Motion of Solid and Fluid Bodies. 175 
du du 
Nar an tee at 
du du du 
ay A Pa Wi ugg 
du du du 
ay ih ge? Ae 0 
4th. This case, in which we have two media in contact, is the most important 
and difficult of all the cases of conditions at limits. In order to discuss it, it is 
necessary to go back to the general equation of motion and equilibrium (12); in 
the case at present considered, the function v, must be conceived to be composed 
of two parts (v,’, v,”), v,/ belonging to one body, and vy,’ to the other. Let 
J (2, y,2) = 0 represent the surface passing through the molecules common to both 
media in their state of rest. There will then be three general equations of mo- 
tion, belonging respectively to the molecules, for which 
S( 2s Ys 2) > Os S( 2,45 2) <9, S( 2, YZ) =) 
1. e. according as the molecules belong to the first medium, for which y, = v,’; 
or to the second, for which v, = v,’; or to both, for which case we have 
v,=v,/+v,". These three equations are 
. 
(iS -(F rie — aya = § ) dedyde = Sev dedyde (36) 
Na : (= Se ani = x On + “La )dadyde = Wev,"dadydz (37) 
SSS ec (GE ce ee by +— TE ee dedydz 
+ Mf - Ea (= eee = — ie”) dadydz 
(38) 
= Wav /dedydz + \Wiv,"dadydz ; 
the single and double marks denoting that the quantities belong to the first or 
second body respectively. As the laws of propagation of waves and the limiting 
conditions arising from fixed or free surfaces respectively have been already consi- 
VOL. XXI. 2B 
