Mr EARNSHAW, ON FLUID MOTION. 223 



and then introduce fixed screens so as to inclose a portion of any pro- 

 posed sliape. Now the effect of a fixed screen is simply to prevent the 

 particles in contact with it taking any normal motion; and this being 

 the only effect, we may suppose the screen removed, providing we intro- 

 duce into the expression for </, such a function as sliall represent the 

 condition of the normal velocities being zero. In one case this can 

 be very simply done, and may serve as an example of what is to be 

 effected, and the process to be followed in other cases. . 



Suppose the medium bounded by a plane whose equation is 



* = '• (1), 



and let a plane-wave be transmitted through this medium, such that 



'P = Fijx + gy + /iz - at) (2). 



This equation supposes the medium infinite; and we must now intro- 

 duce the condition, that at every point in a plane whose equation is (1) 

 the velocity in the direction of w is zero. Let, therefore, the value 

 of ,p which fulfils this condition be expressed by the equation 



this is the proper* form of assumption, for otherwise the equation of 

 continuity would not be satisfied. Tlie velocity in direction of x is 

 d,<p =f.F\fx +gy + hz- at) +fF,' \f {x - a) ^ g' (y - /3) + // {%-y)~ at\ , 

 which must be zero when x = c. 



.-. ^=f-F'(Jc+gy + hz-at)+fF,'{f{c-a)+g'{y-ii) + h'{z~y)-at\. 

 Now it is impossible that tliis equation should always be true, seeing 



• The general integral of the equation of continuity shews that ,/, must consist of the 

 sum of a series of functions of the form 



F\f{^-a)+g{y-ft)...h{z-y)-al). (Art. 26-.) 

 Hence we might add to the right-hand men.ber of equation (2) several functions, but as we 

 have only one condition to satisfy, one additional function is sufficient. Hence, that assumed 

 for ,/, ,n the text is the proper form for the problem under consideration. 



