318 



WILSON AND MOORE. 



this quantity is interpreted as the differential of arc of the Gaussian 

 spherical representation of the surface. No spherical representation 

 of the same simple sort as obtained in three dimensions exists for higher 

 dimensions, though (78) is common to all dimensions. ^^ 



31. Geodesies. ^^ The shortest lines on a surface are determined 

 by means of the first fundamental form alone and might properly 

 have been treated before. We shall however take them up at this 

 point. To minimize 



we follow the ordinary procedure of variation : 



55 = 



dsj I 



sdarsd.XrdXs + 2^rsClrs5dXrdXs 



ds dxt 

 Now by (61) and X^ = ^sdrs^^, 



Srst -^ dXrdxJbXt — 2Zirsd { ttra ^ ]8Xr 



dXs 



ds 



)\(^^8Xt-2^r^dXr 



ds 



By (34) 



dX _ y, dXrdxs _ ^ dxs . ^ 

 as oxs ds 05 



/• s 



X a<^p<i) ^ 

 ds 



S.XrA^*) + 2 



sq 



r s 



X(9^X(«>. 



Hence the condition ds = gives, when we set t for r in the second 

 term and r for q in the third sum, 



31 To have a spherical representation which will generalize we should mark 

 on the unit sphere the great circle which is the trace upon the sphere of the 

 diametral plane parallel to the tangent plane of the surface instead of the point 

 which is the trace of the normal. This representation would therefore be the 

 polar of the ordinary spherical repi-esentation. 



32 In this section we merely follow Ricci's Lezioni. 



