Mr WeatJierbuni, On the Hydrodij navvies of Relativity 83 



In the present case* the integral of the equation of motion 

 found in § 5, viz. 



takes the form 



c^/, + /,;'F=0, 



or, in terms of the rest-densit}^ //, 



cH-' + Vk\;\/c--v' = 0. 



§12. l^^teady motion in two dimensions. Supposing the fluid 

 of minimum compressibility, let its steady motion be parallel to 

 one plane — the plane xy. Introduce a function y^ satisfying the 

 relations 



dy\ 



dyjr 

 KV = ^ 



on; 



•(32), 



u, V being, as in § 1, the components of velocity parallel to the 

 axes of a; and y resjjectively. Such a function -v/r exists, the 

 equation of continuity 



div («v) = 



being satisfied identically. The function yfr is proportional to the 

 flux of matter across a line AP drawn from a fixed point A to the 

 variable point P (x, y). For owing to an infinitesimal displacement 

 Sx of F the increment in the flux of matter is 



kvSx = ku'icvSx = k,' ~ Sx. 



ox 



Thus if ^ denote the flux 



-,^ o.i: = /in : - dx. 



ox ox 



Similarly Jy ^^ ^ ^'" Jy ^^' 



showing that '^P = kj-yjr, 



as stated. The part played by this function yjr is exactly similar 

 to that of the stream function in the two-dimensional motion of a 

 liquid in the classical theory. The present function also is a true 

 stream function. Its value is independent of the path chosen from 

 -4 to P provided the region is simply-connected. For, if ^4PP and 



* Lamia considers only the case of free motion (F= const.) ; loc. cit. p. 71*5. 



