20 Lord Kelvin on the 



§ 30. Let AA', BB', CC, be the ends of the greatest, mean, 

 and least diameters of an ellipsoid. Let Ui U 2 U 3 U 4 be the 

 nmbilics in the arcs AC, CA', A'C, C'A. A known theorem 

 in the geometry of the ellipsoid tells us, that every geodetic 

 through Ui passes through U 3 , and every geodetic through 

 U 2 passes through U 4 . This statement regarding geodetic 

 lines on an ellipsoid of three unequal axes is illustrated by 

 fig. 1, a diagram showing for the extreme case in which the 

 shortest axis is zero, the exact construction of a geodetic- 

 through Ui which is a focus of the ellipse shown in the 

 diagram. U 3 , C, U 4 being infinitely near to U 2 , 0, U! 

 respectively are indicated by double letters at the same points. 

 Starting from Ui draw the geodetic I^QI^ ; the two parts 



of which UiQ and QU 3 are straight lines. It is interesting 

 to remark that, in whatever direction we start from TJ 1 , if we 

 continue the geodetic through U 3 , and on through JJ X again 

 and so on endlessly, as indicated in the diagram by the 

 straight lines U 1 QtJ 3 Q , U 1 Q // U 3 Q /// , and so on, we come very 

 quickly to lines approaching successively more and more 

 nearly to coincidence with the major axis. At every point 

 where the path strikes the ellipse it is reflected at equal 

 angles to the tangent. The construction is most easily made 

 by making the angle between the reflected path and a line to 

 one focus, equal to the angle between the incident path and 

 a line to the other focus. 



§ 31. Returning now to the ellipsoid : — From any point I, 

 between U l and U 2 , draw the geodetic IQ, and produce it 



