Mr Sharpe, On the Reflection of Sound at a Paraboloid. 103 



on within and reflected by the reflector. I confine myself to the 

 case where the motion is symmetrical round the axis, and the 

 same in every plane passing through the axis, the expressions for 

 the sound vibrations being functions of x and (y 2 + z 2 ) . 



Two different cases now present themselves. 



(I.) We can suppose the origin of the sound at an infinite 

 distance along the axis, and the sound falling upon and being 

 reflected by the parabolic surface. The problem here is to 

 discover the law of the magnification of the sound in the 

 reflector, and the position, if they do exist, of foci of reflection. 

 This problem, which may be called that of the Sound-Receiving 

 Reflector, is treated in Arts. 19, &c. 



(II.) We can suppose the origin of the sound at a finite 

 distance within the reflector. Here however I confine myself to a 

 special case, viz. that in which (see fig. 2) OL is a line of sources, 



L U O 



Fig. 2. 



RL being the reflector, and L its vertex. Here, as in the former 

 problem, we have to consider not only the sound from the source, 

 but the reflected sound, and we have to find the law of intensity 

 at different distances, especially at great distances. This problem, 

 which may be called that of the Sound-sending Reflector, is con- 

 sidered in Arts. 33, &c. 



It fortunately happens however that, before engaging one's 

 self with the difficulties of these Articles, a comparatively simple 

 case presents itself which will be treated in Arts. 1 — 17. 



This will include simple cases of both I. and II., but the 

 results so obtained are not so striking as some of those obtained in 

 the later Articles. 



5. In fig. 2 let LR represent a section of half the reflector, 

 whose semilatus rectum is I, the origin of the xyz coordinates 

 being the focus. Or perpendicular to Ox we may call a sort of 



