After a historical review of the theory of the Earth's magnetism, the author indicates how the theory may be improved and gives results of his own calculations for the Epoch 1842

Fritsche, H.

English

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1NASA TECHNICAL MEMORANDUM
	
NASA TM-77100
THE ELEMENTS OF THE EARTH'S MAGNETISM FOR THE
EPOCHS 1600, 1650, 1780, 1842 AND 1885, AND
THEIR SECULAR CHANGES DERIVED FROM OBSERVATIONS
CALCULATED USING THE COEFFICIENTS OF THE
GAUSSIAN "GENERAL THEORY OF EARTH'S MAGNETISM"
Dr. H. Fritsche
Translation of "Dic Elemente des Erdmagnetismus fuer die Epochen
1600, 1650, 1700, 1780, 1842 and 1885, and Ihre Saecularen
Aenderungen, berechnet mit Huelfe der aus alien brauchbaren
Beobachtungen abgeleiteten Coefficiented der Gaussischen "All-
gemeinem Theorie Des Erdmagnetismus"," St. Peterburg, A. Yakobson
Printer, 1899, pp. 1-9.
d
of
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
WASHINGTON, D.C. 20546
	
NOVEMBER 1982
(NASA-Ta-77100) THE ELEMENTS OF THE EARTHIS
!MAGNETISM FOR THE EPOCHS 1600, 1650, 1700,
1780, 1842 AND 1885, AND THEIR SECULAR
CHANGES DERIVED FROM OBSERVATIONS CALCULATED
USING THE (National Aeronautics and Space	 G3/46
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NASA TM-	 100
4. Two as *ANNA THE' ELENH;NTS OF THE EARTHS MAGPE- s, Ram e
TISM FOR THE EPOCHS 1600 0 1450, 1700, 1780, 1842 November 1982
a. r«am"M a,«a/.uM C•aAND 1885, AND THEIR SECULAR CHANGES DERIVED FRCle
OBSERVATIONS CALCULATED USING THE COEFFICTMTS OF
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translation of 'Die Elemente des Erdmagnetismus fuer die Epochen 1600, 1650,
1700, 1780, 1842 and 1885, and Ihre Saeculo- •en Aenderungen, berechnet mit
Huelfe der aus allen brauchbaren Beobachtungen abgeleiteten Coefficienten der
"Allgemeinem	 ","
t
ssischen	 Theorie Des ErdmagnetismusSt Peterburg, A.
0bson Printer, 1899, pp 1-9-
/i I^N•M'
After a historical review of the theory of the Earth's
magnetism, the author indicates how the theory may be improved
and gives results of his own calculations for the. Epoch 1842.
M R« We,& (MIMM RI	 IL 01
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OF POOR QUALM
THE MHOUS OF THE EARTH'S NRGWISM FM THE EPOCHS
1600, 1650, 1700, 1780, 1842 AND 188S, AND THEIR SECULAR
C[IAi MS I1 UM FPO4 OBSERVA71OW CALCUI ATED USIM THE
COEFFICIENTS OF THE GMMIAN 't
	 iL THEORY OF EARTH'S
MAGNETISM'
Dr. H. Fritsche*
CHAPTER 1
Introduction
In the atlas for the Earth's magnetism published by Gauss
and Weber in 1840, on page 32 of the text one finds the following
citation "this means that the Earth's magnetism has been completely,
determined for the first time by its elements (just like planetary
and cometary trajectories), that is,by the 24 numbers given in
Chapter 40. Similar determinations will be repeated in the future
and because of the changes which the magnetic state of the Earth'
experiences, their details will be researched. 	 Data based on
experience will be collected to 
- wrtite the history of
the Earth's magnetism. This is a goal which would be impossible
	 -
without any theory by gust experience alone. Experience alone does
include the changes in individual effects but never effects which
abt on the whole. The correctness of these observations will become
apparent during the course of the following paper, not only for the
future but also to a great degree for the past, that is, for the
centuries between 1600 and 1900. The Earth magnetic observations
which were performed during this interval are often scanty, inaccur-
ate and incomplete (for example, only one element, the declination,
has been observed). This means that understanding the overall mag-
netic state of the Earth (nine elements, the potential, the ideal
	 }
distribution of the magnetism on the Earth's surface, the forces X,
Y, Z. the declination, the inclination I; the horizontal intensity T
Director Emeritus of the Royal observatory in Poking.
1
and the total intensity J) for 1600-1900 can only be based on the
Gauss theory. Other methods to reach this goal only will be
incomplete. For example, the attempt made by Dr. L. A. Bauer 0 to
empirically determine the secular motion of a freely movable magne-
tic needle at its center of gravity is based on long term observa-
tions at a few locations (London, Paris, Rome, etc.) and this covers
only a small part of the Earth's surface. As we will see later on,
this leads to an incorrect result that everywhere on . the Earth's
surface the North Pole of the needle rotates in a clockwise direct-
ion as seen from its point of suspension.
In addition, I believe that the publication "on the analytical
form of the magnetic attraction of the Earth expressed as a function
of time" by the Carlheim-Gyllenskoeld Stockholm, 1896, is
