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



where the coefficients i b 4 2 , &c, B 1} B 2 , &c. can all be determined 

 in terms of A and B Q which are arbitrary according to the laws 



2rA r + {(r-l)r + l}B r _ 1 = 0) 



2rB r +{(r-l)r+l}A r _ 1 = 0\ (80) - 



We know that when (29) is made to coincide with V 1 the non- 

 logarithmic solution of (12) A = B Q = ifi. But when (29) is made 

 to coincide with (28) they will have different values, which we 

 proceed to find. 



14. From (18) putting v for u and A = 



2 [i n 



V 2 = — cos (v cos 6) . (log v + 2 log sin 6) d6 



TTJO 



4 fi* 



= F a log v + — cos (v cos 6) . log sin 6d6. . .(31). 

 ttjo 



The approximate value of V 1 for v large is known from (26), and 

 we have to find the value of the last definite integral. If we 

 divide it up into two parts, integrating first from up to a small 

 finite angle 6 1 and then from 6 X up to \ir, we shall find when v 

 becomes large that the first part ultimately varies as v~l, and the 

 second part as v~\ Consequently the first part is of most import- 

 ance, and we shall get in the limit 



lr log 2 , . . 4 /2\* 



Vo = ! (cos v + sm v) H — - x 

 {irvf y 7r\vJ 



I cos (v — x 2 ). logxdx (32), 



Jo 



from which we notice that logv as a multiplier has disappeared. 

 If then the value of V from (29) is to coincide with F 2 the 

 logarithmic solution of (12) we must have 



log2 4x2^ f ) 



. log 2 4 x 2® r 

 A = -^- — I cosar. log^cfe 



7T2 tt Jo 



D log 2 4 x 2* f 00 . , . 



ij = i + sm x 2 . log xdx 



IT' 2 T JO 



^ (33). 



15. To evaluate the Definite Integrals in the last equations it 

 is best to break them up into three parts, integrating from up to 

 1, from 1 to e, and from e to oo . It is thus easy to shew that each 

 of the integrals in (33) lies between ± (1 + 1/e). We shall find 

 that 



J COS ^Aogxdx = - ¥V2 + ¥]i - WV6 +&c. 



