272 Mathematical Theories of the Earth. — Woodward. 
line can sometimes be accounted for by visible masses, but on the 
whole it must be admitted that we possess only the vaguest notions 
of their cause and a most inadequate knowledge of their distribution 
and extent. 
What is true of plum-line deflections is about equally true of the 
deviations of the intensity of gravity from what may be called the 
spheroidal type. Given a closely spheroidal form of the sea level and 
it follows from the law of gravitation, as a first approximation, without 
any knowledge of the distribution of the earth's mass, that the increase 
of gravity varies as the square of the sine of the latitude in passing 
from the equator to the poles. This is the remarkable theorem of 
Stokes, and it enables us to determine the form of ellipticity of the 
earth, by means of pendulum observations alone. It must be 
admitted, however, that the values for the ellipticity recently obtained 
in this way by the highest authorities, Clarke and Helmert, are far 
from satisfactory, whether we regard them in the light of their dis- 
crepency or in the light of the different methods of computing them. 
In general terms we may say that the difficulty in the way of the use 
of pendulum observations still hinges on the treatment of local anoma- 
lies and on the question of reduction to sea level. At present the case 
is one concerning which the doctors agree neither in their diagnosis 
nor in their remedies. 
Turning attention now from the surface, towards the interior, what 
can be said of the earth's mass as a whole, of its laws of distribution, 
and of the pressures that exist at great depths ? Two facts, namely, 
the mean density and the surface density, are roughly known ; and a 
third fact, namely, the precession constant, or the ratio of the differ- 
ence of the two principal moments of inertia to the greater of them, is 
known with something like precision. These facts lie within the domain 
of observation, and require only the law of gravitation for their veri- 
fication. Certain inferences also from these facts and others have long 
been and still are held to be hardly less cogent and trustworthy, but 
before stating them, it will be well to recall briefly the progress of 
opinion concerning this general subject during the past century and a 
half. 
The conception of the earth as having been primitively fluid was the 
prevailing one among mathematicians before Clairaut published his 
" Th^orie de la Figure de la Terre " in 1743. By the aid of this con- 
ception Clairaut proved the celebrated theorem which bears his name, 
and probably no idea in the mechanics of the earth has been more sug- 
gestive and fruitful. It was the central idea in the elaborate investi- 
gations of Laplace, and received at his hands a development which his 
successors have found it about equally diflacult to displace or to im- 
prove. From the idea of fluidity spring naturally the hydrostatical 
notions of pressure and level surfaces, or the arrangement of fluid 
masses in strata of uniform density. Hence follows, also, the notion 
of continuity of increase in density from the surface towards the center 
