inside a Hollow Unlimited Boundary. 
229 
But by De Morgan’s Biff. Calculus , p. 608, it is known that 
the series in brackets is equal to otj/2, if a x < 7 t/ 2 and (17) becomes 
/L67T = c sin + (^fjL + a x (18). 
If we put for shortness r/a = p, (9) the equation for finding OB 
becomes 
0 Je_ 
p i+p 2 
( 19 ). 
From (6), (8) and (16) the equation to the inner part of the 
boundary ABA' is 
cp sin 6 + pi0 + 52 
(— l) n+1 x sin 2nd 
^...( 20 ). 
From this we at once get OE = ya7ra/2c, whence we must have 
par <2 c ( 21 ). 
If we could suppose c = 0 it would greatly simplify (18) and (19), 
but then it would be impossible to determine the value of OE 
from (20). The difficulty arises from our having chosen a special 
case of equation (5), where only odd multiples of 0 are involved, 
but we see from Art. 5 that there are an infinite number of 
possible forms for (5) where this difficulty does not occur. Sum- 
ming the series in (20) we get for the equation of ABA', 
1 -P 2 
cp sin 0 + pu (6 
’ r )+l 
0 — tan" 
1 +p- 
tan 0 
So from (7) the equation of AD will be found to be 
cp sin 0 + p, (0 - tt) + g 
0 + tan -1 ( J tan 0 
= 0 ...( 22 ). 
= 0. ..(23). 
(18) and (19) contain 2 arbitrary quantities puff and b/c. found 
from (18) must be 
< 7r 1 2 (24). 
From (19) as p must be < 1 we must have 
pulc< 1 (25), 
so pb/c and b/c have to satisfy the 3 conditions (21), (24) and (25), 
but it will be found that these are perfectly compatible, and that 
so we get an infinite number of boundaries of a canal-like form 
(see Figs. 4 and 5) to which the remarks in Art. 4 apply. On 
account of the comparative simplicity of equations (18) and (19) 
and the other conditions it may be well to point out an interesting 
result that readily comes from them. The case is illustrated in 
17—2 
