on an Oblate Spheroid. 



331 



ter'sectioii of the meridian C B by the geodesic line for which V is 

 consecutive to the point C ; and its position will be in this way 

 presently determined. I was anxious, with a view to the con- 

 struction of a drawing and a model, to obtain some numerical 

 results in relation to a spheroid of considerable excentricity, and 



I selected that for which — =1 (polar axis =| equatorial). 



A 



Before proceeding further, I remark 

 that Legendre's expression " reduced 

 latitude " is used in what is not, I think, 

 the ordinary sense; and I propose to 

 substitute the term "parametric lati- 

 tude : " viz., in figure 3, referring the 

 point P on the ellipse by means of the 

 ordinate MP Q to a point Q on the 

 circle, radius OK( = OA, fig. 1), and 

 drawing the normal P T, then we have 

 for the point P the three latitudes, 



X = Z. P T K, normal latitude, 

 X" = Z p K, central latitude, 

 A/ =^-QOK, parametric latitude ; 



viz. X' is the parameter most convenient for the expression of the 

 values of the coordinates x, y (% = A cos \',y = C sin X') of a point 

 P on the ellipse. The relations between the three latitudes are 



C C 2 



tan X"= j- tan X' = -rg tan X, 



so that X", X', X are in the order of increasing magnitude. I 

 use in like manner /, I', l n in regard to the vertex V. The course 

 of a geodesic line is determined by the equation 



cos X' sin a= const., 



where X' is the reduced latitude of any point P on the geodesic 

 line, and a is at this point the azimuth of the geodesic line, or 

 its inclination to the meridian. Hence, if V be the parametric 

 latitude of the vertex V, the equation is 



cos X' sin a. = cos V 



(whence also, when X'=(>, « = 90° — V; that is, the geodesic line 

 cuts the equator at an angle =V, the parametric latitude of the 

 vertex). The equation in question, cos X ; sin a = cos /', leads at 

 once to Legendre's other equations : viz. taking, as above, A, C 

 for the equatorial and polar semiaxes respectively, and 8 for the 



excentricity 



icity,8=^/l- 



j2 ; and to determine the position of P 

 Z2 



