and the Laws of Plane Waves propagated through them. u\ 



to bodies which transmit normal pressure in every direction. Before deducing 

 the equations of motion, it may be useful to show, in another manner, the neces 

 sity for reducing the function of („, ^, y) to a function of (a + H + ^) By the 

 theorem proved in my former memoir, (a, ft y) are transformed by the same 

 equations as (.^ f, .-^) ; hence, V^ F(a, ft y) may be represented by a surface 

 whose equation contains only the squares of the co-ordinates. It is evident 

 that such a surllxce will be symmetrical with respect to the co-ordinate planes • 

 and If this condition be satisfied for every system of co-ordinates, the s^irface 

 must be a sphere ; hence, 



V=F(a,ti,y) = F(a + (i + y). 



The equations of motion deduced from (20) wiU be the equations commonly 

 used in hydrodynamics, and for this reason it may be useful to state them in 

 their most general form. Let(.,y,.) denote, as in equations (1) and (2) the 

 actual positions of the molecules ; then, since 



\dx'^ dy'^ dz]' 

 we shall obtain from equations (2) 



-'IE- X ^^" '^^ du du 



\dp__ dv dv dv dv 



~pdy-'-Jt-''Tx-''Ty-''d-z' ^^S, a) 



\dp_y dw dw dw dw 

 pdz-^ 'dt~''dx~''d^-'^Tz'^ 



dV 



assuming^ = ^. and recoUecting that {u', v\ w') are the total differential co- 

 efficients o^uv,w), taken with respect to (0- These are the mechanical 

 equations of the problem, and, combined with the equation of continuity, 



dp d(pu) d(pv) d(pw) 



dt-^~lhr+-df-+dr = ^^ (24,a) 



VOL xxn. „ 



