406 
MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
Tlie former gives 
Now 
TT \/(f— "A • 
X — — - -sin ^ • 
Z a j) 
P 
where s is the arc. Therefore 
7r v/(l — «“) ■ _i 
a? = - -i-sin '■ ae \ 
2 a 
When a = 0 we have 
TT 
X = ~ 6“^ 
which is the form given by Kayleigh (‘ Phil. Mag./ 1876 ) for Kirchhoff’s case. 
Example 11 .—Tube projecting far into the bottom of a vessel of given breadth. 
The tube is supposed to project so far into the vessel that the motion at its bottom 
may be neglected. 
iiiere are here two singnlar points only, u ~ — a, u — a, corresponding to the 
coalescing angles of the bottom of the vessel. 
The appropriate transformation is, therefore, 
'P ^1 — + \/(l — r^-") \/(1 — 1 + exi -E sj (1 (1 
tiw Vj — it to “h Ct 
— A ^ - ft" - + 2 v/(l - a~) p'{] - 7r) ^ 
'ur — P 
Since, when u = co, dzjilw = — i, we have A = i. 
So that 
dz _ . 2 -• ad - id + 2^/(1 - P) y/fl - id) 
dio ^ v? — P 
dw 
dil 
as before ; 
u 
and 
