Mr Sharpe, On the Reflection of Sound at a Paraboloid. 145 



become in this case, we have to find the values of J 5 , I 6 and 

 dI 5 /dA, dI 6 /dA when .4 = 0. To do this, it is best perhaps to 

 expand the right-hand sides of (117) in powers of A for a term or 

 two, to differentiate with regard to A, and then put A = 0. We 

 shall get I 5 = I 6 = 77-^/4, 



cU = -*{l + 2 l0 Z 2 -*)™ d dA = -T{i + 2 l0 Z 2+ *)> 



C 1= .C 2 = ir-l and G s = 2*-* [~\-\ log 2 - Jj , 



o- 



These values of (7 2 , C 2 , &c. satisfy condition (104). Suppose 

 now the reflector to be a sound-senefoV^ one and OL (Fig. 2) to 

 be a line of sources, and that we wish to find the whole strength 

 of the source. It is shown in Art. 34 (only now N.B. using u and 

 v for the original pu and pv, see Art. 6), that the strength 

 belonging to a length 2du/p at point U (Fig. 2) is the limit when 



v = of — x vdF/dv where F is the velocity-potential. We 



will suppose (see Art. 5) that 



F = [(E t V, - E 2 V 2 ) cos 2pat + (E 1 V 2 - E 3 V,) sin 2pat] x U x , 



where V 1} V 2 are got from (15) and (16), Art. 7, by putting v for u 

 and changing the sign of A, and E 1} E 2 , &c. are got from 6\, C 2 , &c. 

 in (105) by changing the sign of A there and also in I 5 and I 6 , 

 from which suppositions it follows that 



E^ + E^ = -(e^ + l) (120), 



7T 



and also that, when v is large and Ajv small, 



(v - j log vj 



cos 



E^-E 9 V* = 



v i 



sin v 



Ejr 2 -E 3 V x = 



jlogv 



} 



.(121). 



From the above it follows that the strength of source belong- 

 ing to the length 2du/p of OL is = the limit, when v = 0, of 



27r£ r 1 dw 



P 



x v [(E.V,' - E 2 V 2 ) cos 2pat + (E, V 2 ' - E z Vf) sin 2patf] 



?^i^ [- E. cos 2wa£ + ^ sin 2pat]. 

 p 



VOL. XIII. PT. III. 



11 



