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. Geodesics.^^ 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 



S = j[l,raarsdXrdXa] 



we follow the ordinary procedure of variation : 



8s = ~ 

 ds 



is J [ 



1,rs8aradXrdXa + 22; 



attraBdXrdXs 



f[ 



- 'Erst dXrdXabXt — 21^rsd ( tt-s -~ jSXr 



cw oxt \ ds I 



Now by (61) and Xr = 'Lgars^^, 



Sttra 



By (34) 



8s = 



M' 



'TSt 



dxt ds 



d\r, 1 



dK _ y dXdxa _ ^ dxs 

 as dXg as ds 



r s 

 . 9 J 



X„a(p«) 



dxa 

 ds 



= Xa\ra'\^'^ + 



•'sq 



r s 



X(3^X(^\ 



Hence the condition 8s = 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 representation. 



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



