(^.-^/)t+(^3-n')|" + (^.-^/)^ (!)• 



330 Mr Glazehrook, On the Reflexion [Feb. 9, 



Then the expression for the work becomes 



A 1 M — A f^^'' ^^ ^^'^\ 9 R (^^ ^^''^ 



^ \doC dy dzj " \dy dzj 



rp _ -n (dll dw 



^ \dz dx 



T = B (~ + — 

 ^ \di/ dxj 



Moreover, since there are no normal waves, 



du dv dw _ - 

 dx dy dz ' 



Also u = 11, v — v', w = w , when x = 0. 



Any of the equations at the boundary surface being true for all 

 values of y, z and t, we may differentiate with reference to y, z or t. 



Substituting, we have for the work 



{ fdw dxC\ j^, [dw du'\\dw [0/^^^ ^'^^\ 

 \ \dx dz) \dx dz j) dt [ \dx dy ) 



j^, /dv dic\) dv , „ j^,. (du diu du dv 

 \dx dy J) dt [dy dt dt dy 



du dw du dw 

 dz dt dt dz 



The coefficient of 2 {B — B') vanishes whenever ti, v, w are 

 functions of the same function of x, y, z and t : this is the case 

 in the problem before us. 



Therefore the condition that the work should vanish is 



-pidw du\ j^Jdw du'\\dio [-nfdv du\ p, fdv du'S) dv 

 \dx dz) \dx dz )\ dt \ \dx dy) \dx dy)} dt 



= (2). 



This, with the three equations 



u = u', v = v', w = w' (3), 



gives us four surface conditions to determine the intensities of the 

 reflected and refracted waves, and the directions of vibration in the 

 wave-fronts. 



