226 Length of a Degree of the Terrestrial Meridian. 
the meridian, where it cuts the equator, having a greater degree of 
curvature than the equator ; which is demonstrated by a comparison 
of the radii of curvature of the meridian and equator at the points 
where they intersect each other. 
9. Expanding the radical part of formula (4) into a series we 
(20 —a?)*sin.4)+&c. 
Now if « be an exceedingly small fraction, its powers which are 
higher than the first, may be neglected, as being too small to affect 
materially the first two terms of the series; and if we denote by 
L/ the length of the degree of the meridian of the equator (see 8), 
formula (4) may be written under the form, L/=L/+3asin.2L/, 
(4’). The first term of the second member of this formula, being 
the length of the degree of the meridian at the equator, the second 
term is the increment which the length of that degree receives to 
make up the value of L in going from the equator towards the poles ; 
and hence the theorem as given by Laplace in his Mécanique Cé- 
leste ; and which we shall enunciate by translating his own words. 
“The increment of the degrees of the meridian in proceeding from 
~ the equator to the poles, is therefore proportional to the square of the 
sine of latitude.” 
10. It will now be proper to explain the methods by which the 
values of « and a, have been determined: Four distinct methods 
have been used for the purpose of solving the problem: First, by 
_adverting to formula (4) it will be perceived that the second mem- 
ber contains, besides the sine of latitude, the quantities « and a; and 
therefore, if the lengths of two different degrees be determined by 
direct geodetical admeasurement, and the latitudes of their middle 
points be determined by observation, by placing each of the meas- 
ured lengths, and the observed latitudes of the degrees, in formula 
(4), we can determine « and a, since we should have two equations 
with only two unknown quantities: We shall not here describe the 
geodetical operations by which the degrees have been measured, nor 
. the method of determining the latitudes of their middle points ; but 
will exhibit in a tabular form the results of such of these operations 
as are deemed to be the most correct, 
cided 
shall obtain 1+ 12 (2a —a?)sin.? | — 
