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



The most important part of this integral is near <£ = 0. Evalua- 

 ting in the usual way, we shall finally get when z is large 



Z=^— h ov V 1 = et ... v (46). 



We may compare (46) with (39). (46) is true only if ifiJA^ is a 

 small fraction. For some further observations on (46) see Art. 32. 



21. The following is another way of getting the result (46). 

 We see from (41) that V x can be expressed thus, 



V 1 = Z + v 2 Z 2 + v*Z t + &c. + v n Z n + &c, 



where n is even, and Z Z 2 , &c. are functions of z. Putting this in 

 (12), having for brevity dropped the dashes, and being careful 

 always to substitute z/v for A, wherever A occurs, it will be found 

 that we shall get 



zZ " + Z '-Z = (47), 



*%" + Z 2 ' (3 + 2z) + (4 - z) Z 2 = 0, 



and generally 



n*Z n " + Z n '(n+l+nz) + (n*-z)Z n + Z n _ 2 =0 (48). 



Now we know from (41) that Z , Z 2 , &c. are all series ascending 

 by positive integral powers of z, but (48) gives us the law of their 

 connexion and formation. Comparing (47) with (43) we see that 

 Z = Z x as defined in (43 a). The rest of the argument is the 

 same as in Art. 20. 



22. But yet another way must be mentioned of obtaining the 

 result (46), as it throws important light on the whole subject. 

 In (12) dropping the dashes, put 



V=Ce w (49), 



where G is a constant whose value will presently be found, and w 

 is a new function of v. 

 We shall have 



{d 2 w (divV) dw , . n /KAN 



n* i + U)| + a» + "-^ =0 < 50 >- 



Assume, to satisfy this equation 



w = 4* «_i + »o + 'J 1 ! + 5 + 2t + &c-, 



and, remembering that A is supposed large, determine the func- 

 tions v~ 1} v , v x , &c. by the condition that the successive powers 



