330 Prof. Cayley on the Geodesic Lines 



according to the distance between the two parallels) : at a point 

 of contact with the parallel the 

 curve is, of course, at right 

 angles to the meridian; say 

 this is V, a vertex of the geo- 

 desicline, and let the meridian 

 through V meet the equator 

 in A ; the geodesic line pro- 

 ceeds from V to meet the 

 equator in a point N, the 

 node, where A N is at most 

 = 90°; and the undulations 

 are obtained by the repetition 

 of this portion VN of the 

 geodesic line alternately on 

 each side of the equator and 

 of the meridian. 



I consider in the present paper the series of geodesic lines 

 which cut at right angles a given meridian A C, or, say, a series 

 of geodesic normals. It may be remarked that as V passes 

 from the position A on the equator to the pole C, the angular 



distance A N increases from a certain determinate value (equal, 

 n 



as will appear, to - 90°, if C, A are the polar and equatorial 

 A. 



axes respectively) up to the value 90° ; and it thus appears that, 



attending only to their course after they first meet the equator, 



the geodesic normals have an envelope resembling in its general 



appearance the evolute of an ellipse (see fig. 1 and also fig. 2), 



Fig. 2. 



the centre hereof being the point B at the distance BA= 90°, 

 and the axes coinciding in direction with the equator B A and 

 meridian B C : this is in fact a real geodesic evolute of the meri- 

 dian C A. The point a is, it is clear, the intersection of the 

 equator by the geodesic line for which V is consecutive to the point 



A (so that Z.B A= (l - j)^0°); and the point 7 is the in 



