THE SEMI-GRAPHIC METHOD OF ANALYSIS
FOR 7 DAYS OF TIDAL OBSERVATIONS
by Vice Adm iral A. D os S a n t o s
Brazilian Navy
F ranco
1. — INTRODUCTION
In the Adm iralty Tidal H andbook No. 3 Commander S u t h o n s has just
developed a sem i-graphic method o f harm onic analysis o f tides fo r observa­
tion periods as short as 24 hours.
The author considers one w eek’s hourly observations as in fact a set
o f groups o f 24-hour observations whose analysis requires that the relations
between the principal constituents for neighboring stations be known. The
mean o f the values for H and g found through these 24-hour analyses is
then taken as the best answer obtainable.
I
believe, however, that if a continuous 7-day observation recording is
available, it is better to analyse it directly, reducing to the utmost the num ­
ber o f hypotheses to be admitted. W ith this in mind I have developed an­
other m ethod o f analysis, based also on the tidal surface, but having an
entirely general character since the hypotheses to be accepted are the same
as those w hich allowed me to establish the m ethod I described in the Inter­
national Hydrographic R eview , July 1964 (Vol. XLI, No. 2).
Taking advantage of the properties o f the tidal surface, I w ill show
that the com putations are simpler than those o f the num erical method.
2. — THE DAILY PROCESS
If w e take seven days o f hourly observations, it w ill be possible to cut
the tidal surface by six planes as shown in fig. 2 A. 24 points chosen along
each o f these sections, w hich w e shall call “colum ns” , w ill then give us 144
observations to analyse.
In m y article in the IHR (July 1963) the expression (2 j) represents the
intersection o f plane ( 2 <j0 with the tidal surface assumed to be generated by
0
1
2
I
+
t
F
ig .
2A
î 3
I
4
5
one single constituent. But that article showed that if the sections are chosen
to satisfy the condition
a /b = 1/24
we shall have
q — p a /b — 15° n
n being the constituent’ s suffix. That being the case, if S 0 is considered a
constituent for which q = p = r = 0 , the contributions from all the consti­
tuents w hose suffix is n could be written in the form
y ' = 2 R cos (15° nt -f p c /b — r0)
cn
If the distance between the colum ns, measured along the d axis, equals
one day, we will have c /b = d, d being the num ber o f days from a suitably
chosen origin. Thus the preceding expression m ay be written in the form
y ; = I R cos (15° nt + p d — r0)
or, by developing,
y„' = E R cos ( p d — r0) cos 15° nt
cn
— 2 R sin (p d — r0) sin 15° nt
cn
or, b y putting
2 R cos (p d — r 0) = X '
( 2 a)
cn
and
— 2 R sin (p d — r0) = Y '
(2 b)
we w ill have
y'„ = X ' cos 15° nt + Y ' sin 15° nt
Consequently, if we consider all the constituents, the height y o f the tidal
surface at hour t will be expressed by
y — 2 (X ' cos 15° nt -)- Y ' sin 15° nt )
(n = 0, 1, 2, 4)
(2c)
C
If we assign y suffixes indicating the hours corresponding to the heights y,
we shall see that expression ( 2 c) can be expressed in m atrix form by the
redundant system in table 2-1. If we call M the matrix o f this system and
(y ) and {Z } the colum n vectors of the know n and unknow n terms respectiv­
ely, the system of normal equations will be expressed by
MT (y ) = MT. M. { Z }
(2d)
But the product of each row vector o f MT and the colum n vectors o f M will
always be expressed by one o f the four follow ing expressions :
23
1
23
2 cos 15° nt cos 15° n't = — 2 [cos 15° ( / i + n ') f -)- cos 15° (n— n ')t]
f=o
23
2 t= o
1
23
J^ cos 15° nt sin 15° n't = — ^ [ s i n 15° (n -(-n ')f — sin 15° (n— n ') t ]
23
2
1
(2/)
23
2 sin 15° nt cos 15° n't = — 2 [sin 15° ( n-\-n')t 4- sin 15° (n— n ')f]
¢=0
2 *=o
23
(2e)
(2o)
23
— ^2 sin 15° nt sin 15° n't — — 2 [cos 15° ( n-\-n')t — cos 15° (n— n')#] (2 b)
t=o
2 <=°
where n' allows us to identify the rows (0, 1, 2, 4) of the transposed matrix.
