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



preference being perhaps given to certain large positive values of 

 A (see Art. 18, &c.) which are nearly proportional to the squares 



v w 



Kg. 1. 



of large successive integers. We shall also want the non-log- 

 arithmic solution of 



*§ + s + ^ + ^= < 2 >- 



As in these solutions the ratio xj A may vary from very small 

 to very great, it is obvious that here alone we have a large order 

 in pure mathematical investigation, as it seems likely beforehand 

 that these solutions would take very different forms according to 

 the various magnitudes of the ratio. It will be found (Art. 25, &c.) 

 that special interest, in the case of the equation (1), attaches to 

 the case x = A, as this point will be found to determine a focus 

 of reflection in the case of sound falling on the reflector from a 

 distance. 



3. From time to time I have published in the Messenger of 

 Mathematics Papers discussing equation (2). A good deal of what 

 was there said will apply also to equation (1), but a good deal is 

 inapplicable, and will be supplied, supplemented, or superseded in 

 what follows. To two of these Papers I shall occasionally refer. 

 One in No. 19 of the year 1881 I shall call Paper No. 2, and one 

 in No. 149 of the year 1883 I shall call Paper No. 5. 



4. Suppose we have an infinite Parabolic Reflector — I say 

 infinite because it is only to an infinite one that the present inves- 

 tigation applies. I shall call the semilatus rectum L Suppose 

 the origin of three rectangular axes to be at the focus, the axis 

 of x being coincident with the axis of figure, and the positive 

 direction not cutting the surface. Suppose sound motion going 



