Tlie decomposition of (fQ into two parts is therefore the same, 

 whether we use cjab, g"-'' or %"^. 



It is further of importance that when the system of coordinates 

 is changed, not only öQdS is an invariant, but that this is also 

 the case with ö^QdS and d^QdS separately.") 



We have therefore 



V-d V-i 



(46) 



§ 36. For the calculation of rf j Q we shall suppose Q to be 

 expressed in the quantities yj"'" and their derivatives. Therefore 

 (comp. (43)) 



6,Q-=^{ah)M^^ó^o^, ...... (47) 



if we put 



Now we can show that the quantities Mnh are exactly the 

 quantities Gah defined by (40). To this effect we may use the 

 following considerations. 



We know that ( q*^^ ) is a contravariant tensor of the second 



we infer 



UdQ) dS = CffQ dS 

 C{ö,Q)dS=: fd.QdS. 



As this must hold for every choice of the variations ^Qab (by which choice the 

 variations Si]"^ are determined too) we must have at each point of the field-figure 



{d,Q) = d,Q 



~) This may be made clear by a reasoning similar to that used in the preceding note. 

 We again suppose ^gab and ignb,e to be zero at the boundary of the domain of 

 integration. Tlien Sg'ab and 3g'ab,e vanish too at the boundary, so that 



(d,Q'dS' = , id^QdS=0. 

 From 



CdQ'dS'= jdQdS 



we may therefore conclude that 



I d.Q'dS' z=^ Cd.QdS. 



/■s this must hold for arbitrarily chosen variations Sgab we have the equation 



d.Q'dS' = d^QdS. 



