Mr Weathei'burii, On the Hydrodynamics of Relativity 73 



Using a dash t(j refer in every case to the rest-systeui tS", we 

 have for the rest-density 



37»o _ hm^ _ k 

 ~BV'.~y8V y ^"^• 



The function F is defined by the integral 



i'=rt (4), 



•' Pa f^ 



and in terms of this function k is given by 



-^=^(1 + ^) <'''■ 



For the rest-system *S" the quantity y has the value unity, 

 while K becomes 



k'=1+-^ (5'). 



The constancy of normal i-est-mass leads, as in the classical 

 theory, to an equation of continuity 



¥ + fl.<'-«> + a^<''"> + 3i*^^''> = ° <">■ 



§ 2. Using F and v for the force and velocity vectors, we 

 may write the equations of motion more conveniently 



|(«v) + v.V(/cv)-[--VP=F (7). 



ot y 



Then because the gradient of the scalar product of two vectors is 

 given by 



V (a • b) = b • Va + a • Vb -f- b x curl a + a x curl b, 

 the second term of (7) is equivalent to 



- V (k-v") - V X curl (kv), 

 Ik 



while, in virtue of (5'), VP = c'-'^/c'. Hence the equation may be 

 exjjressed in the form 



r) 1 



^^ («v) -I- „ V {k-y- -I- cV'-) - V X curl («v) = F. 

 ot 'Ik 



But again the second term is equal to 

 1 



VOL. XIX. PARTS II., m. 6 



2.^ 



