486 The Open Organ-Pipe Problem. 
Taking first an integration with respect to x, we have 
[a 2 e— thee dy =) = ( — eth) dy 
, V(?—1) J, tke /(? 1)’ 
in which 
if f dv & " dy wasp 
1 vy/ (v?—1) i. v/(1—v-?) [= 2 ‘ 
Again, for the second integration with respect to 2, 
“aol a 0? i aon (1—e7#) dv 
J, to JP —1) J, Che) 2/(e—D’ 
in which 
Thus 
AE EON care! Qi ( reed 
(Vda dy=y! = ry} 
ik BR 
When z is great, the outstanding integral in (15) may be 
treated in the manner already explained. The integral 
(15) 
( —rwdat 
J wa/(1+ wv?) 
will yield the same value, whether taken from w=7 to 
w=i+in, or from w=itow=x +7. The first gives 
(“ eda 
re, RS 3 ; 
tie e VAC em di) 
the second gives 
tae e—Me-irdy 
Jo (u+2)/(2iut ut)’ 
Accordingly 
eo ent ay e-ir ig e8B-=dB 
1 vr/(v?—1) ee (Qtr), f, ae je} 
(14 2).4/(14 95) 
and the latter may be expanded in inverse powers of 7 and 
integrated as before. For the leading term we have 
ir ¢-8B-? dB=,/n. 
