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



41. We will next suppose A finite. We have now to use the 

 equations (98) and (99) — but we will drop the dashes and no 

 mistake will arise if we are careful to remember the remark at the 

 end of Art. 39. 



In (98) and (99) A is one of the roots of the equation in A 



dU/du = 0, when u = l (100). 



Suppose, if possible, A^ and A^ to be two different roots of this 

 last equation, and when A = Ay, A2 respectively, let U become 

 Ui, U2 respectively. Then 



d^Uy dU, , , ^ rr ^ 



d^Uo dU» , . . T~T 



It is easy to shew that 



'^^^'^^ +{A,-A,)UJJ, = 0. 



du 



^ du ^ du) 



Integrate this with regard to u between the limits and I, 

 when we get 



^ UyU,du = (101). 







This shews that the equation (100) considered as an equation in A 

 has all its roots real, for otherwise, if Ai and Ao were two conjugate 

 unreal roots, (101) could not be satisfied. 



Jo 



42. We will next suppose that in (99) u is much larger than 

 A, then it is evident that, in the neighbourhood of points which 

 satisfy this condition, U does not differ much from Jo(u), and 

 the curve whose ordinates give dU/dii is a wavy or periodic curve 

 Avhich cuts the axis. This is corroborated by Art. 23, where it is 

 shewn that if w > ^ we have approximately 



U=—cos(u + ^A \ogu + a) (102), 



u^ 



where B and a are constant functions of A. 



[N.B. In Art. 23 we have V, v and —A, but it will be found 

 that exactly similar reasoning may be applied to the case IT, 

 u and + J..] 



We will next suppose that v is much larger than A, then we 

 can shew in like manner that, near points which satisfy this con- 

 dition, the curve whose ordinates give dV/dv is a wavy curve 



