Length of a Degree of the Terrestrial Meridian. 223 
2. Let the annexed ellipse represent the generatrix which being 
revolved round its minor-axis.pp, would generate the earth: during 
2’ the revolution the extremities ee, 
/* of the major axis, generate the 
circumference of the terrestrial 
equator ; and any point m, of the 
ellipse, generates the circumfer- 
ence of a parallel of latitude: 
py The axes ee, and pp, would re 
resent respectively, the equatorial 
and polar diameters of the earth. 
Let the former of these axes be 
denoted in length by 2a, and the 
latter by 20; and the abscissa CP, 
of the point m, by x; and the 
corresponding ordinate Pm, by y. The angle eNm, made by the 
normal Nz, with the plane of the equator, is called the latitude of 
the place m; and we shall represent this rein by J. The ob- 
A 
lateness of the earth is measured by the ratio —, which we will 
put equal to « ; and then we shall have 5=(1—a)a, which being 
combined with the equation of the ellipse as found in treatises on 
conic sections, will give y?=(1—«)?(a?—«2?), (1), for the equa- 
tion of our generatrix. 
3. From the properties of the ellipse, we know that the subnor- 
2 
mal is expressed by -;7; hence, by substituting the value of 6 in 
terms of a and a, as above expressed, we have PN=(1—«)?z; 
and since tang. L=py by replacing PN by its value just found, and 
a? -— gy? 
y by its value as given by (1) we shall have tang. )= Vis + a)2a2 
From the last expression we immediately deduce the equation, 
a? 
* = T+ tang.?L(1 —2)? (2). 
4, Every terrestrial meridian being an ellipse equal in all respects 
to that which, by its revolution, generates the spheroid to which we 
have assimilated the figure of the earth, it follows that the law of 
the curvature of this ellipse, will be the same as that which governs 
the curvature of any meridian of the earth. Of all the circles that 
