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



result is that the point g (fig. 7) in LO of greatest source-strength 

 is given by 



Uu)i = ^ (85), 



so that the larger A is the more nearly is the strength concentrated 

 in the geometrical focus of the reflector. 



Next to get the whole strength of the source we have (Art. 34) 



to find the value of I JJ x du. This cannot be obtained directly 



from (82) because (82) is not an approximation when u = 0. The 

 only way apparently is to integrate U 1 from the ascending con- 

 vergent series (15) first from u = to u = u x where u x is the value 

 of u given by (85) and then from u = u 1 to u = I using (82) 

 for CTj. 



For the first part, we get from (15). [N.B. Before (15) can 

 be used here we must first put in it pu for u and A/p for A.] 



£ W» = «, - 1' A + g^ (* - p>) - ^ (A- - 5p°A) + &a, 



and remembering from (85) that Au x is finite, and that A is large 

 this becomes in the limit 



/, 



U x du = -Q- A~* 



o 8 



mm 2 m 3 « 



2 + 372 !) 2 ~ 473!) 2 + 



(86), 



where m = — . Of course to make this result correct we suppose 



A incomparably larger than p. 



To get the second part of the whole strength, we have 

 from (82) 



lz~* cos (2^* — \tt) du 



= -j {z l * sin (2,8* - |tt) + fsri cos (2«* - -|tt) + &c.}, 



by extending the series, which descends by powers of z~%, the 

 integral can be got with any desired accuracy. The 2 first terms 

 are accurate, the error being of the order z~^. Next taking the 

 integral from u = u x up to u = I we get by (83) and (84) 



"i l 



U x du = j-j [(Al)-i cos (n a 7r) — (Au^-i cos 7r]. 



Whatever value the integer n a has, the result cannot be of 

 higher order than A~ T , so that when A is very large we see that 

 (86) gives us approximately the whole strength of the source, but 

 for this to be true A is supposed much larger than p. 



f 



