Mr Sharpe, On the Reflection of Sound at a Paraboloid, 121 



where /a is a new variable and v and A are both supposed to be 

 large and fi of moderate value, (12) becomes 



aW dV 

 dfx? dfi 



A 2 V(l-fx) = 0. 



.(63). 



In this equation put 



V= Ge z . 



.(64), 



where z is a function of //., and C a constant to be determined. 

 (63) becomes 



*{S + (i)K;-^ (i -^° (66) - 



Assume, to satisfy this equation 



z = Az_i + z + -j + -~ + &c. 



.(66), 



where z_ lf z , &c. are functions of lc. Putting this in (65) and 

 equating to the various powers of A we shall have 



/ x(/_ 1 ) a -(l-/^) = x 



(fid 2 + d) z_ x + 2fMZ_ 1 z' = 



(fid 2 + d)z + y {(z\) 2 + 2z'_ lZ \} = I (67), 



(fid 2 + d) z-l + 2fi {z'-^z + z' Q z\\ = 

 (fid 2 + d)z 2 + fi {(z\) 2 + 2 (z- x z' z + z' z' 2 )} = 0, &c. ) 



where d = d/d/j, and dashes denote differentiation with regard 

 to fi. When A is large, (66) practically reduces to its first term, 

 so that in (12) for v and A both large and v = Ay, (fx moderate) 

 we may approximately put V=Ce Az ~ 1 . But from the 1st of 

 equations (67) z _ x vanishes when //, = 1, so that V seems to be 

 a maximum when fi=l } but we shall presently see that this is not 

 strictly true. 



Solving the equations (67) [and putting " exp. z" for t z when z 

 is complicated] we shall get 



V= 



G 



-£7= a x exp. 



^(1-^)4 



± {sin->* +/**(!- fif\ A + — x 



V- 12,1+ 3 1 1 



fl(\-Lif _ A 2 



.(68). 



We get two solutions, the upper signs going together and the 

 lower together. It will presently be shewn that the upper signs 

 in (68) correspond to the 1st or non-logarithmic solution of (12). 



