on an Oblate Spheroid. 339 



where the columns marked with an * show respectively the 

 longitude of the node, and the length (or distance of node from 

 vertex), for the geodesic lines belonging to the different values of 

 the argument V, 



The remarks which follow have reference to the stereographic 

 projection of the figure on the plane of the equator, the centre 

 of projection being the pole (say the South Pole) of the spheroid. 

 It is to be remarked that if a point P of the spheroid is projected 

 as above, by means of an ordinate into the point Q of the sphere 

 radiusOK( = OA),then projecting stereographicallyas to the sphe- 

 roid and the sphere from the south poles thereof respectively, the 

 points P and Q have the same projection. And it is hence easy 

 to show that an azimuth a at a point of the meridian (parametric 



Q 



latitude X', normal latitude X, and therefore tan A/= -r-tan X) is 



projected into an angle («) such that 



: sin X' 

 tan (a) = - — — tan a. 

 v ' sinX 



In fact in fig. 3, if we take therein O K, C for the axes 

 of x } z respectively, and the axis of y at right angles to the 

 plane of the paper, and if we have at P on the surface of the 

 spheroid an element of length P R at the inclination u to the 

 meridian P K, then if x, y, z are the coordinates of P, and 

 x + 8%, y -f By j z + Bz those of E, we have 



Bx= p cos a sin X, 



Bz = — p cos a cos X, 



By = p sin u, 



and thence 



tanc* = 



VBx' + Bz^' 



Now, if the meridian and the points P, E are referred by lines 

 parallel to z to the surface of the sphere radius A, the only 

 difference is that the ordinates z are increased in the ratio C : A ; 

 so that if the projected angle be (a), we have 



tan(«)=- By 



and then projecting the sphere stereographically from its south 

 pole, the angle in the projection is = (a). And according to the 

 foregoing remark, the angle («) thus obtained is alsQ the projec- 



