Tao 
$ 5. The equations of motion may be derived from (12) in the 
following way. When the variations dv, have been chosen, the 
varied motion of the matter is perfectly defined, so that the changes 
of the density and of the velocity components are also known. For 
the variations at a fixed point of the space S we find 
= Òyas y 
ity a err Het eh ree gs 
where 
Nab — WOE adj Apt a PN BS (14) 
(Pherelore : Hbo Hair Noa = 0): 
If for shortness we put 
PES V S(ab)dapvarrs ss hetere en DEED 
soshat D= SP and 
ONE U RENE ni Ae (16) 
we have 
dL = — X(4) = dw, = — & (ab) ue ah 
> / 0 Ua =f >( / 0 Ur 
= — ») ——| — Yab = (ad) yab = — |, 
i Oes pe 0) ab 025\ P/ 
so that, with regard to (14), 
0 (Ue \ 
dL + D(a)Kede, = — (ab) a (} 10 = | 
0 (ua 
4. S(ab)(wsdta—wadas) —(—" ) + S(@) Kae 
0a, \ P 
If after multiplication by dS this expression is integrated over 
the space S the first term on the right hand side vanishes, 4,5 being 
O at the limits. In the last two terms only the variations de, occur, 
but not their differential coefticients, so that according to our fun- 
damental theorem, when these terms are taken together, the coeffi- 
cient of each dr, must vanish. This gives the equations of motion °) 
K S(0) 0 (us 0. [tg (1s 
Ye =e Wh 5 z a a ae : ’ . ’ . 8 
‘ ‘ Oaq r Ox Te 
which evidently agree with (4), or what comes to the same, with 
a (ae Oe eo ade a A 
In virtue of (18) the general equation (17), which holds for 
1) In the term 
Slabyord A ois. 
ae LD Wy OL h —— | — 
Ve aa PB 
the indices a and } must first be interchanged. 
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