(c^ + l)*/"* U x du (122), 



Jo 



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



Therefore the greatest value of the strength at U (Fig. 2) 



= (ES + E£f x 1 — = 7T-* x (€ nA + If x — , 



K p P 



therefore the whole strength in the line-source LO (Fig. 2) is 

 = 2ttJ / _ a . _, fP l . 

 P 



where TJ± is given by (15) in Art. 7. Now suppose A = 0, then 

 by (15) U 1 becomes J (u) and we will further suppose pi to be so 

 large that we can practically put oo in the upper limit of the 



Integral. (For the possible range of values of p see Art. 5.) (If, 



rpl 

 for a smaller value of pi, a more exact value of I Jo (u) du is 



JO 



required, it can be got by integrating the semi-convergent series 

 that, for large values of u, represents J (u).) Then, since by 

 Art. 450 of Todhunter's Laplace's Functions it is known that 



/•OO 



I J (u) du=l we have the whole strength of source in LO 



= 2M/p nearly (123), 



and by (40 a) of Art. 19 and by Art. 5, at a point V on the axis 

 distant v/2p from the geometrical focus of the reflector (Fig. 2) 

 (that distance being supposed great) the velocity in the axis of 

 the reflector = 2pdF/dv (u, v being here used N.B. for the original 

 pu, pv, Art. 6). But by (121), putting A = 0, at a distant point V 

 we have 



cos (v — 2pai) 



F= 



$ 



so that the velocity at V = — 2p — t— — — - nearly, v being 



large. But here it must be remembered that v is used for 

 px 20V. We will, in Fig. 2, for clearness put 0V = v , so that 

 the maximum velocity at V now becomes 



(2p/v )i (124). 



From (123) and (124) we see that the ratio of the maximum 

 velocity at V to the whole strength of the source in LO (which 

 we might consider a fair measure of the sound magnification in 

 the case of a sound-sending reflector) 



-(&)' (125) - 



We see from this that if we suppose V to be a fixed point 

 (and therefore v constant) and if we make experiments with 



