ON LINES OF INDUCTION IN A MAGNETIC FIELD. 
325 
When u = tank l (b/a), V* = V,, and when u 
u. 
V, = Y e ; this gives 
and 
Aa — aA, -f- 6B,. 
A, cosh w, + B, sinh u x — A e cosh u v — A* sinh w. 
Again, when u — tanh 1 (6/a), we must have 
and when u = u { , 
These two conditions give 
. .0V O 0V ; 0V, 
^ ^ du ” dn P 0?t ; 
, SV 0 8Y f 0 V, 
{/X ~ 
— (/x — 1)6H = 6 A — /x6A, — fxa B, . 
(/x — 1) H sinh u x 
— A e sinh u l — A e coshw, — /xA, sinh u x — /xB cosh u 1 
Solving equations (4), (5), (6), and (7) for A, A,, B,, and A e , we get 
(/xo — 5) (a sinh u x — b cosh «,) 
• (4) 
.(5). 
•( 6 ), 
•( 7 ) 
A = 
A, — 
B, = 
(/xo 2 — 5 2 ) (/x sinh v x + cosh xx,) — (/x — 1) ab (/x cosh w, + sinh «,) 
(/xo 2 — 5 2 )sinhxx, — ab (/x cosh m, + sinh w,) 
(/xo 2 — 5 2 )(/xsinh ?<, + coshxt,) — (/x — 1) ah (/x cosh u x + sinh ?q) 
. (/x — 1)H, 
• (ft - 1)H, 
a 
(A - A,), 
, A, cosh if, + B, sinh v, 
A-e — 
e ~ Ui 
These expressions may be reduced to a somewhat simpler form by putting 
y/a 2 + X, = o„ .y/6 2 + X, = 6,, so that a,, 6, stand for the semi-axes of the 
external elliptic bounding surface. Remembering that u x = tanh _1 (6,/a,), we get 
A 
A t 
A e 
(/Xff - b)(fl — 1) 
fA~ct b fjL (cl -f- b 1 
cici x — 55, 
o5, — a, b 
. H 
/xa — (« + 5) 
55, 
o.5, — a,5 
(o — 1) 
fJU'CL + 5 + /x(o -f- 5) 
oa, — 55, 
. H 
a 5, — o,5 
c5(o, + 5,)/(«5, — a,5) 
A + & + A*(« + 5)^1 
55, 
(/x- 1)H 
«5, — o,5 
(/xoa, + 55,)/(a, — 5,) 
/x~o b -i- /j. (a -f- 5) 
OO, 
55, 
• (/x-l)H 
«5, — o,5 