T able
n = 0
QOS 0°
r
co s 15°t
n = 4
n = 2
n = 1
sin 15°t
c o s 30°t
sin 30°t
co s 60“t
sin 60°t
y» -
1
1, 000
0, 000
1, 000
0, 000
1, 000
0, ooo
yi
1
0, 966
0, 259
0,866
0, 500
0, 500
0, 866
y?
1
0, 866
0, 500
0, 50C
0, 866
-0, 500
0, 866
y3
1
0, 707
0, 707
0, 000
1, 000
-1, 000
0, 000
y*
1
0, 500
0, 866
-0 ,5 0 0
0, 866
-0, 500
-0,86 6
y5
y6
yi
1
0, 259
0, 966
-0 ,8 6 6
0, 500
0, 500
-0,86 6
1
0, 000
1, 000
-1 ,0 0 0
0, 000
1, 000
0, 000
1
-0 ,2 5 3
0, 966
-0 ,8 6 6
-0 , 500
0, 500
0, 866
- 0 J 500
0, SSC
—G, 500
-0 , 866
-Q, 500
0,866
0, 707
0, 000
-1 , 000
-1, 000
0, 000
ye
1
y*
y io
1
-0 , 707
1
-0 ,8 6 6
0,500
0 ,5 0 0
-0 , 866
-0, 500
-0 ,8 6 6
y ii
1
-0 ,9 6 6
0,259
0, 866
-0 , 500
0, 500
-0 ,8 6 6
y«
(a)
(b)
(a)
(a)
(a)
à to
L
2 -1
y 23
(b)
XJ
x ; ?•
(a)
De t = 12 à t = 23 les valeurs se répètent
(a) avec même signe
(b) avec signes con tra ires
F rom t = 12 to t = 23 the values are
repeated :
(a) with the sam e sign
(b) with contrary signs
_ Y1
Applying the general expression
sin 15° ( T + l ) N /2
cos (a-j-15“ NT)
sin 15° N /2
to the second m embers of expressions (2 e) through ( 2 h), it is easy to
conclude that the results o f the sums for T == 23 and N = n ± n'
0 are
equal to zero, but if n = n', N — n — n' = 0, and we shall have :
2
t=o
cos (a + 1 5 ° NO
23
2
cos (a + 1 5 ° Nf) = 24 cos a
¢=0
Under these circum stances expressions (2 f) and (2 g), where a = ± 90°,
w ill always be zero, while expressions ( 2 e) and ( 2 h), where a = 0 °, will
equal 12. Then noting that expressions (2c) and (2h) for n = n ' correspond
to the diagonal o f matrix MT M, we logically deduce that the product is a
scalar matrix w hose expression is
MT • M = 12 U
w here U is the unit matrix. Expression (2 d) may then take the form
2MT • { y } = 24 U {Z }
(2i)
This expression represents the rigorous solution to the problem and, as
expression (2i) shows, this solution requires m ultiplying the colum n vector
made up o f the values o f y by matrix 2MT whose elements are 2 cos 1 5 °nt
and 2 sin 15 °nt. Under these circum stances we would have to make quite
lengthy com putations with the help o f a standard desk computer. Thus
we must look for a less rigorous solution but one whose results are
acceptable in practice and w’hose com putations are not very lengthy. A
reasonably satisfying solution is found by replacing matrix 2MT by another
m atrix whose elements will be 0 , ± 1 and ± 2 , corresponding to the best
approxim ations o f the values o f 2 cos lb °n t and 2 sin lb °n t. If we call
this new m atrix KT, we w ould norm ally find matrix K by rounding off
values o f 2 cos 15°7if and 2 sin lb °n t taken from matrix 2M. Table 2-II
com prises the elements o f m atrix K where the colum n vectors correspond­
ing to n = 0 have not been m ultiplied by two. Actually this duplication
was not necessary since all the cosines have, in this case, the value -f- 1
and all the sines the value 0. In table 2-II we see in addition that symbols
sn and c„ have been used to designate the colum n vectors derived respectively
from the values of the sines and o f the cosines.
