994 MATHEMATICAL AND PHYSICAL SCIENCE. (Diss, VI. 
observing the earth’s magnetism and its changes. 
The instruments devised by them were a De- 
clination Instrument and a Bifilar Magnetometer, 
The first is a heavy magnetized bar some feet in 
length, suspended by a bundle of parallel silk fibres. 
It is suffered to place itself in the magnetic meridian, 
and the small displacements due to hourly, annual, 
or irregular fluctuation, are ascertained by viewing 
with a telescope the reflection of a fixed scale of equal 
parts placed at some distance in a small mirror 
which is attached to the magnetic bar. The bifilar 
instrument is for ascertaining changes of intensity in 
the earth’s magnetism. It is a bar like the last, sus- 
pended by two parallel threads or wires. By twist- 
ing round the points of double suspension, the bar 
is forced by torsion into a position at right angles 
to the magnetic meridian, where it is held in 
equilibrium by the force of the earth’s magnetism, 
and that of the torsion of the wires. Supposing the 
latter force to be constant, if the former vary, the 
bar will change its position, which is observed by the 
mirror and telescope as before. 
~ (900.) To these instruments, Professor Lloyd, of Dublin, 
added another for measuring the vertical component 
ofthe magnetic intensity, It isa magnet suspended 
horizontally on knife-edges like those of a balance, 
Its deflection from the horizontal line indicates va- 
riations in the vertical portion of the earth’s mag- 
netism. 
(901.) The dip is determined in the usual way ; the va- 
riations of dip, however, are ascertained by comparing 
the variations of horizontal and vertical intensity, 
(902.) Gauss did not content himself with suggesting 
The Got- new forms of apparatus, and recommending them to 
bc others. With the active assistance of M. Weber, he 
bberva. erected, in 1833, at Gottingen, a magnetic observa- 
tions. tory free from iron (as M. de Humboldt and Arago 
had already done on a smaller scale), where he 
watched with patience the incessant movements of 
the newly-constructed needles or bars. It was from 
the same observatory that he sent telegraphic 
signals to the neighbouring town, thus showing the 
practicability of an electro-magnetic telegraph (856). 
He farther instituted an association (magnetischer 
Verein), composed at first almost entirely of Ger- 
mans, whose continuous observations on fixed Term- 
days extended from Holland to Sicily.1_ The maryel- 
lous coincidence of the occurrence of even the minute 
irregularities of the earth’s magnetism was thus 
more fully established. 
(903.) The Mathematical Theory of Terrestrial Magnetism 
; of Gauss? is intended to replace all arbitrary assump- 
tions whatever as to the distribution of magnetism Gauss’s 
in or over the earth. It proposes to express by an 
empirical law, involving as little of hypothesis as 
possible, the direction of the freely suspended mag- 
mathema- 
tical theory 
of terres- 
trial mag- 
netic needle and its directive force at any point of the netism. 
earth’s surface, the data being of course derived from 
a limited number of observations. 
The theoretical assumption with which Gauss 
starts is merely this,—that the elementary force of 
magnetism varies inversely as the square of the dis- 
tance, and that it is distributed over the matter of 
the terrestrial globe in a way or according to a law 
presumed to be entirely unknown. 
Following up the methods employed by Laplace 
and others, for representing the ai ons of a 
sphere or spheroid, he proposes to discover the form 
of that remarkable function (sometimes called the 
Potential)* whose differential coefficients express the 
resolved components of the total magnetic force. 
This quantity V, and also its differential coefficients 
(representing the attractions in given directions), 
may always be expressed by a series with indetermi- 
nate coefficients, which is known to converge more 
or less rapidly; and since the component forces 
or attractions are given by observation, the coeffi- 
cients of the terms of the series representing them 
may be deduced from a comparison with the data. 
The convergency is greatest if the magnetic matter 
be disposed towards the centre of the sphere, least if 
it reside near its surface. 
Retaining quantities of the fourth order, there are 
twenty-four constants, which, rigorously speaking, - 
may be deduced from complete observations of the 
three magnetic elements (888) at only eight stations 
anywhere situated on the surface of the globe. With 
great patience and skill, Gauss collected as manyavail- 
able data as possible for determining these constants 
with accuracy, and he thence deduced by calculation 
the values of V, those of the declination, horizontal 
and vertical force, and that of the dip for all points 
of the globe, The beautiful charts which he caused 
to be constructed show a remarkable general ac- 
cordance with the complex facts of magnetism as 
then known. But in many instances more accurate 
data have since been obtained applicable to the im- 
provement of the empirical theory. Perhaps the 
most surprising fact which Gauss considers to be 
demonstrated is this, that the average amount of 
magnetic force associated with every cubic yard of 
the earth’s volume (supposing the distribution uni- 
form) is equal to that contained in six saturated steel 
bars each a pound in weight. 
1 The volumes of their publications, from 1836 to 1839, contain much interesting information on the history of this subject. 
2 Allgemeine Theorie des Erdmagnetismus. Leipzig, 1839. 
8 See Art. 877. This function V represents the integral 
S = were dm is the attracting or repelling element, and » its 
distance from the point acted on. Gauss represents in his charts the values of the function V at different points of the earth’s 
surface, The lines of equal yalues of V are everywhere perpendicular to the direction of the needle, and the horizontal inten- 
sity is inversely as the distance between two adjacent lines, 
(904.) 
(905.) 
Theorems 
concerning 
attractions, 
906.) 
