Mr Sharpe, On the Reflection of Sound at a Paraboloid. 141 

 Put y = iz. Then 



irA iri r <x> I Ai 1\ , 



Now we know that if k and n are real 



^ 1 (. /l )_ /"" -*as+(n-l)log* v 7 

 > _ io 



But in this formula n may be imaginary. 



Comparing tlie last two equations we will suppose k = 2 and 

 n = (l+Ai)/2 and putting 



r(i + ^-*) = 2 1 i + 2tg (109), 



(A \ f A \ 



g- log 2 J and s for sin ( — log 2 ] we shall get 



after some reduction 



irA 



I = \e 4 {—qc+ps+pc + qs + i(qc — ps+pc+qs)} ...(110), 

 from which ( putting B = A log 2* — — 1 we get 



7 5 = I cos (2.r 2 + A log #) c&c = 2 _i e 4 (# sin 5 + p cos 5) 



,.00 ir4 



/« = I sin (2x 2 + A log x) dx = 2 _ * e 4 (g cosB-p sin 5) 

 The above is Stokes's result. From them we get 



ttA 



(i 5 y + (i 6 y = y (p*+q*). 



But it is known that if m be real and < 1, 



7 Siii7n7r 



But here ??t may be a mixed imaginary, so put m = - + — , and 



we get 



„/l iA\ „ /l t'J.\ 27r ., „ „, .,,,_. 



r (2 + 2-) r (2-TJ = ^^2 = MP + ?)-.. (U2), 



e 2 + e 2 



so we arrive again at the formula (106). It is interesting to 

 notice the two entirely different ways by which (106) is es- 

 tablished. We proceed to find the value of F I - + — J when 

 A <1. 



,(111). 



