184 
MR, W. M. HICKS ON THE STEADY MOTION AND 
Let the stream function for the non-circular section be \//+X where 
1 R 
X= /(c_ C ) 2 (X* cos nv +Yn sin nv) ~ 
and X , 0 Y„ are also small. The necessary condition is, that when k has the value 
given above, t//+x mil st be constant, say i// 0 +e. Then neglecting squares of £ 
V o+«=fo+^ 
1 
v/(C 0 -r) 
2(X„ cos nv-\- Y n sin nv) 
The value of e is arbitrary, since with any given surface conditions the circulation 
remains undetermined. We shall choose it so as to make the circulation zero. It 
would be impossible to determine X„, Y n in the general case where both £ and bxp/bK 
are infinite series; but in the case required in the present paper A„/A „_ 1 is of the 
order k, k being small, and the terms in A n are neglected after A 2 . This simplifies the 
calculation, and it is easy to determine the terms X„, Y„ in terms of M„, N„. But it is 
further greatly simplified by the fact that in the case to which we have to apply it the 
velocity along the surface is already uniform to the first order—in other words 
U_ (Cq—c ) 2 /ty\ dK 
e 2 S 0 \bn )o du 
whence 
(^±\ _ TT 
Wo k(C 0 —c ) 2 U 
Hence the equation determining the X a , Y u is 
n sq i 
e= — ° x 3 W+' 77 r , _ . 2(X„ cos nv+Y n sin nv) 
"Wo c ) V Wo c 
But since in our applications k is itself so small that k s has been neglected compared 
with unity, the above becomes 
e= — 2 « 2 (l + 3 P+ 4 k cos v-\ -6k 2 cos 2 / y)U£ 
+ y/(2k)(l + 5 ^+^ cos y-j-§ k z cos 2 v) 2 (X„ cos nv-\-Y n sin nv) 
The various normal functions will therefore be composed of a set of principal terms 
in cos nv, &c., each corrected by an infinite convergent series of small terms of the 
others. The principal will be given by 
e= — 2 a 2 US 1 (M„ cos nv- N„ sin nv) -f- y / ( 2 /. - )S(X ;i cos nv-\- Y n sin nv) 
Therefore 
