Geodesic Lines on the Earth's Surface. 359 



and the azimuth of the vertical plane exceeds that of the geo- 

 desic by 



2 



2 



§a= — cos u t sin a y {H cosw y cosa y + K sinw,}, 

 H = l 



(13) 



tan cr 

 K = o- — 2 tan \ <r. 



5. 



It is interesting to consider the case of geodesic lines starting 

 from a point on a spheroid of small excentricity and diverging 

 in all directions. First, to confine our attention to a single line, 

 it is well known, and may be readily inferred from the auxiliary 

 spherical triangle, that a geodesic touches alternately two paral- 

 lels equidistant from the poles — the difference of longitude be- 

 tween the successive points of contact being constant and some- 

 thing less than 180°, depending on the angle at which it cuts 

 the equator. Now suppose a line starting from (for simplicity) 

 a point on the equator with the azimuth a ; the osculating plane 

 at that point cuts the equator again at the opposite point N. As 

 a point P moves along the geodesic towards N, the angle tn of 

 the auxiliary spherical triangle increases from 0; and when it 

 becomes it, then a also becomes =ir, and P has reached the 

 equator, its longitude being, by (2), tt— \e*7r sin a. Since in (12) 

 H is negative for all values of a from to it, the geodesic lies 

 wholly on the south side of the osculating plane at the initial 

 point ; and its distance south, when <r = 7r and P is on the equator, 

 is \ e 2 cos air. We infer from this that all geodesies proceeding 

 from the same point on the equator have an approximately equal 

 length \ ire 2 (about 36' in the case of the earth) intercepted be- 

 tween the meridian through N and the equator. Consequently 

 the ultimate intersections of the geodesies will form an envelope 

 like the evolute of an ellipse, or the hypocycloid #*-f y*z=sk*. t N 

 being the centre of the curve. (Mr. Cayley informs me that this 

 property of geodesies is referred to in some remarks in Jacobins 

 Vorlesungen iiber Dynamik, Berlin, 1866, which, however, I 

 have not seen. It is noticed that if geodesic lines, starting from 

 a given point, intersect so as to form an envelope, then each line 

 is a shortest line only up to its point of contact with the envelope 

 and no further.) If the lines diverge from a point not on the 

 equator, the diameter of the envelope will vary as cos 2 w. 



6. 

 We shall now apply the results we have obtained to the geo- 

 desic line joining Cadiz and St. Petersburg. Referring it to 



