510 



and 



= :E (a 771 1) -^ dx' u" o"> f 2{am) c/,,,,, j«" dv" + v'» dw\ . (3) 



Tiiese are llie equations wliioli must be anplicable to dn", dv". . . 

 etc. in the traiislaticjns mentioned. It is not difticnlt to find espressions 

 satisfying the equations. We can add to (1) the identity 



d.vi 



dx'^ 



0//,,/ dgi„ 



0^^ ^ (aitn) ( ^^ ^ I dx" d.r'w» , 



and a similar identity to (2) 



= ^^ (aim) , , 



Replace at tiie same time the index b in the tirst term, right-hand 

 side of (1) by /, and ni in the second term, rigiit-hand side of (2), 

 by b, and we get 



divl u" 



= :i^ (a) 



and 



= V (a) 





d.i;" 



^(im) — — - — d.vi i("> -f ^(6) 2a„/, du'^ 



' 0.1-' O.r'" O.r'' ' 



d.v", 



u". 



Dividing by 2 \A'e can reduce the e(j nations to the form 



dn'' 4- ^(Im) c/.c'j<" 



= .2' (a6) <hth d.r' 



= :S (ab) Onu '<" 

 Similarly, we can put for the third ecjnation 



^ MM 



= S{ab) g„b 



n"\dr^' t ^-j [rf.f''-'" I f '•■' {dn'' 



A^' 



^b^ J 



So we can satisfy the imposed condition by taking 



ilvi I 

 b \ 



(1') 

 (2') 



(3') 

 . (4) 



and similar expressions for dr^', dw'' .... 



The equation (4) is covariant. It will retain its form wathever be 

 the choice of coordinates. 



It defines the position of the points of the small rigid when, by a 

 first approximation, they have all covered the same infinitesimal 

 distance. 



It is seen, from (1), that in developing yab into a series we have 

 neglected terms with products u'" u". The squares of the distances 

 covered therefore can diflTer from PQ by an amount of the order 

 g' A', so that the distances may only be taken as equal up to an 

 amount of the order t" L, which we shall neglect. 



