401 
rent, having thus yielded to the first onset, collected its strength to force 
its opponent back. On the 9th there was a lull, followed by a north 
wind lasting up to the 11th, during which time the barometer rose 
7-46 lines, and the thermometer fell 15°4. On the evening of the 11th 
there was a fresh lull; and on the 12th the equatorial current set in 
again, as is shown by the wind getting round to 8. W., the thermometer 
rising above Zero, and the barometer falling 6-06 lines.’ ; 
‘<Yours very sincerely, 
“‘Rogert H. Scorr. 
“¢ Rev. Professor Haughton.” , 
Proressor Hennessy, F.R.S., read the following paper :— 
ON CLAIRAUT’S THEOREM. 
Laprace has shown that this theorem follows whatever may be the 
density of the interior parts of the earth, provided it consists of similar 
concentric strata, and that the form of the outer stratum is ellipsoidal. 
In the “Philosophical Transactions” for 1826, Mr. Airy (the present 
Astronomer Royal of England) has presented an equivalent result; more 
recently, Professor Stokes has shown that we can deduce the law of va- 
riation of terrestrial gravity without any hypothesis whatsoever as to the 
earth’s interior structure. He assumes merely that its surface is sphe- 
roidal, and that the equation of fluid equilibrium holds good at that sur- 
face. In vol. yi. of the ‘(Cambridge Mathematical Journal,” Professor 
Haughton presented a demonstration, founded upon the same assump- 
tions as those of Professor Stokes, and in which he uses certain propo- 
sitions relative to attractions which had been enunciated by Gauss and 
Mac Cullagh. While studying the labours of those mathematicians, it 
appeared to me that the question could be entirely divested of the hydro- 
statical character, and that Clairaut’s theorem may be directly deduced 
from the equations to the normal of any closed surface, without any 
considerations as to the physical condition of the matter forming that 
surface. Thus every surface concentric with the earth, and perpendi- 
cular 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 Z=0. Then from the equations of the normal we have 
aL aL aL aL 
Ln Ly vat dx “A dz 3 
when the resultant of these forces is perpendicular to the given surface. 
If we represent by V the potential of the earth on the particle in ques- 
tion, by w the angular velocity of rotation, we have 
dV 
Ge oe OW 4 % 
Aaa, t we, gt a Tn’ 
PROC, RB, I. AA—VOL. VII. 3 L 
