Mr Skarpe, On the Reflection of Sound at a Paraboloid. 135 



Now studying equations (46), (50) and (51), we see from (51) 

 that there are two solutions of (50), which are approximately 



e 1(Av$ 6 -2(Av)i 



The latter corresponds to what Stokes calls (in his Paper on the 

 Discontinuity of Arbitrary Constants, &c.) " the inferior function." 

 The result is, that, whether there be a source in OL or not (see 

 Fig. 2), as long as v < A, in the region so defined, the vibration 

 will be nearly stationary. This is, I think, what we would 

 expect, and we may compare this with the simple cases above 

 considered. 



Again in Fig. 9 suppose there is no source at S. We can have 

 for possible velocity-potentials, 



F 1 = (sin mr sin mat)/r, F 2 = (cos mr cos mat)/r. 



Either represents stationary vibration, and F 2 + F 1 gives a pro- 

 gressive wave from source at 0. 



F 1} F 2 here correspond exactly to V u V 2 , the two solutions of 



d 2 V dV , . XTr _ 



and a linear combination of these two gives the outwards going 

 wave for large values of v. 



40. We proceed now to consider in more detail the case of 

 .4 = 0, which was considered in Arts. 10 — 13 of my former Paper. 

 I was not aware when I wrote that Paper of all the details that 

 are known with regard to the zeros and turning points of the two 

 functions J Q (x) and J 1 {x). We proceed to supply the deficiency. 

 It may be observed that (though perhaps finite and large values of 

 A give us the most interesting cases) the case of A — has this 

 advantage, that it can be completely worked out to any required 

 degree of accuracy. 



41. In what follows, and till further notice, we shall use 

 u and v in the sense of the first part of Art. 6, that is (Fig. 2) 



u=20U, v = 20V. 



And first we will suppose the paraboloid u = I a sound-sending 

 reflector, LO (Fig. 2) a line of sources, giving rise to a progressive 

 wave. By Art. 17 corrected as above Art. 38, or by Art. 34 the 

 law of the strength of source at any point U in LO (Fig. 2) is U 1 

 where 



^,,4 



^ = 1 - V + 2*. 4* ~ &C - " /o ( ^ 



