
stones. T h e stones a r e clearly not c r u s h e d
in t h e course of t h e f l u v i a l processes a n d
t h e r e f o r e t h e y follow t h e r o u n d n e s s developm e n t lines r a t h e r closely. On t h e o t h e r h a n d ,
the glacier can c r u s h stones w h i c h h a v e
a l r e a d y b e e n r o u n d e d (cf. H o l m e s 1960,
p. 1648). As a r e s u l t of this n e w stones a r e
f o r m e d w h i c h in t u r n a r e s u b j e c t e d to t h e
a b r a s i o n activity of t h e r o u n d i n g processes.
Discussion
As g e n e r a l r u l e it m a y be said t h a t t h e
degree of r o u n d n e s s of stones is d e t e r m i n e d
by t h e process a n d t h e distance t h e y a r e
transported.
At p r e s e n t t h e r e a r e t w o t y p e s of m a t e r i a l
w h i c h a r e t r a n s p o r t e d by t h e ice a n d w a t e r s
of T a k u Glacier.
First, t h e r e is m a t e r i a l
deposited on t h e p e a k s a n d slopes of m o u n tains d u r i n g t h e Wisconsin glaciation and,
second, m a t e r i a l r e c e n t l y b r o k e n off t h e rocks
as a r e s u l t of periglacial w e a t h e r i n g processes
(Hamelin 1964).
It is not k n o w n h o w f a r t h e f i r s t t y p e of
m a t e r i a l has b e e n t r a n s p o r t e d a n d w h a t
stages it has passed t h r o u g h b e f o r e being
deposited on t h e n u n a t a k s .
T h e originally
fluvial material may have been carried by
c o n t i n e n t a l ice f r o m d i f f e r e n t d r a i n a g e a r e a s
to t h e n e w s u r r o u n d i n g s . In s t u d i n g m a t e r i a l
t h a t m a y h a v e b e e n t r a n s p o r t e d by t h e cont i n e n t a l ice it is t h e r e f o r e necessary to b e
e x t r e m e l y c a r e f u l in d r a w i n g conclusions
a b o u t r o u n d i n g t h a t has r e s u l t e d f r o m glacial
a n d glaciofluvial processes.
T h e stones f r o m t h e Wisconsin glaciation
m o r a i n e on t h e T a k u Glacier a r e noticeably
m o r e r o u n d e d t h a n stones t a k e n f r o m t h e
t a l u s sample. These m o r a i n e s a r e small in
size a n d t h e i r m a t e r i a l m o v e s only slowly
d o w n t h e slope u n d e r t h e i n f l u e n c e of
solifluction. C o n s e q u e n t l y t h e a m o u n t of
m a t e r i a l m o v e d n o w is v e r y small w h e n compared with the material that crumbles away
by f r o s t w e a t h e r i n g a n d is t r a n s p o r t e d by
t h e glacier a n d its w a t e r s .
S e d i m e n t a t i o n on t h e coasts of A l a s k a is
e x t r e m e l y r a p i d at t h e t e r m i n u s of glaciers
(cf. J o r d a n 1962). Stones of the p u s h m o r a i n e
of t h e T a k u Glacier w h i c h lie close to t h e
s u r f a c e at t h e edge of t h e glacier h a v e p r e s u m a b l y been t r a n s p o r t e d t h e r e only a f e w
y e a r s earlier.
T h e m a x i m u m s t r a i g h t - l i n e distance over
w h i c h m a t e r i a l can h a v e been t r a n s p o r t e d
f r o m t h e source of t h e T a k u Glacier to its
p r e s e n t foot is 60 kilometres, a n d t h e direction of t r a n s p o r t is d o w n w a r d s along t h e
glacier valley. T h e stones h a v e been c a r r i e d
e i t h e r f r o z e n into t h e glacier or g l a c i o f l u v i a l ly by m e l t w a t e r s . In this case t h e m e t h o d
b y w h i c h this m a t e r i a l has a r r i v e d at its
p r e s e n t site m a y be t e r m e d a o n e - w a y
t r a n s p o r t a t i o n system.
On t h e basis of one rock t y p e t h e source
of w h i c h is k n o w n it is possible to a r r i v e at
a f a i r l y reliable p i c t u r e of t h e r o u n d i n g
qualities of t h e stones a n d t h u s a r r i v e at an
idea of h o w g r e a t a distance t h e y w o u l d h a v e
h a d to t r a v e l to become completely r o u n d e d .
Stone roundness of moraines connected with T a k u Glacier, southeastern Alaska
It is, of course, impossible to m e a s u r e u n d e r
n a t u r a l conditions t h e total distance t h e
stones t r a v e l l e d b e f o r e being deposited.
In
this s t u d y w e k n o w only t h e place in w h i c h
t h e stones h a v e b e e n deposited: w e do not
k n o w h o w long a stone m i g h t h a v e b e e n
rolled back a n d f o r t h on one spot, for e x ample, in a glacier moulin. T h e r e are p l e n t y
of g r a n o d i o r i t i c rocks in t h e T a k u Glacier
area, in t h e b a t h o l i t h s of t h e Coast R a n g e
(Forbes 1959), so t h a t t h e stones could h a v e
o r i g i n a t e d f r o m several d i f f e r e n t sources
along t h e sides of t h e glacier.
By selecting only g r a n o d i o r i t i c stones f o r
t h e m e a s u r e m e n t s it w a s possible to k e e p
one v a r i a b l e (rock type) c o n s t a n t (cf. K i n g
a n d B u c k l e y 1968, p. 200).
