Length of a Degree of the Terrestrial Meridian. 223 



2. Let the annexed ellipse represent the generatrix which being 

 revolved round its minor-axis jop, would generate the earth : during 



^' 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 : 

 The axes ee, and pp, would rep- 

 resent respectively, the equatorial 

 and polar diameters of the earth. 

 Let the former of these axes be 

 denoted in length by 2fl, and the 

 latter by 26 ; and the abscissa CP, 

 of the point m, by x', and the 

 corresponding ordinate Pm, by y. The angle eN?n, made by the 

 normal Nz, with the plane of the equator, is called the latitude of 

 the place m ; and we shall represent this latitude by 4-. The ob- 



a — h 

 lateness of the earth is measured by the ratio , which we will 



put equal to a ; and then we shall have 6=(1 — a)a, which being 

 combined with the equation of the ellipse as found in treatises on 

 conic sections, will give 3/== (1 — a) ^(a^ —a;^)^ (1)^ for the equa- 

 tion of our generatrix. 



3. From the properties of the ellipse, we know that the subnor- 



^' 

 nial is expressed by —x ; hence, by substituting the value of h in 



terms of « and a, as above expressed, we have PN = (1— a)^a;; 



y 



and since tang. 4^ — plvj by replacing PN by its value just found, and 



=v^ 



y by its value as given by (1) we shall have tang. 4-= v n-a^a-c 

 From the last expression we immediately deduce the equation, 



^' "^ 1-1- tang. -4(l-(x)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 



