122 Mr Sharpe, On the Reflection of Sound at a Paraboloid. 



It will at once be seen that, whilst [x must lie between 1 and 0, it 

 must not lie excessively near either of them or the solution (68) 

 would fail. The case of v/A being very small was treated in Art. 20. 

 The case of v/A being very near or equal to 1 will be treated in 

 Art. 30. 



Since from the 1st of equations (67), (/_0 2 = ^ _1 — 1, which 

 makes the resulting value of z"_ x negative, we see that for points 

 for which /u, increases from near to near 1, the curve for V 

 (Art. 25) is concave to the axis, and z_ x continually increases. 

 Again, looking at the outside factor in (68) we notice that fi = \ 

 makes /jl(1 — /j,) a maximum, and that as fi increases from \ to 

 near 1, /a(1-/a) continually decreases. For a double reason 

 therefore, as fi approaches l, the leading term of V continually 

 increases. But it is important to notice that we cannot conclude 

 that For V is actually a maximum when fi = l, because in that 

 case (68) fails. We shall see what really happens further on 

 (Arts. 29, 30). 



Next, to distinguish between the two solutions (68) and 

 to find the value of G we will compare it with the result (46). 

 To do so, we must suppose in (68) //, to be much nearer zero than 

 one. Keeping to the upper signs and considering only the leading 

 term we get 



fA* v* 



Comparing with (46) we see that if (68) is to represent V 1} the 

 non-logarithmic solution of (12), we must have in (68) 



C = 1/2ttU* (69). 



27. We will next examine the value of V x at a point A 2 (fig. 4) 

 to the right of A. Proceeding exactly as in Art. 26, we put in (12) 

 v = Afi, but fM is now greater than l, and it will be found that the 

 solution, instead of taking an exponential, will now take a trigono- 

 metrical form, in which (for the 1st time since leaving (fig. 4) 

 and moving along the axis) we are able to distinguish between the 

 incoming and the outgoing wave. In (64) z instead of being real, 

 is now complex, and we shall find that z , z 2 , z 4 , &c. are pure 

 imaginaries. The final result may be written in the following 

 form 



V— x e -{M- 2 +«4^- 4 +&c.} x 



B 2 cos (Az_, + j + x + &c ' 

 + C 2 sin(^_ 1 +| + | 3 + &c.)l (70), 



