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



It is obvious at once from this that v = A makes vdV/dv a maxi- 

 mum or minimum, and since V, or rather the leading term of it is, 

 on the left of A, always positive, we can readily shew that v = A 

 gives a maximum. Suppose whilst the abscissas of the curve in 

 fig. 5 represent vs, the ordinates represent the values of vdV/dv. 



Fig. 5. 



Fig. 6. 



When v = OA = A there is a maximum ordinate. Now make a 

 new curve, fig. 6, in which the ordinates represent dV/dv. On the 

 first curve take two points Pj , P 2 near F, on either side of it, such 

 as to be equally distant from Ov. Let p 1) p 2 be the corresponding 

 points with the same abscissas on the second curve. It is obvious 

 that p 2 is nearer to Ov than p x is, and so in fig. 6 the maximum 

 ordinate has been shifted to the left of A. Again, when V" = 0, it 

 is easily seen that vV" = V{(v-Af-l}. If then V" is to be 

 negative (which it must be to make V a maximum) we must have 

 (v — A) 2 < 1, and since v and A are both supposed large, the point 

 required is very near A (fig. 4). It is for this reason that it is 

 difficult or perhaps impossible to get the required point from the 

 solution (68) because that solution fails for values of //. very near 

 unity. We proceed therefore to another investigation which does 

 not seem open to any serious objection. 



