Mr Sharpe, On the Reflection of Sound at a Paraboloid. 113 



Case of A large. So-wce Distant. Sound Receiving 

 Reflector. 



19. In order to get the large roots of equation (19) in A we 

 must first find what (15) becomes for moderate values of u and 

 large values of A. Fortunately we are able to do this with com- 

 parative ease, without even referring to the definite integral solu- 

 tion of (13). Drop the dashes in (13). Divide (13) by A, put 



Au = z (34), 



and we get 



d*U dU tz 



^ + ^ + b + 1 r = ° (35) - 



Now as we here suppose u/A, that is zJA' 2 , small, we see that 

 the smaller u/A is, the more nearly does the last equation pass into 



. '■%+%**-* <» 



This gives us the clue to the required solution of (35), for if we 

 call Z the non-logarithmic solution of (36) we know that 



But if we look at the assemblage of first terms in the brackets in 

 equation (15) we shall see the series (37). In my Papers in the 

 Messenger referred to in Art. 3, I shew that equation (15) can be 

 arranged in the following form : 



U, = [1 + uP (id + K) + iri ( T l 8 -# + ^d 3 + ^d*) + &c] Z. . .(38), 



where d stands for d/dz. (38) is obtained thus : First collect into 

 one sum all the 1st terms in the brackets in (15) and we get Z, 

 then all the 2nd terms into one sum and we get u? (±d + \d?)Z, 

 and so on with the 3rd terms, till the series (15) is exhausted. By 

 Art. 408 of Todhunter's Laplace's Functions (TLF) we know that 

 for large values of A, and therefore of z, Z can be expressed in a 

 semi-convergent series of descending powers of z. The leading 

 term in this expansion is 



cos {2z* — \tt) . jz^ir' 2 , 



and from the form of this we see by performing a few differentia- 

 tions that the larger z or A be, the more nearly does (38) coincide 

 with the value of its 1st term Z. We see then that if A be taken 

 large enough, we can write down 



c ost2(^-^} 



