Intelligence and Miscellaneous Articles. 397 
Astronomer Royal of England) has presented an equivalent result ; 
more recently, Professor Stokes has shown that we can deduce the 
law of variation of terrestrial gravity without any hypothesis what- 
soever as to the earth’s interior structure. He assumes merely that 
its surface is spheroidal, and that the equation of fluid equilibrium 
holds good at that surface. In vol. vi. of the ‘ Cambridge Mathe- 
matical Journal,’ Professor Haughton presented a demonstration, 
founded upon the same assumptions as those of Professor Stokes, 
and in which he uses certain propositions relative to attractions 
which had been enunciated by Gauss and Maccullagh. While 
studying the labours of those mathematicians, it appeared to me 
that the question could be entirely divested of the hydrostatical 
character, and that Clairaut’s theorem may be directly deduced from 
the equations to the normal of any closed surface, without any con- 
siderations as to the physical condition of the matter forming that 
surface. ‘Thus every surface concentric with the earth, and per- 
pendicular to gravity, will possess the property of exhibiting this 
relation in the intensity of gravity at its various points. 
Let X, Y, Z represent the components parallel to the rectangular 
axes of the forces by which a point is retained at rest on a given 
surface whose equation is L=Q. ‘Then from the equations of the 
normal we have : 
y te xo =0, 7 il yah 
dx dy da dz 
when the resultant of these forces is perpendicular to the given sur- 
face. If we represent by V the potential of the earth on the-par- 
ticle in question, by w the angular sa of rotation, we have 
dV dV 
X= — wy “he ’ scape Ale 
and the above equations become 
Ee a Ce 
dy dx dx dy dy “dx}’ 
dV dl_dVdb_, al 
dzdx dx dz "ie 
If, in conformity with General Schubert’s* recent determinations, 
we assume the earth’s surface to be that of an ellipsoid, with three 
unequal axes, we should substitute for L 
Pe y 2 ies 
et ae 
or 
dL _ 20 di_y dl pind’ 
de @' dy @ dz @ 
* Memoires de ’ Académie Impégriale des Sciences de St. Hee, sér. 7, 
tom. i. 