erroneous because Dr. Carlheim-Gyllenskoeld made almost no compari-
son of his hypothesis and the observed magnetic elements. For
instance, his provisional theory for the Epoch 1600 results in a
declination for England which is 16 0 different from
what was observed. 	 Because the author did not use the
Gauss theory formulas themselves but differential formulas in order
to go back from the 19th Century, because he did not give the
numerical details of his calculation (his data are incorrect). The
hypothesis of Dr. Carlheim-Gyllenskoeld about the changes in the
coefficients in time wa s examined by me later on in Chapter 3 and
was found to be untenable. Also the results of the paper "founda-
tions of the Gauss theory ► and phenomena of the Earth's maanetiam
in 1829" written by A. Ermann and H. Peterson at the request of the
Royal Admiralty in Berlin in 1874 are not usable because their work
is incomplete. They only calculated 24 coefficients of the theory
instead of 46 as I have proven are required to satisfactorily des-
cribe the observations. The residual average deviation of the cal-
culation from the observations for the forces X. Y, Z are too large
in these calculations, that is, t0.115 GE (see page 24 of the paper
mentioned above). My calculations for thb same-area over which the
Contribution to the knowledge of the secular variation of the
Earth's magnetisi by Louis A. Bauer, doctoral dissertation Berlin,
1895. Dr. Fritscte, Elements of the Earth's N agnetism 1600-1885.
2
^^ s
i
observations extend (longitude from zero to 360 0 , latitude from
about +70 to -70°) and for the Epoch 1842, only amount to tO.055
GE, that is, barely 1/2 of the above.
The papers of Dr. A. Schmidt (Goths.) about the Earth's magne-
tism are not yet completely published unfortunately. However, I
would here like to discuss a few aspects of publications no* avail-
able to me with which I do not agree.
The majority of scientists who deal with the Earth's magne-
tism have tolerated the view for many decades that the Gauss theory
of the Earth's magnetism is only an interpolation method and not
a theory. This opinion was especially reported by the unsuccessful
calculations of Erman, Peterson, Quintus, Julius and Neumeyer.
They all determined only 24 coefficients gh and since they did not
obtain satisfactory agreement between theory and observations in
this way, they expected a future improvement according to a method
not yet discovered. I expressed my views in 1897 in a paper called
"The Determinations of the Coefficients of the Gauss General Theory
of the Earths Magnetism for 1885 and the Relationship of the Three
Earth Magnetic Elements" On pages 1 and 2 and I would :tike to add
the following here because Dr. A. Schmidt is a proponent of this
improvement method of coefficients gh according to the writings
(page 21C) published in the Annals of Hydrography and Maritime
Meteorology, January 1898. He only considers this theory as an
interpolation method (cf. Dr. A Petermann, Geographic Communications
1898, vol.vli, p.4). The general theory of the Earth's magnetism esta-
blished by Gauss is, therefore, a theory because Gauss developed the
law of the effect of the magnetic elements of the Earth on the
points of the Earth's surface (sphere or ellipsoid surface) (accord-
ing to their position with respect to changes), using the fundamental
law of interaction of magnetic masses (proportional to the masses
and inversely proportional to the square of the distances). From
this there results a very substantial simplification of the problem
In the following I will call this paper 111897 Paper".
3
when the theory is applied to reality. The theory gives the anal-
ytical expressions for the determining equations for the coeffi-
cients gh (see paper 1897 page 10). If one were to not use the
analytical expressions from the theory and if one were to repre-
sent the three force components X. Y, Z by pure interpolation for-
mulas as a function of longitude and latitude, then more than 200
coefficients from observations would have to be calculated, that is,
four times as many as when one uses the Gauss theory which only
rejuires 46. This estimate is based on the following:
For example, the sine series (51) (Bessel series) page 73 of
my paper 1897 are of the same kind as the formulas (10-21) for X.
Y, Z given in Chapter 2 below, because the latter can also be given
in the form of equations (51) in paper 1897 * . Now, the general
interpolation formula (52 and 53) (page 74 of the paper 18975 de-
rived from (51), which represent the horizontal intensity T as a
function of longitude A and the inclination i contain about 50
coefficients and exponents which were calculated from the (observed)
formulas (51). But since (52) and (53) already close with three A,
