Mr Sharpe, On the Reflection of Sound at a Paraboloid. 115 



20. Remembering (Art. 7) that V 1 is derived from (15) by 

 changing u into v and A into — A, we will next see what V x 

 becomes for moderate values of v and large value of A. Here we 

 follow closely the method of reasoning used in the preceding 

 article. We have 



+ (!])^ 4 " 1 ^ 42 + 9) + &C (41) - 



For brevity drop the dashes in (12). Divide (12) by A, put 

 Av = z and we get 



d*V dV (z A Tr A ,, aN 



-3* + -& + \&- 1 ) r -° < 42 >- 



Now the smaller we suppose v/A to be, the more nearly does 

 the last equation pass into 



d*V dV 



z 



+ ^77-^=0 (43). 



dz 2 dz 



[If we here put z — a new variable -r,(43) reduces to an equa- 

 tion so elaborately discussed by Sir G. Stokes in Art. 20 of his 

 Paper " On the Discontinuity of Arbitrary Constants that appear 

 in Divergent Developments."] 



If we call Z the non-logarithmic solution of (43) 



**l+* + ^ + ^ + *^««M2(-*W (43a), 



and this represents the assemblage of the 1st terms in the brackets 

 in equation (41) and V x takes the form 



Fi = [1 + v 2 {-Id + 1#) + v* (£# - Ud* + 3L d*) + &c] Z . . .(44). 



For moderate values of v and large values of A this reduces to 

 its first term Z. We can easily shew that 



Z = - e~ oJ cos <l> x dd> (45). 



7T Jo 



To evaluate this integral for large values of z break it up into 

 two parts, from up to 7r/2 and from 7r/2 up to it. It is readily 

 shewn that the 1st integral = 1/2^ and that the second integral 



IT 



Jo 



