Mr Sharpe, On the Reflection of Sound at a Paraboloid. 101 



On The Reflection of Sound at a Paraboloid. By the Rev. 

 H. J. Sharpe, late Fellow of S. John's College, Cambridge. 



[Received June 20, 1899.] 



1. A good many years ago I made some experiments on the 

 Reflection of Sound at the surface of two conjugate Parabolic 

 Reflectors, placed coaxially at various distances apart, with their 

 concavities turned towards one another, a watch being placed at 

 the focus of one, and the ear applied at the focus of the other. 

 Fig. 1 (which as regards the paraboloids themselves is drawn to 

 scale) represents the arrangement. For convenience the parts of 

 the reflectors to the left of LL' and to the right of IV were made 

 removable. The dimensions were as follows in inches, 



DD' = 44, WA = 26, LL'=l 6, 



da" =11, ea = ll, IV = 4. 



A watch placed at W was heard ticking at e — on one occasion 

 at the extraordinary distance of 186 feet apart. This was on a 

 still night in winter. 



2. In the year 1877 I published in No. 57 of The Quarterly 

 Journal of Mathematics a Paper on the Mathematical treatment 

 of the subject. That treatment however was very imperfect, and 

 the results, few as they were, were incompletely given. 



It should be clearly understood at the outset that neither in 

 the Paper already alluded to nor in the present one is any attempt 

 made to solve the most interesting case of all, viz. that of a 

 single source of sound in the focus of the reflector, a problem 

 which is probably a long way beyond the power of our present 

 methods. 



But the present Paper appears to give the solution of the next 

 most interesting case, viz. that in which FIT is a line of sources, 

 fig. 1, V, being the vertex of the reflector. 



It will presently be seen (Arts. 6, &c.) that the cases here 

 treated require for their full development a complete discussion 

 of both the solutions of the differential equations 



*S + S + ^- 4 >= «• 



for all real positive values of x and for all real values of A, 