whereas the formulas for X. Y and Z go up to five A inclusive,
general empirical interpolation formulas which represent X. Y, Z
as functions of longitude and latitude would contain about
3.r0 =Jia constants to be determined from observations. In the
paper 1897 I gave a fast and easy method for calculating the required
number of coefficient gh (46) and I demonstrated high degree of
agreement between the Gaussian theory and experience. This means
that one can no longer raise the objection of incompleteness as
	 k
long as we do not have much more and better observations than now. 	 1
The predictions for the these new observations at the present time
are not optimistic.
I would like to discuss the method mentioned in the atlas for
the Earth's magnetism by Gauss and Weber paragraph 43 page 32. It
suggests plotting differences between the calculation (using 	 yr.
For example, ,t.+tl^ ':,1K •yat^if^.}•iwltt;'•.^^l►^+R^t(W'+J
4
i
iI	
`•
ORIGINAL. PAGE 9
OF POOR QUAMY
approximate coefficient gh and observations on maps or to present
them in tables in order to make the foundation for an improved
calculation of the gh. I remarked about this in the paper 1897
pages 1 and 2. I belitve that this is unmanageable and this sug-
gestion of Gauss and Weber made my predecessors take the wrong
course when further developing the Gauss theory. Weber and Gauss
intended *
 to improve the six elements t, 42 tl 'fV :F !s
of a planetary trajectory which is calculated from three complete
observations. In the event where more observations are made, the
quantities are improved by the amount at, &4 &is fit, •tr sts
using differential equations. One obtains the elements j, is *, 44L*r,4
by a tedious calculation using a large number of formulas without
the method of the least squares because no excessive determination
equations are present. The improvements ♦t, At, .60, ^fy;
however, are found by using the method of least squares from six
determination equations which are relatively easy to treat. This
method not only represents a simplification but is a necessity for
planetary calculations.
The problem of improving the coefficients gh of the Earth's
magnetism of fourtb order and the special calculation of the gh of
the fifth and sixth, etc., orders is quite different. 'The coeffi-
cients gh themselves are obtained from linear equations of the form
W. tt)
which are given in Chapter 2. The corrections •j .!A
if one wishes to carry out such an operation, have to be calculated
according to exactly the same formulas (X, Y, Z), by substituting
A.! &A
In his introduction to his atlas, Gauss states that Weber wrote
the explanation for the maps, that is, the text, with the except-
ion of paragraphs 21-25 which were written by Dr. Goldschmidt.
Regarding his improvement methods, Gauss expresses himself more
carefully in paragraph 33 of his general theory than does Weber.
Gauss says there "there is no doubt that an improvement calculation
according to theseobservations would result in a much better
agreement and such a calculation would not be difficult. The length
of this calculation would still be frightful, even if one would use
similar measures for abbreviations as are used by astronomers in
order to improve the elements of the planetary and cometary traj-
ectories".
5
f^^P^ ft
for gh and .ill, ^, s ah, a	 gmLRy►
for klmKLM. By collecting differences between the calculations
and the observations in maps or tables, which Weber, Neumeyer, etc.
recommend, and for which A. Schmidt makes available his so-called
normal elements, one has gained nothing. It is much better if the
desired unknowns gh are calculated from the beginning if suffi-
cient material is available. The comparison of the elements of the
	 }
i
Earth's magnetism with those of geodesy which Schmidt carried out
is not appropriate because the elements of the planets and geodesy
i
are much more constant over time and their phenomena follow much
more regular laws than those of the Earth's magnetism which in mauy
aspects is similar to meteorology.
CHAPTER 2
A. Formulas for the calculations
	 5
I will now give the formulas which I have used and the obser-
vations which are the foundation of my calculations. The formulas
were almost all given in the paper 1897 which is why I will only
recapitulate here. The symbols are those used by Gauss. The poten-
tial V divided by the Earth's radius R is at^P'" lP
The northern component X,-the western component Y and the vertical
Z are expanded in the following cosine and sine series*
11 • X.+iii ^V^ ^; ^ a^t^ eW !^ ^^K !^ ^k^ilill^^
where the quantities	 are functions of geographic lati-
tude 0 and are determined b y the following
 formulas (X). (Y). (Z).
111/!t s Ln a s	 is used for abbreviation, and we use the letters
it -812 IF
	
