Mr Skarpe, On the Reflection of Sound at a Paraboloid. 123 



where *_, = p (p - 1)* - log ^ +1 _^_ 1) } , | 



B>= l x J^i+L, I ( 7l). 



1 48 ^{fi-lf ' 1 



The two solutions are of course indicated by the two arbitrary 

 constants B 2 and G 2 . First we notice that, keeping A constant, if 

 we increase /u,, that is, if (fig. 4) we move from A outwards or to 

 the right hand, V is diminished, again showing us that the maxi- 

 mum value of V must be somewhere in the neighbourhood of A. 

 The point A is a kind of critical point. On the left of it the 

 solution takes an exponential, on the right a trigonometrical form. 



By comparing (70) with (56) we may determine B 2 and C 2 in 

 (70) so as to make V in (70) coincide with V 1 the non-logarithmic 

 solution of (12). This is shewn thus. By a method exactly 

 similar to that employed in my Paper in the Messenger referred to 

 in Art. 3 (viz. New Series, No. 149, Sept. 1883) it can be shewn 

 that (70) is not only true for /x moderate and A and v large, but 

 also for two other cases (1) for /jl large and A moderate, (2) for /x 

 large and A large. We will apply it to this second case — that is, 

 to a point like P 2 (fig. 4). Supposing then //, large in (70) and 

 confining ourselves only to the leading terms, outside and inside 

 the square brackets, we shall get 





B 2 cos A {/a — log (2/u*)} + C 2 sin A{fi — log (2/**)} 



Ai 



vh 



= 4> 



B 2 cos {v - \A log v + A log (&A*)} + G 2 sin {ditto} 



which practically agrees with (56). Comparing this with (60) we 



get 



B 2 - (2ttA)-1 x (e^ A + e-*" A ) (cos C x + sin Cj) 



C 2 = (2ttA)-$ x (eW + <r** A ) (cos A - sin G x )) ( } ' 



where G x is given by (59). 



28. We will next find the values of V 1 and dVJdv at A 

 (fig. 4), i.e. for v = A, both v and A being supposed large. This is 

 best got from the definite integral solution (57). 



Put W x for I cos ivcosx — A logcot^j dx (73) 



We make an investigation exactly like that of Art. 24. In the 



