4 



derivatives and we shall determine the variation it undergoes by 

 arbitrarily chosen variations (Jcjab-, these latter being continuous functions 

 of the coordinates. We have evidently 



ÖQ bQ 'óQ 

 (fQ = :S{ab) - — dffab + ^{abe) -^ fft^ao.e f ^ {ahef) (fflahef. 



Ogab Ogab,e ^9ab,ef 



By means of the equations 



Ö d 



(f9ab,e/=^^—(fgab,e and (fgab,e =^-^— (fgab 

 OXf OXe 



this may be decomposed into two parts 



dQ=(f,Q + J,Q, (42) 



namely 



Ó,Q = 2 {ah) \^-^{e) — -^+ 2{ef) ~~- — ^- ög,, . (43) 

 f Ógab OXe Ogab, e ÓXeÓXfÖgab, ef\ 



ö / öQ \ ö / dQ * \ 

 (f,Q = :^ {abe) ^- (f ga b]-j- ^ (abef) -— ögab, e — 



ÖXe\Ogab,e J ÓXf \dgab,ef J 



- -(«^/)— |— -(-^^V^„J (44) 



The last equation shows that 



ö,QdS=() . (45) 



/' 



if the variations ó<jab and (heir iirst derivatives vanish at the boundary 

 of the domain of integration. 



§ 35. Equations of the same form may also be found if Q is 

 expressed in one of the two other ways mentioned in § 33. If e.g. 

 we work with the quantities c)"^ we shall tind 



m) = {ö,Q)^{ö,Q), 

 where (rf, (2) and {<\^Q) are directly found from (43) and (44) by 

 replacing g^b, ffab,e, (Jab,ef, dyab and ögab,e etc. by 9"^ n"^'^ etc. If the 

 variations chosen in the two cases correspond to each other we 

 shall have of course 



{ÖQ) = öQ. 

 Moreover we can show that the equalities 



{ö,Q) = ff,Q, {d,Q) = ö,Q, 



exist separately. ^) 



1) Suppose that at the boundary of the domain of integration igab = and 

 ^9ab,e = 0. Then we have also ^q"^ = and rh]ab,e = o, so that 



({(f,Q) dS = 0, Cd.QdS = 



and from 