for the following functions which only depend on
X is the eastern longitude from Greenwich and u is the angular
separation from the astronomic North Pole so that u + ^ a 900.
6
twxll-
OIAL PAGE A
OF POOR WAM
even powers of the quantity it which are simple symbols, that is:
t^4i►;slat • ^ ts.4,Ji^• ^' 	 •	 t+i0b^
t=41f)3^	 ,
t^+- ^,/yam s i "Ulf rtst^M'Ka^ t	 '^i
tR•+RJIJJ s	 -22 is#-4Ar1f%A
it
t04im a	 ^•-	 .Ojlijt;ti! • fir
04 im
0.46 .^'	 j^• i+d	 •M7a i^
Values of Mt f :r Gauss, units of the third decimal..
M. a' ts' !l' w' »' aN • 7r• t0' !0' Iw' /M' i!i' ^^^ Ny' /JO' 1i^ 1!i
s -»t -^i -t 9-11	 0." +U -1 -ta -H -17 -l0 +!► +16 +u +18 *71Karl -!! -t7 -8 at +09 #4 ♦til -t -31 41 -10 Al +t +ii +60 01 *01
1700 NN -ti -t +7t +18 #" +11 -! It -bt -W -#f	 +t7 ++1 +t0 #.#A
1700 .111 -41 Of +" 07 +#A +N . -N -11 a" to -li	 +11 of +6f +41
1161 -u7 -a -30 +:i ♦It +tt +37 + t' -11 -)f -3t -/! N► +40 414 +tt +ItMt-131 -t0 41 01 441 +004 -M -09 -att -40 -17 +11' Of 01 +40 +30.
Values of m,-St'#1 s!, Gauss. units of the third decimal.
u. 1!' at0' 3(t' 1y' f0' tat' Y►' t0^ !!' 1aM' 100 /t0' /^^ li0' /1r' KI' 1!0' ,
>t+N -fN •!fl •1►3 *N►7 +ttt +1 f1 +t7 • 33 ^I t7 ^K f-li►ar -4t +)! +^ +^Mrt +^ + h'
/Ot1 -*tt •t7! -4++N►1►+^1;1111 +K •71'+!11-h1'*rt! •H {)L NIt+IUf 11!11 a#.
JtWil! its -1t ws 0011 N!) + #: -N -137 .17)-Its -1l +17+mtMro+1!! +1t!178# I-M -)N -11 +/A +#!/ +110 + 70 -4t -at -01-J8 -Os -4 +17 +nl *01 tu)
18#1-or-no - 0+ll#tt41 +t" my + $ -110 -t7) -j07 -r 1 +1►!	 to • J
JJts'-6il--3p -4 #i77+IK±tft v-) -,lt-tK-!#1 .tft -it +w+!)t+ avi • #r
Values of	 Gauss . units of the third decimal.
We /0' Ito )0' ♦0' l0' bat' 1f' f e M. OW a 1W 1)0' 1W 1W 1W 1W
Kh -I87 -lit -W OOM -N t!# +M+10#1" +0 -tat q :40	 +1t
I•t>r -M -ft0-l7Y-w -tf +f# +fail #41 ++18# +#J •W-4)444 .01 -410'0 + Al
1711.19 -891- y0.11f -wmt #AV #M +100 +!# -4f .4144#0 in .0 -i0 ♦ !1700 at" 438 49 fit - y +Mt #M +in +Nt +34 -#1-M,M) •*I -M -tj -M
Jt1►t -J11- aii-tt! •ft + tt +#/ +IK ++131 +3at .0444 .4" • 0 •149 ♦ )iJ#tS-!#at-3i# -Kt •i8#-i7 raitJ +#7#+1!1+!11+ •180-t17-uM)t fM+14 f!#
7
QUMM
As to be expected according to the equations above for . ,F As
 
