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 aJ2, if a 2 < 7r/2 and (17) becomes 



fiir = c sin a ± + (p, + -) a x (18). 



If we put for shortness r/a = p, (9) the equation for finding OB 

 becomes 



0=c-*- 



bp 



•(19). 



P 1+P 2 



From (6), (8) and (16) the equation to the inner part of the 

 boundary ABA' is 



op sin 6 + fx6 + b'E 



(-iy 



x ^— sin 2nd 



2n 



[LIT 



...(20). 



From this we at once get OE — /iira/2c, whence we must have 



/jltt< 2c (21). 



If we could suppose c = 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 6 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', 



cp sin 9 + p, (6 — 7r) + 



6 — tan" 



\ — £ tan 6 

 1+p 2 



= 0...(22). 



So from (7) the equation of AD will be found to be 



cp sin 6 + p, (0 — it) + 



+ tan-^ 



-1 



J tan 0Yj=6... (23). 

 «! found 



(18) and (19) contain 2 arbitrary quantities fi/c and b/c. 

 from (18) must be 



<tt/2 (24). 



From (19) as p must be < 1 we must have 



fi/c<l (25), 



so fife 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 



