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 



1.3 . 1.3.5 



shall obtain 1-f ^^ (2a — a^^sin.^ ■^j-' — T-^-5(2a — a2)-'sin.*4'-f &c. 



Now if a 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 

 \jf the length of the degree of the meridian of the equator (see 8), 

 formula (4) may be written under the form, L'=L' + 3asin.^%j^L% 

 (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 Mecanique Ce- 

 leste ; and which w^e 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. Tt will now be proper to explain the methods by which the 

 values of a 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 a and a ; and 

 therefoi-e, 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 a 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 l^ .iations 

 as are deemed to be the most correct. 



