Mr Sharpe, On the Reflection of Sound at a Paraboloid. 127 

 30. V having the value given in (41), put 

 V=f(A+v-A)=f(A)+f'(A).(v-A)+ f ^(v-Ay 



+£^j^{v-Ay + &c (78). 



Here of course f n (A) means what d n V/dv n becomes when in it v 

 is put = A. This series is of course always convergent, but when 

 A is large, and v not differing much from A, it is rapidly con- 

 vergent, for by continually differentiating (12) with regard to v we 

 can express f" (A), f" (A) &c. as linear multiples of f(A) and 

 f (A), and these series proceed in ascending powers of A" 1 . More- 

 over f{A) axidf (A) have been found in Art. 28. It is thus easy 

 to get from (78) as approximately as may be desired the values of 

 (v — A) which make V' and V" vanish. 

 We get 



V =/' (-4) +/" (A) . (v - A) +^2T } (v - Af + &c. 



V"=f"(A)+f"(A).(v-A) + 8zc. 

 V'"=f'"(A)+ko. 



On the whole it will be found that these series descend by powers 

 of A~% sometimes more rapidly. The value of (v — A) which 

 makes V" vanish is given approximately, using (77), by 



— f-f-GHIH© ^ 



and then V" =/'" (A) which from (12) is negative, so that we 

 have found a maximum value of V . This shews that the position 

 of the point on the axis of greatest sound intensity is a little to 

 the left of A, fig. 4. This verifies Art. 29. Similarly we can get 

 approximately the point where V is a maximum. V vanishes 

 when 



f 



v — A = — jr, = A from (12) or v = 2A, 



and then V" =f" (A) which is negative, shewing that a maximum 

 has been obtained. The point obtained is of course on the right 

 of A, fig. 4. It should however be carefully noticed that what has 

 been obtained here is an Algebraical maximum for V, and 

 remembering (Art. 27) that on the right of A (fig. 4) V may be 

 negative it is quite possible that this algebraical maximum may 

 be a numerical minimum. But we are not much interested in 

 maxima or minima values of V. Of course the whole of this 



10—2 



