Mr Watts on the Elliptlcity of the Earth. ^89 
In order to find the elliptical figure of the earth, in which the 
greatest deviation in the measures of the lengths of the seconds'* 
pendulum is less than in every other elliptical figure, it will 
be requisite to seek such values of the unknown quantities in- 
volved in the problem, as correspond in the best manner pos- 
sible with the whole of the observations. This is what MM. 
IMathieu and Biot have done, by means of the method of least 
squares ; and i shall adopt the same method in the following 
investigation, on account of its superior advantages over every 
other mode of computation, in questions of this nature. 
Let Zj, Zj,, ^ 5 , &c. represent the lengths of the seconds’ pen^ 
dulum, as determined upon the different points of the British 
meridian ; let &c. denote the squares of the sines of 
the corresponding latitudes ; suppose that in the ellipse sought, 
the length of the seconds’ pendulum is expressed by a function 
of the form Z — a; -f- ^ sin ^L, or Z == a? -f ^ ; in which equation 
w represents the length of the seconds’ pendulum at the equa- 
tor, where the latitude L is nothing ; and y is the excess of the 
length of the seconds’ pendulum at the pole, above that at the 
equator : since in the elliptical hypothesis, the length of the pen- 
dulum is known to vary, as the square of the sine of the lati- 
tude; then by calling the errors of observations, &c, 
we shall have the following equations of condition : 
h — 
for if we compute the values of Sn ,' or sin ®L, for each of the 
places where the observations were made, the formula 
a; sin^L, will represent, in each place, the length of the se- 
conds’ pendulum ; then if we subtract this expression from the 
observed length of the pendulum, the difference jswill represent 
the etror of the elliptic hypothesis. 
By proceeding in this manner, with the results of the obser- 
vations at each station, as specified in the preceding table, w© 
shall form the seven following equations of condition : 
39,17146 — a; — 0,76136670 .y = 
39,16159 — a7 — 0,71419880 .y - Zg, 
