1900.] 



on the Dynamical Theory of Heat and Light. 



379 



TJ 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 U 1 which is a focus of the ellipse 

 shown in the diagram. U 3 , C, U 4 being infinitely near to U 1} C, U 2 

 respectively are indicated by double letters at the same points. 

 Starting from U^ draw the geodetic \J 1 Q U 3 ; the two parts of which 

 U^ Q and Q U 3 are straight lines. It is interesting to remark that 

 in whatever direction we start from U x if we continue the geodetic 

 through U 3 , and on through U \ again and so on endlessly, as indicated 

 in the diagram by the straight lines U 1 Q TJ 3 Q' XJ 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 



Fig. 1. 



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 

 Uj and U 2 , draw the geodetic I Q, and produce it through Q on the 

 ellipsoidal surface. It must cut the arc A' C A at some point 

 between U 3 and U 4 , and, if continued on and on, it must cut the 

 ellipse A C A' 0' A successively between "O^ and U 2 , or between U 3 and 

 U 4 ; never between U 2 and U 3 , or U 4 and U 1 . This, for the extreme 

 case of the smallest axis zero, is illustrated by the path I Q Q' Q" Q'" 

 Q IV Q v in Fig. 2. 



§ 32. If now, on the other hand, we commence a geodetic through 



2 o 2 



