MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
397 
Then the general formula reduces to 
dw Ai 
~(h ~ (- _ a) {z^ - 
where A is real, supposing that there is no singular point in the field. 
The nature of the multiplier Ai is obtained by considering that between 
X = — b and x = h = 0 . 
cLc 
The inteoral of this is 
lu = Ai log 
- U’A 
h{z — a) 
supposing the potential of the conductor is zero. 
(6) Now let the conductor be two semi-infinite planes, with a gap from a? = — h 
to X — h between them, and let there be a line distribution at a; = a where 
h> a > — h. 
x = a 
- 1 . 1 . 1 -- 
X = — b X — h 
Then 
where A is real 
And 
dw 
dz {z — a) 
w — Ai log 
- c,- -E v/(5^ - e"-) y - z^) 
b{z — a) 
the potential of the conductor being zero. 
(c) If we put 0 = 6 + z' in the result of [a), and then make 6 = oo, we get for the 
case of a semi-infinite conductor a? = 0 to .x = — co with a line distribution at aj = a, 
i • 1 -v/^ + \/^ 
10 = Ai log -V- , 
*= x/s - 
the potential of the conductor being 0. 
These results (a), (6), (c), could of course be deduced from the known formulae for 
elliptical conductors. 
