4.12. Gauss on Interpolation
251
sin (trr - ,x)x. (These follow since, e.g., cos (pn + ,x)x = cos (7T/2 + ,x)x =
cos 7Ta·cos (7T/2 - ,x)x + sin 7Ta·sin(7T/2 - ,x)x for the 7Tvalues of x we are
considering.) But Gauss notes that by calculation k sin 7Ta/2 = 1cos 7Ta/2, and
hence
~ (k
+ k cos 7Ta +
~ (l
+
1 sin 7Ta)
=
k,
k sin 7Ta - 1cos 7Ta) = I.
Thus in X" he finds that yA leads to the term k cos (7T/2 - ,x)x +
I sin (7Tf2 - ,x)x, and that oA leads to I cos (7T/2 - ,x)x - k sin (7T/2 - ,x)x.
To illustrate these ideas Gauss took data on the position of the asteroid
Pallas from Baron von Zach's tables. The data tabulated below from Gauss's
paper give the declination X of Pallas as a function of the right ascension x.
(The abbreviations Austr. and Bor. stand for the adjectives australis, southern,
and borealis, northern.) We notice there are twelve values of x which span
the period of 360°. Thus 7T= 12.
X
x
0°
30
60
90
120
150
180
210
240
270
300
330
6°48' Bor
1 29
1 6 Austr
0 10 Bor
5 38
13 27
20 38
25 11
26 23
24 22
19 43
13 24
= + 408'
+ 89
66
+ 10
+ 338
+ 807
+ 1238
+ 1511
+ 1583
+ 1462
+ 1183
+ 804
Now, to form the finite Fourier series for this function, Gauss broke up
the values of x into three groups of four each as shown below.
0°
a=
b= 90°
e = 180
0
d = 270
0
a' = 30°
b' = 120°
c' = 210
d' = 300
A = + 408
10
B=+
C = +1238
D = +1462
0
0
A' = +
89
= + 338
= +1511
D' = +1183
B'
C'
all
b"
c"
=
60°
= 150°
= 2400
d" = .3300
A" =-
66
B"
= +' 807
C" = +1583
D" = + 804
and then formed the functions
X'
= I' + 1" cos x + 1''' cos 2x
+
for each group. CP- = 4,
v
0' sin x
+
0" sin 2x
(4.65)
= 3; this is case (II) above where n = 1, m = 2.)
252
4. Laplace, Legendre, and Gauss
Thus he found three Fourier expansions:
Pro periodo
ubi y = 4x
prima
secunda
tertia
0
120
240
0
0
0
r
y
y
+779.5
+780.2
+782.0
-415.0
-404.5
-413.5
8'
yn
8"
-726.0
-721.4
-713.3
+43.5
+ 9.9
+11.7
0
+17.1
-20.3
-
one for each group.
Gauss now viewed each coefficient y, y', a', y", a" as a periodic function
of the variable y( = 4x) and formed its Fourier series. This gave him the
expansion for y,
.
j (779:5 + 780.2 + 782.0)
2
+ '3 (779.5 +
+ ~ (780.2 sin
780.2 cos 120°
+
120°
+
782.0 cos 240°) cos 4x
782.0 sin 240°) sin 4x,
and after simplification he found the expansions
y
780.6 1.1 cos 4x - 1.0 sin 4x,
y' = -411.0 - 4.0 cos 4x + 5.2 sin 4x,
a' = - 720.2 - 5.8 cos 4x - 4.7 sin 4x,
(4.66)
y" = + 21.7 + 21.8 cos 4x - 1.1 sin 4x,
a" = - 1.1 + 1.1 cos 4x + 21.6 sin 4x.
Now when these are substituted into the formula (4.65) above for X', and
a little manipulation is performed, there results the expansion
780.6 - 411.0 cos x - 720.2 sin x + 43.4 cos 2x - 2.2 sin 2x
- 4.3 cos 3x + 5.5 sin 3x - 1.1 cos 4x - 1.0 sin 4x + 0.3 cos 5x - 0.3 sin 5x
+ 0.1 cos 6X.84
Recall from our previous discussion that in the present case (II) Gauss
showed that the (fLn + m)th term in X" was -!-(k - I') cos tpn + m)x +
-!-(I + k') sin (fLn + m)x, where ym, am contained the terms k cos pnx +
I sin unx, k' cos unx + I' sin unx, respectively. Hence in this case, where
fL = 4,v = 3,m = 2,n = 1, wehavek = 21.8,1 = -1.1,k'
= 1.1,1' = 21.6;
and we see that the expression above reduces to 0.1 cos 6x, since I + k' = O.
Next Gauss chooses fL = 3, v = 4. Then we have his case (III), and 3n +
m = 6 + m, and n = 2. In this situation he divides up his 12 values into four
groups of three each and writes X' = y + y' cos x + a' sin x. He finds now
Pro periodo
prima
secunda
tertia
quarta
84
ubi y = 3x
0
90
180
270
0
0
0
0
y
+776.3
+786.0
+785.0
+775.0
y
,
-368.3
-414.5
-453.0
-408.2
8'
-718.8
-676.0
-721.1
-765.0,
Gauss III TI, p. 310. (The signum of 411.0 is erroneously given there as +.)