04
the quantities sl and 92
 are much larger than s on the average and
all three cannot be ignored.
B. Observations forming the foundation of the calculations
The observations which I used for the Epoch 1885 are
given in my paper 1897. The observations which I used as a basis
for my calculations about the theory for the Epoch 1842 are taken
from the known collection of Sabine. In all of the previous and
following tables and formulas, we have the following definitions:
X eastern longitude from Greenwich; 4 the geographic latitude, u the
angular separation from the astronomic North Pole so that *faZM;j?'• *
the potential divided by the Earth's radius R 94*4pj •t , the Ideal
distribution of the magnetism on the Earth's surface; X the northern,
Y the western, Z the vertical component, d the declination, i the
inclination, T the horizontal intensity and finally J the total
intensity.
TABLE 1. Epoch 1842. Observed declination 4
-	 is' '6e jr la#? lam' /W ta Soe 1W i##' 
i
^'	 i0i +^',r'« ►'.^°
to # iv ar # #4 *-rt #-n u •1 #•! #-* #1.* ^^ 	 im s
jo +a +ti off ff-* #,N tm $#-1 *-a at-u Jf •4-04 #.n t##.N
+rS	 Y M.	 ♦ 	 •
♦
	 MLV-
#0 W+8 0-1
 
M •t #III M00P.M0.0 104T NO !mot W+W #
to #aa it Ji## i0 8 -y tf+J 1-2 1%4	 0•1 a" W#4.1
yo +lt +4 uM Jf+i 1 -J No a 4-4 "-a a •3 of •i 4 M evil ' *Wir
JN -to WK up* I0 1W WI 31+0 #.y 16-1  JI-A 1-1 Is l
Jt# •io low "M We *0 u-1 34-0 »-t "-! 1-4 9-0 IsA 43
l^ 1	 I N 1'o*" I^ ""W" - 0-0 64%#•34." # M
8
MEN
OF Pw Qm=
TABLE 2. Epoch 1842. Observed inclination•
^t fN 41" V4 74i
* tll^,J tA1 ^i
:iq 3
iitW
its
	 }
fft
as
TABLE 3. Epoch 1842. Observed horizontal intensity
W ne• OW t	 u^' ^a' WK
N^ ♦  ii! .3}t
N
^t1t RN iii
w Nr 4	 •^,
+Pt 4%1 l.lx U Nt I%%% U /	 4A1
Dr. Fritsche. Elements of the Earth's magnetism 1600-1885.
(end)
A
9


The elements of the Earth's magnetism for the Epochs 1600, 1650, 1700, 1780, 1842 and 1885, and their secular changes derived from observations calculated using the coefficients of the Gaussian "General Theory of Earth's Magnetism"

The elements of the Earth's magnetism for the Epochs 1600, 1650, 1700, 1780, 1842 and 1885, and their secular changes derived from observations calculated using the coefficients of the Gaussian "General Theory of Earth's Magnetism"

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