509 



2{h)g„bg-^^z= 



1 for n =z a, 

 O for n =1= rt 



For the sake of brevity we sliall write // for ilie determinant 

 formed of the tj^'K Further we shall avail ourselves of Christoffei/s 

 well-known symbols: 



lm 

 h 



lm 

 a 



=-k 





Ö-f" 



r/mi 



The definition of the line-element entails the definition of the 

 length of a vector v, with components v" : 



r» = 2:{ah) gab v" ^'^ 

 and the definition of the angle between two vectors v and w. 

 vw cos {viv) = ^{ab) gab ''" ^o^'- 



7. Let the points of a small rigid be given by their coordinates 

 relative to the centre: u", r«, lo" . . . {a = 1, 2, 3, n), these being 

 the components of vectors Ut v, w . . . which are of the order of 

 a vanishing quantity b. If the centre shifts from P to a neigh- 

 bouring point Q, determined bj the infinitesimal increments in the 

 coordinate» <h'" (of the order A), then we require the new coor- 

 dinates of the points of the rigid i-elative to Q in order to satisfy 

 the definition and first condition of section 2 : the points are to be 

 points of a rigid, and must each cover an equal distance. 



Denoting the new relative coordinates by u" -f- du", v" ~\- dv" . . . 

 etc. it is easy to formulate the latter half of the condition. For the 

 increments of the coordinates of the point designed l)y u will be 

 dx'' -j- du'', and the starting point of the line-element through which 

 it runs lies beside P, at a distance defined by u. So, if this line- 

 element is to be equal to that from P to Q, i.e. 



ds" = :E{ah)gabdx'^ dx^, 

 we necessarily must have 



^gab 



= :S (ab m) ~ "" ii"< dw" dx^ + ^iab) 2g„b d.v" du^ 

 da,'"' 



(1) 



If the aggregrate of points is to form a ligid, both the lengths of 

 the relative veclors u, v, w.... and (he included angles must be 

 constant, and expressions such as 



u* = 2!(ab) g,„u "" ""'• » "'' cos (uv) =z ^ ((un) g„,„ n" r"' , 

 must have the same value in P and Q. This implies 



^9"> 



= ^ (am/) -^^ dx^ n" u'" 4- ^{am) 2gn,n u" du>' 



(2) 



