Mr Sharpe, On the Reflection of Sound at a Paraboloid. 107 



As in what follows we shall not consider the case of u = being 

 a line of sources, we shall not want the second solution of (13), but 

 as we shall consider, among other cases, the case (Arts. 13, &c. 

 33, &c.) of the line v = (or OL, fig. 2) being a line of sources, we 

 shall there require the second solution of (12). For U then we 

 may always put (7=0 in (17) and (18), and then the condition of 

 reflection (10) gives that A should be one of the roots of the 

 equation 



ri* 



cos 0sm(l cos + ^llog cot \0)d0=O (19). 



These are all real (Art. 8) and if u 1} u 2 , &c, v l} v 2 , &c. be the values 

 of U and V corresponding to these various roots, we may put 



P = chUM + a 2 u 2 v 2 + &c.| -on} 



Q = biU^Vy + b 2 u 2 v 2 + &c.J 



where a 1} a 2 , &c, b 1} b 2 , &c. are arbitrary constants. 



8. From (13) it is easy to shew that if u.- b , Uj are two values of 

 U corresponding to two different roots A{, Aj of (19) then 



| iiiUjdu = (21), 



Jo 



from which it follows in the usual way that (19) has all its roots 

 real. 



9. It is important to remark that in the present Paper I 

 make no attempt to deal in detail with the problem of combining 

 together a number of different solutions given by different values 

 of A as suggested by (20), but having selected a particular value 

 of A generally large, I trace the varying values of a single term of 

 (20) for various values of u and v. As we shall generally confine 

 ourselves to points on the axis for which u = 0, the chief thing we 

 have now to do is to find the various values of V in (12) for various 

 values of v from to oo . 



Simple Case when vl = 0. 



10. The most interesting results are obtained, as we shall 

 presently see (Art. 19, &c), by using large values of A from (19), 

 but before discussing these perhaps it will be well to see under 

 what conditions we may have A =0 in (19). Remembering that 

 after Art. 6, I here means I' we must have 



f^cos 6 sin (V cos 0) dd = (22), 



Jo 



