511 



8. Tlie "corrections" given by eq. (4) are of the order As. In 

 order to see if tlie solution defined by tliein is tlie only one, we may 

 ask if we can satisfy the equations by "corrections" differing from 

 chi^, such as r/w" -\- du'\ wliere (hi^ is of the same order as (hia- 



If these are to satisfy eq. (1), (2), (3), we must evidently have. 



= ^{ab) gab dx" (fu^ , 

 — 2{ah)gab "" <f"'' y 

 = :E{ah) gab «" (h'> -f gab v^ ^"^ , etc. 

 If the rigid, besides the centre, consists of /> points, we shall have 

 2/> -f- ^ p{p — 1) eqnations for pn variables (hi^, (fv^. . . etc. For 

 p = 271—3 we have as many homogeneous linear equations as 

 there are \ariables. If no set of n, vectors u,v,w. . . are situated 

 togetliei' in an {n — l)-dimensional space, then these equations only 

 permit a solution of the form (e.g. for //. = 3): 



öW — --- S (ij) '^ d.^•' 7Ü = — ! , . . (5) 



l/.V Pbj gcj I K<7 ■ "6 "c 



where (o is an arbitrary number, and by b, c, a, we mean a set 

 of three indices which form an even permutatio)) of 1, 2, 3. We 

 denote by (Lvb and U/, with lowered index the covariant combinations: 



cLcb = ^{i)gbi '^•'i"' , U(j =: 2{j)g,,j uj. 



It can easily be ascertained that the expression given for (hi", 

 together with similar expressions for (fv", (f>v". . . satisfy the equations. 

 They must define the displacements in the case of an infinitesimal 

 rotation about dx as an axis ^). For all ve(!tors u,v,w . . . keep their 

 lengths unchanged and both the angles included with dx and the 

 mutual angles remain unaltered. 



Since the condition imposed thus far appears not to be sufficient 

 to define a displacement without ambiguity, we must recur to the 

 condition of reciprocity of section 2. 



Shifting the centre of the compass-rigid from F to Q the particle 

 designed by u might come from a position R into the position 

 defined by the coordinates 



^« 4- dx^ + u" -f du" -f Sti" , 



or, 



\lrn} to \ dxb dxc I 



x'^ f rf;r" -V u" — Eilni) Idxhi'» -| 'I. 



^ f a \ \/g\ Ub Uc ' 



If we now shift the centre from /-* to R, and we then wish to 

 find what will be the new position for the particle from (2 according 



1) Through an angle r.) dx' . 



