Mr Sharpe, On the Reflection of Sound at a Paraboloid. 109 



and we must substitute for it the semi-convergent series (see 

 Todhunter's Laplace's Functions, p. 318), 



ttvJ V 4/ 4.8 \2v 



1 2 .3 2 .5 2 .7 2 /iy . 

 + 4.8.12.1 6UJ- &C ' 



or approximately, 



r "-(sM-s) (26) - 



but for this to be true we must have, to make the series initially 

 convergent, v much greater than 3 V2/16. 



In this case we have stationary vibration, the case of two equal 

 waves which, restoring the old notation, are expressed by 



(2^)" l cos i pv " ¥ " 2 ^) + cos { pv ~l + 2pat ) } ' • " (27) 



moving in opposite directions and continually reflected at OL 

 (fig. 2). The intensity of the sound will vary roughly inversely as 

 the distance from 0. Also for a given reflector and a given distance 

 from 0, low sounds will be more magnified than high ones. Again, 

 we see that on the right hand of a point (see fig. 2) determined, 

 roughly speaking, by pv = 3 \/2/16 (using the old notation, Art. 6), 

 it is possible to discriminate between the two waves, but not on 

 the left, at any rate by the present method. 



13. We will next consider the logarithmic solution of (12) 

 which is from (16), putting v for u and A = 



Here LO (fig. 2) is a source of sound. We will presently find its 

 strength, but before doing so we will find what (28) becomes when 

 v is large. We know by Todhunter's Laplace's Functions, Art. 403, 

 that the general solution of (12) when A =0 takes the form 



V=v~ i (A + AjV- 1 + A 2 v~ 2 + &c.) cos v 



+ v~$ (B + B^- 1 + B 2 v~ 2 + «Sz;c.) sinv (29), 



VOL. X. PT. III. 9 



