248 Mr. Ivory on the Properties of a Line of shortest 
In this equation s and z increase together from zero. If it 
be expanded we shall obtain a formula for computing s as in 
the foregoing instance. I shall not however take the trouble 
of further developing the expression, because it is not proper 
to be employed in the research of the figure of the earth. The 
reason is, that the small arc z is determined. by its cosine; so 
that a minute error in the latitude ~ would occasion an exces- 
sive variation in x When a geodetical line is perpendicular 
to the meridian, the variation of latitude is at first proportional, 
not to the length measured, but to the square of the length ;- 
and therefore it cannot safely be employed in so delicate a 
research as the deviation of the figure of the earth from a 
sphere. 
In the most general case when the geodetical line is inclined 
to the meridian in any proposed angle, we must make, 
sin « = sin f cos z. 
And here it is evident that the determination of the are z will 
be liable to the same objection as in the- perpendicular to the 
meridian, unless sin « is considerably different from sin 6. It 
follows therefore that a geodetical line must not make a great 
angle with the meridian, at least if we employ the difference 
of latitude in the research. The inclination to the meridian 
ought not to exceed 45°. On this supposition the are z will be 
tolerably well ascertained, and the formula (C) will be suffi- 
cient for finding the length of s by means of that arc. The 
expression would however be a little complicated on account 
of the number of quantities that enter into it; but as an in- 
stance of such an oblique measurement has neither actually 
occurred, nor can any good reason be given for carrying it 
into execution, I shall not pursue the subject further. 
I have now considered very particularly the problem of the 
figure of the earth as it depends upon the lines measured on 
the surface, and the observed differences of latitude. It fol- 
lows that observations made in the direction of the meridian 
are the most advantageous for obtaining the values of the 
quantities sought. When the lengths measured extend only 
to a few degrees, we may use the differential equation before 
found, viz. d (l+e2)du 
s= —, 
(1+ e? cos? wu)? 
instead of the integral. In this case, ds or s is the length 
measured; du, or u — J, the difference of latitude in degrees ; 
and if m denote the degrees in the arc equal to the radius 
(57°°29578), then — , Will be the radius of a circle in which 
an are equal to s contains w —/ degrees. Hence if P be the 
polar 
