BY SMALL OBSTACLES OF CYLINDEICAL AND SPHERICAL FORM. 257 



The second term of (8) gives the refractivity of the medium as modified by the 

 wires. If S be the retardation due to the wires of the stratum dx 



8 = 



Hence, if fi be the refractive index as modified by the wires, we have approximately 



- 1 }p ........ (10) 



where p denotes the ratio, assumed small as before, of the volume occupied by the 

 wires to the total volume. 



If the waves of sound traverse a medium in which a number of parallel wires are 

 arranged, then the reciprocal of a will determine the distance which the waves will 

 travel before the intensity of the sound is diminished in the ratio of Ife. For sound 

 waves of wave-length 10 cm. passing through a medium, in which there are 100 

 parallel wires of radius 10~ a cm. per unit area of a section perpendicular to the wires, 

 this distance is 47 cm. For greater wave-lengths the distance is greater. It seems 

 hardly probable that any arrangement of wires could improve the acoustic properties 

 of a room unless some other factor than viscosity is taken into account. Of course, if 

 n is made sufficiently great, the reciprocal of a may become very small ; but it seems 

 probable that it would be difficult to arrange the wires so closely that n should be 

 greater than 10 3 . If it was possible to arrange wires of radius 10~ 3 cm. so closely that n 

 was 10 4 , then the intensity of sound of wave-length 40 cm. would be diminished in 

 the ratio l/e after passing through a thickness of less than 4 cm. Such a contrivance 

 could hardly, however, be carried out in practice. 



6. Problems Relating to Spherical Obstacles. We require first a solution of the 

 equations of motion suitable to such problems. Differentiating the equations of 

 vibration (1), 1, with regard to x, y, z respectively and adding, we obtain with the 

 help of (2), 2, 



y =c 'V 2 S + tvV 2 | ......... (1). 



If we now assume a time factor e iat , this equation takes the form 



(V 2 +h 2 )s = 0, where h* = <r s /(c'+f'o-) .... (2), (3). 



Also the equations of motion (1), 1, may with the help of (2) and (3), 2, be 

 written in the form 



/ 7,2 7 2\ T I \\ 



where 



<f> = t,crh~ 2 s = A~ 2 div (u, v, w) and &* = KT/V . . . (5), (6). 



These equations (4) are satisfied by 



(u, v, w) = grad <f>, 

 VOL. CCX. A. 2 L 



