567 
figure of the earth. 
be solved to the first order, yet by the examination of a 
simple hypothesis (that of a homogeneous fluid whose density 
is very small surrounding a central nucleus), it may easily be 
seen that in this case the spheroid is not so far protuberant in 
middle latitudes as the ellipsoid. If the attraction of the fluid 
be neglected, and the central mass be called m, the force on 
any point in the direction of x = — — ; that in the 
(* + *. + *)! 
direction ofy = 
■ my 
a 4 -?* 
if y s + * 
(x z + y 9 - + 2 *) 2 
Hence U 
f ’ that inz = r - 
m z 
(x* + y 1 + 2 1 ) 2 
+ * 2 =C; or 
v/(x l + y z + z z ) 
V 7 
i 2 o 
+ tT 
x z + v 
my 2 my / \ir L w 2 
" — C* l CJT 1 
T a } 
C ; whence 
a : 3 = 
v 2 -f 
6 7 T 4 wi 1 
C 4 T 4 
V 
nearly, 
(‘ 
% 
supposing small ; and this may be put under the form 
(a 3 — v 2 ) [n — p v 2 ), from which the proposition is evident 
I conclude therefore that the results of Captain Sabine’s 
observations, as far as they go, are in strict agreement with 
theory. 
I now come to the comparison of the expression for the 
length of a meridian arc with those arcs which have been 
most accurately measured. The influence of local circum- 
stances on these measures, it is well known, is greater than 
that on pendulum experiments ; and the discordances in dif- 
ferent measures are such that, in order to get a result of any 
exactness, we must confine ourselves to the longest arcs which 
have been measured with the greatest care. I have selected 
the following: l. Bouguer’s arc in Peru: 2. Lambton’s 
whole arc in India : 3. The French arc from Formentera to 
