Mr Sharpe, On the Reflection of Sound at a Paraboloid. 143 

 We thus get 



'S + T)/ r G-T) 



<* ->(■-£) ('-D-('-s^t) . „, m , 



= p p 3 x ?- z x e z ^ . . .(115). 



e -z r ° + J r >-4 r <+ &c - 



Next in (114) giving to n the successive values 

 -1, -3, -5, &c, -(2r-l) 

 and multiplying the numerator and denominator of (115) by 



,( 1+ |+| + ... + _L_) 

 t — e , 



we get 



/l t\4\ 0« * 3 «« 



Or 



r(|-^) e -?*+f*-W te 



r£ + " 



ft l A\ 



V 2 + g / = -■(?r k +|r i +to.to.) x e , logr x 



V2 2 



From (109) the above 



_ p + t'ff _ (y + igf 

 ~ p-iq" p* + q 2 ' 

 Therefore extracting the square root we get 



p+iq -(«r I +^r,+^r,+4o.tooo) ,.,__„ , v 



(p 2 + q 2 )* - v ' 



1 A 3 A s \ 



-i I Ar t - ^-r 3 +-=-r s -&c. to oo l+Ji^logr . . 



= + e v 3 5 / ( r _ qq ^ 



So that if we put 



^n - 4" n + y - &c. to oo - \A log r = E . . .(116), 

 we get 



t£j^= ±(cobE -isixiE). 



(p^ + q 2 )* 



If A = it is known that q = and that jo = -5- , so that in 

 the above equations the upper sign is to be taken, and we get 

 p = (p 2 + q*)h cos i£ and ^ = - (p* + q 2 f sin E. 



