326 
PROFESSOR HELE-SHAW AND MR, ALFRED HAY 
The equations to the lines of induction are :— 
For the central space inside the shell, y = constant. 
For the space occupied by the substance of the shell, 
K cosec 6 = cosh u 
and for the external space 
K cosec 6 = cosh u 
H - A, . n 
——-sinn u, 
b i 
A, — H 
—r smh u, 
K being in each case a variable parameter. 
If we suppose that the impressed field is along the minor axis of the ellipse, 
so that 
V 0 = — H v /cr — Jr sinh u sin 6, 
then, assuming' 
V i — A'y/'a* — />' . sinh u sin 6 , 
V : = x /cr — Jr (A' x cosh u + sinh u) sin 0. 
and Y e = y/a 2 — Jr . A' e e~ u sin d, 
we find, proceeding as before 
A' = 
(/xJj — a)(b cosh ?q — a sinh u x ) 
V = 
(/ ub 2 — o 2 ) (/r cosh ?q + sinh tq) — (/ x, — 1) ab{fi sinh ?q -f cosh ?/-!) 
abe u i 
( fj,¥ — a 3 ) (jx cosh v x + sinh/q) — (/u, — 1) ab sinh ?q + cosh/q) 
P. ' — A ' _ A ' 
Hi _ A h A l3 
. , _ Af cosh /q + Bf sinh iq 
C “ V ' 1 
Or, in terms of cq, b x , 
A' = . (p - 1)H, 
, ab(a x + b x )/{a x b- ab x) 
A i = -*-• (A ~ OB, 
. (y — 1)H, 
. (/r — 1)H, 
B/ = 
fxb — 
A 
aa x {a + b) 
a x b — ab x 
where A = y~b + a — y(a + J>) 
aa x — bl> x 
a x b — a1> x 
(,, - 1 tH, 
A/ = + aa h (ai ~ h) . (p - X)H, 
The equations to the lines of induction are : 