T able
{c 0}
{Cl}
{* 1}
{C2}
{* 2}
{ C4>
{*4}
2
2
2
1
1
1
0
1
1
1
2
2
2
2
2
1
1
1
0
1
1
1
2
2
0
1
1
1
2
2
2
2
2
1
1
1
0
2
2
1
0
1
2
2
2
1
0
1
2
2
2
1
0
1
2
2
2
1
0
1
2
0
1
2
2
2
1
0
1
2
2
2
1
0
1
2
2
2
1
0
1
2
2
2
1
2
1
1
2
1
1
2
1
1
2
1
1
2
1
1
2
1
1
2
1
1
2
1
1
0
2
2
0
2
2
0
2
2
0
2
2
0
2
2
0
2
2
0
2
2
0
2
2
—
—
—
—
—
—
—
—
—
—
—
2-II
—
—
—
—
—
— I
—
—
—
—
—
—
—
—
—
—
1
1
2
2
2
2
2
1
1
1
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
By writing the system in table 2-1 in the form
{y } = M • {Z }
we can write
KT • {y } = KT • M {ZJ
Introducing into this expression the values for the elements o f matrix K
(table 2 -II) and those for m atrix M (table 2-1) we obtain the follow ing
final system :
r { c 0} T
K i}T
(M T
- {C2} T
{M *
{y h
{»}
{y }
{y } ►=
{!/}
{y }
24.000
0
0
0
0
0
0
0 24.250
0
0
0
0
0
0
0 24.250
0
0
0
0
0
0
0 25.856
0
0
0
0
0
0
0 25.856
0
0
0
0
0
0
0 24.000
0
0
0
0
0
0
0 27.712
r x '°i
x;
Yi
-< X '2 ►
Y'a
X '4
. Y 'J
This expression shows that we have arrived in the end at a system o f
equations whose m atrix is diagonal, w hich means that the use of the table
2 -II m ultipliers permits us to isolate com pletely the groups with the same
suffix. If we had also considered the third-diurnal and sixth-diurnal
species, w e w ould have seen that they also would have been isolated.
Designating by G any one o f the elements of the diagonal matrix we
can express the equations in (2 j) by :
(2* )
{cny • {y } =: c x ;
or
( 21)
{*n}T • {y } = C Y '
as the case may be. But if we make
(2m )
{ c « r • {y } = x n
and
( 2 n)
(s« )T ' {y } -(2k) and ( 2 /) will becom e
( 2 o)
x ; = x „ /c
and
(2 p)
y; = y „ /c
Expression (2 /) also shows that, except for X 4 and Y£, the values of C are
equal for each pair o f functions
and Y£.
At this point it w ill be easy to explain the operations in the daily
process. W e first take the values o f y in fig. 2 A and with them construct
the m atrix seen under (a) in the form given at the end of this article.
W e perform the operations indicated in expressions (2m) and (2n) using
the m ultipliers in table 2 -II and record the results under ( b ) of the form
m entioned. The daily process is then at an end.
3. — THE MONTHLY PROCESS
As in the developm ent o f the num erical m ethod o f harm onic analysis
for seven days of hourly observations, w e w ill use the term m onthly process
for the com bining o f daily values o f Xn and Y n, the first step in isolating
the constituents.
F rom expressions (2a), (26), (2o) and (2p) we can write :
X„ =
j
C 2 R cos (prf — r0)
cn
and
Y„ =
— C I R sin (p d — r0)
cn
If we take as initial day (d — 0) the day w hich corresponds to the inter­
section o f the first “ colu m n ” (fig. 2 A) with the d axis, the value o f the
central day will be 2.5 and the preceding expressions can be transformed
into
X „ = C E R cos [p ( d — 2.5) + 2.5 p — r0]
and
Yn =
- C 2 R sin [p (d — 2.5) + 2.5 p — r0]
or, by m aking
— r 0 + 2,5 p =
(3a)
— r
and by developing, we obtain
Xn =
C E [R cos r cos p ( d — 2.5) - f R sin r sin p (d
2.5)]
cn
and
Y„ =
— CE [R cos r sin p ( d — 2 .5 )— R sin r cos p (d — 2.5)]
c»
These tw o expressions may be written in the general form :
F„ = C E [a cos p ( d — 2.5) -)- (3 sin p (d — 2.5)]
(3i>)
a = R cos r
(3c)
|3 = R sin r
(3d)
where
and
for F . = X n, and
=
(3c)
R sin r
and
(3 /)
cos r
for F„ = Y„
As in the numerical method for seven-day analysis (see IHR July 1964),
we have in the present case daily values o f F„ to com bine. But the six values
we have here have only the contributions o f the constituents o f suffix n.
