184 T. H. Havelock 
We now obtain the result of putting in the integrals of (30) a typical term 
vrpre™? or iv™pre'”4 instead of f(v); we then apply these results to each of 
the terms in (31) and repeat the process until we have obtained all the terms 
of the required order. For the integrations with respect to v which occur 
in the process we use the notation 
© p—2va _ ( = iN) 72 e—2vb 
Pe =| ez vndv, (33) 
2 y—2vd 
Tn = i [—en20d vdv, (34) 
n not being zero in the second case; these integrals may be evaluated in 
finite form. 
We now give the result of this process; we obtain 
f(k) = k—dixpB, ec” —iK2p?B, ce? + Liki By e3 + A Kxpt Bet 4..., (35) 
with B, = 1+kprysin 6 + 4p?(r, — 2rj cos 20) 
+ tkp*{ro(4 sin’ 6 — sin @) + 479(7, — 2r; cos 20) sin 7} 
—tp*{rs cos 20 — 275 cos 40 — 2(r, — 2r} cos 20)?}+ ..., 
By = k—ikp?(r, — 2r}) cos 20+4p%7.sin0+..., ee) 
B, =1+kprysiné+..., 
B,=k+.... 
6. We consider in particular the case in which the circulation is such that 
the fluid velocity remains finite at the rear edge of the plate; the condition 
for this in an infinite stream is 
[= 2nepsiné, ork =1. 
Returning to the expression (21) for the elliptic cylinder, the condition 
requires that the imaginary part of dw/d¢ should be zero for € = +77. 
This gives, after putting & = 0 for the flat plate, 
iT 
ee ee Cn CA) = OF (37) 
From (23) we obtain 
Z(=1)"G, = —4ope” 
0 7n—1 Fl (,-)\ p—2ka _ ( __7\n—1 F)*/( _ ,-) p—2kKb - 
| ir7-1F¥(k)e (—2)"-1 F*(—k)e nud, (pet) 
0 l— e—2Kd 
+2" {ir F( aig kK) = ( —i)jrt F(k)} ed 
aan GUE 
ary 
; Tae nJ, (Kpe*”) ae (38) 
445 
