104 The Rev. S. Haughton on a Classification of Elastic Media, 



If w be an element of the surface 



^ 1 ^ '^^ 



ayaz = w cos \ = kw -j-, 



/ A ^^ 



(ImZ — W cos U = KU) -^r- , 



dy 

 a.rcly = w cos v = kw -y- ; 



these equations will reduce the double integrals (4) to the form 



dV dF dV dF dV dF\ ^^ 

 dai d.T da, dy da^ dz j 



,,,V/r dF dV dF dV dF\ ^ 



=ff 



f/j3i f/.r rf^o dy dji^ dz 

 flY-- iE (K 'IK 'LZ /II.\ ^^li- 



These will be the double integrals resulting from one body, and from them 

 must be subtracted similar terms derived from the body which bounds the one 

 under consideration. F{a; y, z) = is common to both bodies when at rest, 

 also ^, »7, f, will be the same for all the bounding surface, /(.?,?/, z, t) = 0, during 

 the motion ; also 5^, ci;, ?f, are independent ; hence tlie condition at the limits 

 will be 



A' - A" =: 0, 



which is equivalent to three equations, 



dai dai J dx \da,_ rfaa ) dy \ da, da, J dz 



civ;>_djj\dF fdn_djT\dF fcm_d]^y^ 



dp, dpjda'\dp, dpjdy'^\dji, dpjdz ' ^' 



dy, dy, J dx \dyi dy^ ) dy \dy3 dy^ ) dz ' 



V', V" denoting the functions proper to the first and second body, and 1 \ 

 denoting that the values of {x, y, z), deduced from F(x,^, z) = 0, have been 

 substituted in V. 



To the three equations (6) must be added the self-evident geometrical 