T his allow s us to consider each species separately w hich considerably
facilitates the isolation o f each constituent. Let us take fo r instance the
diurnal constituents Kx and O t, taking into account also the disturbing
effect o f Q j on these constituents. By introducing into expression (3 b) the
p values for constituents K1( O x and Qi and the successive values d — 0, 1,
2 , 3 , 4 , 5 , w e shall have a group o f expressions which can be written as
the follow ing single m atrix :
c (F '>
0.999 0.448 : 0.106
1.000 0.787 ! 0.536
1.000 0.796 : 0.944
These three lines
are repeated in
inverse order and
with the same sign
■>
f
an,
ao!
■i
►+
a Ql
»
>
— 0.044 0.894 ; 0.994
— 0.026 0.617 ! 0.844
— 0.009 0.220 ! 0.329
These three lines
are repeated in
inverse order and
with the opposite sign
”
r
(3kt
(3g>
■*
frji
In the m atrix which multiplies colum n vector { a } the lines are the values
o f cos p (d-— 2.5) from d = 0 to d = 5, w hile the lines o f the m atrix w hich
multiplies colum n vector {^} are the values o f sin p (d — 2.5) from d = 0
to d = 5. W e are now in a position to find the multipliers ± 1 w hose signs
are those o f cos pKj (d — 2.5) for com bination {0 } of table 3-1, and those
o f sin pG1 (d — 2.5) for com bination { b } o f this table, as w e have already
explained for the num erical m ethod (see IHR, July 1964). These same
multipliers will be used to isolate the semi-diurnal constituents. As for
isolating the quarter-diurnal constituents, it has been necessary to try
several filters, the most effective being that indicated by {2 } in table 3-1.
In this special case these m ultipliers have been established b y using the
signs and values o f cos pM84 (d — 2.5).
3-1
Daily m ultipliers
T able
d
( 0}
{b }
(2}
0
1
2
1
1
1
1
1
1
1
1
1
1
1
1
1
2
1
1
2
1
3
4
5
—
—
—
In practice the products
( 0 } T • {X 0} = x 00
{ 0}T •
' {Xx} = x 10
{6 }T •
' { X x } = X 1B
{ 0 } T ■ { X 2} = X 20
{ f t p • { X 2} = X 26
{ 2 p • { X 4> = X 42
{ 6 p • { X 4} = X 46
W T • {Yi> = Y 16
W T - {Y i} = Y 10
W T ■ {Y 2} = y 26
{ 0 } T • {Y 2} = y 20
{ f t p . {Y 4} = Y „
{2 p • {Y 4} = y 42
w ould be arrived at by using Table 3-1 and the values o f X„ and Y„ taken
from the form under ( b ). The results, inscribed under (c), are already the
know n term s of the equations to be solved.
4. — FORMATION AND SOLUTION OF THE EQUATIONS
Let us continue with the example o f isolating constituents K x and (^ .
