190 Mr Sharpe, On the Re/lection of Sound at a Pai-aboloid. 



On the Reflection of Sound at a Pai-aholoid. By the Rev. H. J. 

 Sharpe, M.A., late Fellow of S. John's College, Cambridge. 



[Received 22 May 1909.] 



38. In the year 1899 a Paper by the author with the above 

 title was published in Vol. X. Part iii, of the Proceedings of the 

 Camhridge Philosophical Society. That Paper contained some 

 minor and possibly some major Errata, which I should like some 

 day to correct, but that is not my object at present. Rather it is 

 to make an addition to my former Paper suggested by Arts. 10, 18 

 and 23, in which additional results are obtained, shewing a strange 

 analogy with the Retiection of a point of Light at plane surfaces. 

 Thinking it would be better to make this Paper intelligible by itself, 

 I have ventured to repeat, as shortly as possible, some of the early 

 part of the Analysis. Wishing also this Paper to be considered 

 part of the first, I have made the numbering of the Articles, 

 Equations, and figures run on continuously. 



39. Suppose then (fig. 8) LR to be the material parabolic 

 Reflector, its focus and OiV the axis of x. The sound motion is 

 supposed to be such that it is everywhere symmetrical round the 

 axis, and in planes passing through it. 



Let P be any point within the Paraboloid LR. Draw two 

 confocal parabolas PU and PY passing through P with as 



focus. Put 0U=%,0V=~ and 0L = \:. It will be found that 



any sound motion within the Paraboloid must satisfy the following 

 equation 



u + V \ dv' du- dv duj a- df ' 



