1903.] Bending of Waves round a Spherical Obstacle. 41 



coefficient with respect to c. The function is that which occurs in 

 the representation of a disturbance which travels outwards, and ( 323) 

 may be denoted by 



-^/.(i) ........................... (3). 



where 



K = 27T/A, 



and 



The differential coefficient of (3) is 



-!!!" {(l+iKC)f n (iKC)-iKCf n '(iKC)} ............... (5), 



so that the ratio in question takes the form 



(1 + lKC)f n (IKC) - tKC/n (iKC) 



In these expressions n is the order of the Legendre's function P n (p) 

 which occurs in the series representative of the velocity-potential. 



When KC is very great, the ratio expressed in (6) may assume a 

 simplified form. From (4) we see that, if n be finite, 



f n (lKC) = 1, iKcf(iKC) = 0, 



ultimately, so that 



(6)= -i, (7), 



independent of n. 



This is the foundation of the simple result reached by Mr. Mac- 

 donald. Its validity depends, therefore, upon the applicability of (7) 

 to all values of n that need to be regarded. If when KC is infinite, 

 only finite values of n are important, (7) is sufficiently established; 

 but ( 328) it appears that under these conditions the most important 

 terms are of infinite order. I think it will be found that for the most 

 important terms n is approximately equal to KC, and that accordingly 

 (7) is not available. In any case it could not be relied upon without 

 a further examination. 



In Tlieory of Sound, 328, the problem is treated for the case where KC 

 is small, and the calculation is pushed as far as KC = 2. The results 

 indicate no definite shadow. I have commenced a calculation for 

 KC =10, about the highest value for which the method is practicable. 

 But it is doubtful whether even this value is high enough to throw 

 light upon what happens when KC is really large. 