By pre-m ultiplying (3^) by { 0 } T and { b }T w e have :
1
C
and
1
{ 0 }T • {F x} =
(5.998
4.062 : 3.162) {a }
{ b } T • {F i} =
(— 0.158
3.462 ; 4.334) {£ }
(4a)
(410
If in the first o f these equations we replace Fx by X j and by Y x in the
second equation, then from (3c), (3/) and (3/i) we obtain
— X 10 =
C
(
5.998
j 3.162) {R cos r)
(4c)
3.462 ; 4.334) {R cos r}
<4d)
4.062
and
Y 16 =
(— 0.158
Now m aking Fx = Y a in (4a) and Fx = Xj in (4 b) we will have, from (3d),
(3e) and (3/i) :
1
— Y 10 =
C
(5.998
4.062
:
3.162) {R sin r }
(4e)
3.462 ; 4.334) {R sin r}
(4/)
and
X 16 =
(— 0.158
Thus if we designate by I'/C the first m em ber o f either (4c) or (4e), and by
I" / C the first member o f either (4cf) or (4 /), and if further we designate by
z', z " and z " ' either the elements o f { R cos r }, or those o f { R sin r }, we
can write the general system o f equations :
C being the diagonal element o f m atrix (2 j) which corresponds to X x and
w hich is equal to the one corresponding to Y x. Since its value is 24.250
we can write the preceding equations in the follow ing w ay :
1
/ v \ _ z»>f 3162 \
24.250 \ I" J
\
4.334 J
5.998
— 0.158
4.062
3.462
M ultiplying both m embers o f this expression by the inverse matrix o f
the second member and transposing, we obtain :
In the form under ( d ) we again find the figures for the m atrix o f the first
term o f the second member o f this expression w hich is used for isolating
Kj and O x. It is then easy to verify that we have V = X 10 and I” = — Y 16
for the com putation o f the values of R cos r, and that V = Y 10 and V = X 16
when we are dealing w ith the R sin r com putation. The values found by
using these expressions in the form give us the provisional values R cos r
and R sin r which are recorded under ( g ). For constituent Qi they must
be corrected for the second term o f the second member of (4 g) where
z '" equals R ' cos r' or R ' sin r', as the case may be. My description o f the
num erical m ethod already m entioned gives the theory w hich allows us
to obtain the approxim ate values of z ’" and we find under (e) and (/) in
the form the details of this com putation. Having found the values of
R' cos r' and R' sin r' we com pute the corrections to R cos r and R sin r
by the form ula under (h ). These corrections are procured by follow ing (4<j0.
The problem is resolved in exactly the same way for the sem i-diurnal
constituents as for the diurnal ones. Thus we need only clarify the isolation
process o f the quarter-diurnal constituents. Here we are considering only
constituents M 4 and MS4, and that being so, it is not necessary to take the
corrections into account. In addition we must note that according to
expression (2j) the values o f the diagonal elements corresponding to X 4
and Y 4 are different. Consequently when we form expressions similar to (4c)
and (4 /), the values o f C will be different and we will not have the same
repetition o f figures in the form ulae under (d) for M 4 and MS4.
Having found the values for R cos r and R sin r fo r the chosen
constituents, the rest of the com putation presents no particular interest.
However one must note the fact that the central day is expressed by a
fractional number and that the decim al part o f this fraction is introduced
into the com putation o f g in the form 0.5 p. As a matter of fact the quantity
