Mr Sharpe, On the Reflection of Sound at a Paraboloid. 195 



which cuts the axis, also when v>A we have approximately from 

 Art. 23 



V=-^cos(v-^A\ogv + a') (103), 



where B' and a' are constant functions of A. 



43. We will next suppose that in (99) u is much smaller than 

 A. Then divide each side of (99) by A and put Au = z. We 

 then get approximately (see Art. 19 of this Paper), if z is large, 



• TT J- /n ix COS {2 (Au)i - lir] 



This shews that in the neighbourhood of points which satisfy 

 the condition u\A small {A being large) the curve whose ordinates 

 give the velocity dUjdu is a wavy curve which cuts the axis. 



We will next suppose that in (98) v is much smaller than A. 

 Proceed as before. Divide by A and put Av = z. We shall then 

 get approximately (see Art. 20 of this Paper) 



F=^=l+^+^ + ^ + etc. = /„{2(-^)4l. 



It is evident from this, that in the neighbourhood of points 

 satisfying the condition v <A {A being large) no positive value of 

 V can satisfy the condition oidVjdv being a maximum or minimum, 

 and that therefore near such points the curve, whose ordinates 

 give dV/dv, is either not a wavy curve at all, or if it is a wavy 

 curve, that it does not cut the axis. This kind of curve is shewn 

 by the wavy line in fig. 10. Or it is possible, but not likely, that 

 between and A the velocities might uniformly increase or 

 uniformly diminish without there being maxima or minima. 

 Perhaps we might be allowed to speak of this kind of curve as an 

 exponential curve. All this is further corroborated by the fact that 

 when z is large the approximate value of V takes an exponential 

 form (see Art. 20) 



V=~ r (105). 



44. We have now examined the shape of the air-velocity 

 curve in the neighbourhood of four classes of points defined thus 



u> A, u<A, v>A, v<A. 



It remains to examine the same near four other classes of points 

 defined thus 



u does not differ much from A in excess or defect, 



