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



It is proved in Art. 264 of Todhunter's Integral Calculus that 

 if n be real and r an integer, 



12 3 r 



T (n) = limit of —, ' ' '" x r n ~\ 



n (n + I) . . . (n + r — 1) 



when r is infinite. It can be shown that this formula is true 



1 iA 

 even when n is complex. First then put n = - + — , next put 



1 iA 



n = ■= — — and divide the 1st result by the 2nd result. We thus 



2 2 



get 



2 ' 2 // V2 2 

 I iA\ /3 iA\ (h iA\ fir- I iA 



. ,2 2 / V2 2 ) \2 2 7 "*• V 2 2 , 



limit 01 — r-r- j; r-j -= r-r- r - rr— X r'^, 



/l wl\ /3 tA\ fa %A\ /2r-l i^L\ 

 when r is infinite. For shortness put i J. = 0. Then 



,. . l / 1 -->( 1 -I)( 1 -I)-( 1 -^t) ::j . (113) 



= limit of - x ?^...(113), 



a+*)(i+|)'( 



1 +5) ... (1 + 



5/ V 2r-l/ 

 when r is infinite. 



Now, to make these infinite products convergent, use the 

 device of Weierstrass given in Art. 287 of Hobson's Trigonometry. 

 Noting that 



9 1 



)z z 2, z° z i 

 ~n = e "2^ + 3^-4^ +& c- / 114 x 



give to n the successive values 1, 3, 5, &c, (2r — 1) and multiply 

 the numerator and denominator of (113) by 



_* (1+1+1+ + -±-) 

 e \ 3^5 - 2r-iJ = e -zn SU ppose. 



r, , ,11 1 



Further put 1 + ^ + - + . . . + _ ^ - r 2 , 

 1 1 1 



&c, = &c 