6 °.5 w hich is seen under ( e ) when a is being com puted derives also from
the multiplication o f the fraction o f a day by the difference in daily speed
of M2 and N2.
P o r t : B a ie d 'A r a t u (B r é s i l ) - A ra tu h a rb ou r
X
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
126
168
207
228
223
190
142
90
50
33
42
73
115
152
196
222
223
200
153
105
65
47
54
85
1
106
155
198
228
234
216
172
120
70
34
27
46
83
132
175
210
233
218
182
130
81
48
40
65
Date c e n tr a le - ce n tr a l date : 3 ^ /8 /1 9 4 7
2
3
4
5
X0
74
121
170
212
238
234
200
147
94
46
28
30
58
100
153
199
225
232
204
162
110
63
38
42
52
93
142
192
230
238
220
180
122
68
37
20
34
74
47
76
120
168
209
238
239
204
160
99
57
27
28
60
100
146
190
224
228
204
160
113
65
42
45
60
102
147
190
223
239
230
186
140
82
47
32
50
80
124
170
208
226
220
184
140
93
58
3 189
3 203
3 180
3 143
3 204
3 276
Ï21
172
210
232
220
184
136
83
47
36
X
Y
164
254
228
247
225
180
00
10
19 195
-
1 298
-4 8 4
X2
Yx
Xx
-1 07
- 46
- 28
35
115
153
-
lb
-
lb
y
384
1 080
1 786
2 281
2 491
2480
Y*
15
70
44
34
46
9
-1 1 4
- 36
- 20
4
64
82
4b
20
2b
42
-1 0 502
4 625
4 002
8637
316
-3 2 0
10
2b
20
4b
53
3
42
10* R sin r
6 6 7 0 X 10 + 7 825Ylfc
3 0 4 X i0 - 11 555Ylfc
- 11 496Y 2h
6 4 4 5 X 20 + 8 679Y 2h
-6 7 5 8 X „a - 1 0 6 7 5 Y „
8 9 16 X I|2 + 3 357Y „fc
v « = 3 3 5 !8
= -1 2 3 ,6
5 ,9
" w. 2 = 6 ?5
X,
6
122
106 R c o s r
Kx
Ox
m2
S2
M*
MS„
2
2 373
2 316
1 942
1 365
679
- 38
-
7 825 X lb +
11 5 5 5 X lb +
11 4 9 6 X 2h
- 8 6 7 9 X ab+
12 3 2 7 X Hh - 3877X ^ +
t e o i-g o j* »
C C
6 670Y10
304Y10
6 445Y20
5 852Y,2
7 721 Yh2
< *.*“ 8*2 )"
=
<7 =
= 2 1 2 ,8
0 2 = 212, 8
a = 212,8 f^-kJd+W),,. 0,221 c2=k*(l+W)>1= 0,221
(•) Si nous n 'a v o n s p a s d 'a u t r e s r e n s e ig n e m e n ts - If no o th e r in fo r m a tio n e x is ts : g 0i - g 0i = g «2 -
= 0 ,1 9 3
__ ________________________________ <n____________________
o*
R2
R
R* * c R
tan r
r
r* = r + G
37, 83
4 943, 59
6, 2
7 0, 3
1, 37
1 5 ,5 4
0, 235
- 0 , 865
13, 2
139, 1
226, 0
351. 9
cos
r1
- 0 .6 9 5
0, 990
s in r '
R ' cos r '
R 1 sin r*
- 0 ,7 1 9
-0 ,1 4 1
- 0 ,9 5 2
1 5 ,3 8 5
- 0 , 985
-2 ,1 9 1
(8 )
Kx
Ox
M,
s.
M,
MS*
m
il c o s r
0, 309
-1 ,2 3 8
-1 ,2 7 2
0, 436
4, 870
5, 987
-5 3 ,1 6 9
-2 7 ,5 4 5
0 ,2 9 4
1 ,1 7 9
- 1 9 ,5 7 0
6 ,7 0 8
...
...
m R* c o s r*
-
R cos r
R sin r
m ' R ' sin r
R s in r
R2
tan r
4 , 576
7, 166
-7 2 ,7 3 9
- 2 0 ,8 3 7
1, 280
1, 743
0, 861
- 0 , 032
4 6 ,0 0 7
20, 932
- 0 , 304
1, 219
- 2 ,7 8 7
- 0 .9 5 5
0, 557
1, 187
4 3 ,2 2 0
1 9 ,9 7 7
0 ,6 6 8
0, 163
2 1 ,2 5
5 2 ,0 5
7 1 6 8 ,9 3
8 3 3 ,2 6
2, 08
3 ,0 6
0 ,1 2 2
0 ,1 6 6
- 0 , 594
- 0 ,9 5 9
0, 522
0, 094
(h)
So
Kx
Ox
m
2
V
1 /1
C o r r . m o is
C o r r ,jo u r
9 .8
209 , 3
2, 0
143, 7
22, 2
309, 3
153, 5
231, 1
311, 2
u
w
0, 5 P
r
221, 1
7 ,0
-1 5 ,5
0, 5
7, 0
115, 2
8, 1
...
-1 1 ,4
9 ,4
335, 8
- 1 ,8
206 , 1
1, 069
1, 116
1 ,1 9 3
4 ,6
121, 3
1, 111
111, 1
0, 981
1, 111
7, 2
0, 981
84, 6
4
6
s{
1+ w
f (1 + W)
R
H
-1 2 , 2
149, 3
86
Si
M.,
m sh
217, 2
341, 3
285, 1
. ..
-1 7 ,4
0, 0
136, 2
1 1 8 ,8
0, 862
28, 9
34
311, 6
- 3, 6
-2 4 ,4
2 7 ,6
311, 2
0, 962
0, 962
1 ,4
1
3 3 5 ,8
- 1 .8
-1 7 .4
-1 2 .2
5 ,4
3 0 9 ,8
o, m
0 ,8 6 2
0 .8 4 6
1, 7
2
1 2 3 ,6
(2V + u*u 76? 6
(V ■h u^=,2 1 4,8
3 V „ - ■2V., = 40, 2
f.i- 1, 069
1 ,1 7 0
Kx : wf
-1 5 ,5
= 0, 124
Wf,x ■
W = 0 ,1 1 6
Sa : w /
-14*9
w = -1 7 ,4
W /f „ - =-0 ,1 1 8
w ;* -0 ,1 3 8
N, :
w=
1+W=
5?9
1,147