S a m p l e 1 is r e p r e s e n t a t i o n of those stones
t h a t h a v e loosened f r o m t h e rock in t h e course
of periglacial w e a t h e r i n g processes. T h e r e sults of m e a s u r e m e n t s i n d i c a t e t h a t a cons i d e r a b l e n u m b e r of f l a t stones b r e a k a w a y
a n d become t a l u s m a t e r i a l . T h e g r a n o d i o r i t i c
rock has one p r e d o m i n a n t direction of Ass u r i n g a n d t w o o t h e r s at r o u g h l y r i g h t angles
to the f i r s t b u t at d i f f e r e n t angles to each
other. This c h a r a c t e r has its e f f e c t on t h e
original s h a p e of t h e stones.
T h e d e g r e e of r o u n d n e s s of t h e stones of
t h e second s a m p l e is m u c h t h e s a m e as t h a t
observed b y K i n g (1969, p. 294) in analyses
m a d e on B a f f i n I s l a n d of m o r a i n e s of d i f f e r e n t ages. U n f o r t u n a t e l y K i n g does not
r e v e a l w h i c h t y p e s of rock she used in h e r
measurements.
In glacial m e l t w a t e r s t r e a m s m a t e r i a l does
not need to be t r a n s p o r t e d v e r y g r e a t distance
for r a p i d r o u n d i n g to t a k e place because of
t h e e f f i c a c y of the t u r b u l e n t , r u n n i n g w a t e r
of such s t r e a m s (cf. K i n g and B u c k l e y 1968,
p. 211). L a r g e stones b e c o m e r o u n d e d m o r e
r a p i d l y t h a n small ones (cf. K i n g a n d B u c k l e y
1968, p. 209) since t h e y t e n d to be rolled along
t h e b o t t o m of m e l t w a t e r c h a n n e l s in t h e
course of t r a n s p o r t . T h i s can be seen in Fig. 6
93
w h e r e t h e best r o u n d e d stones a r e not t h e
smallest ones.
The l a r g e s t n u m b e r of m e a s u r e m e n t s of
stones w a s t a k e n in t h e t h i r d s a m p l e (Table 1). T h e s e stones h a v e possibly u n d e r g o n e
all t h e d i f f e r e n t t r a n s p o r t a n d a b r a s i o n
phases connected w i t h t h e glacier.
Theref o r e t h e most r o u n d e d stones a n d t h e g r e a t e s t
v a r i a t i o n in m a t e r i a l a r e to be f o u n d in this
sample. A small p r o p o r t i o n of t h e m a t e r i a l
m a y h a v e c a r r i e d on t h e s u r f a c e of t h e glacier
so t h a t it h a s not b e e n a b r a d e d at all.
We do not k n o w e x a c t l y w h a t e f f e c t tidal
w a t e r s h a v e h a d on t h e r o u n d n e s s of stones
in t h e T a k u Inlet area, w h e r e t h e r e a r e s t r o n g
tidal c u r r e n t s (cf. S l a t t a n d H o s k i n 1968, p.
434—435), n o r t h e significance of o u t w a s h
processes in proglacial s t r e a m s (Alimen 1961
according to F l i n t 1971, p. 168).
These
processes m a y h a v e a f f e c t e d t h e d e g r e e of
r o u n d n e s s to some e x t e n t b e f o r e t h e p u s h
moraine was formed.
Conclusions
C o m p a r e d w i t h t h e m a t e r i a l b r o k e n off
t h e original rock a n d t h e stones deposited
e a r l i e r d u r i n g t h e glacial p h a s e t h e r o u n d n e s s
of stones increases v e r y c o n s i d e r a b l y in t h e
course of t h e i r being t r a n s p o r t e d b y t h e T a k u
Glacier to its m a r g i n . In g e n e r a l it m a y be
said, t h e r e f o r e , t h a t t h e r o u n d i n g e f f e c t of
a t e m p e r a t e valley glacier a n d its w a t e r s on
t h e m a t e r i a l it t r a n s p o r t s is v e r y g r e a t , even
w h e n t h e distance over w h i c h t h e materialis t r a n s p o r t e d is short.
P e r h a p s t h e most i m p o r t a n t observation
m a d e in this s t u d y is t h a t t h e r o u n d n e s s of
t h e stones d e p e n d s on t h e i r original s h a p e
a n d size in w h i c h t h e y loosened f r o m t h e
rock face. Of course, t h e f i n a l s h a p e of t h e
stones d e p e n d also on t h e rock t y p e a n d
erosional processes to w h i c h t h e y a r e s u b jected (cf. B l e n k 1960).
94
Matti
Seppälä
It is intended later to study experimentally,
with the aid of a ball mill-like device, changes
in the shape of stones, and so follow the
rounding of stones of a given shape as they
pass t h r o u g h d i f f e r e n t rounding processes (cf.
K u e n e n 1959).
Acknowledgements
— Professor M a y n a r d M.
Miller, Head of the 12th Glaciological and Arctic
Sciences Institute, 1971, on t h e J u n e a u Icefield
m a d e it possible for me to t a k e p a r t in the I n stitute. T h e staff of t h e Institute assisted me in
m a n y ways.
Mrs. Leena Kiiskilä m a d e the f i n a l d r a w i n g s
of t h e figure. Mr. Christopher G r a p e s t r a n s l a t e d
t h e m a n u s c r i p t into English.
My participation in t h e S u m m e r I n s t i t u t e w a s
m a d e possible by scholarships f r o m t h e National
Science Foundation (U.S.A.), Leo and Regina
Wainstein's Foundation (Finland) and t h e National
Research Council for N a t u r a l Sciences (Finland).
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