843 



i^kz* = 2sin^iirf-^^il ^cos^^cf). . . . (21) 



or 



z\/Jc = 2 si?i 4 ff — — — - (2 cotan i- (f — siti 7 ) '), • • • (22) 



óly k 



whence 



u\/k=log tan | </)-f 2 cos ^ ^~^T~7L ^^ ^^^ ^"^* T '/' H" T ^^^^^* \tp-eüs (f - |). (23) 



óly' fC 



Putting (f successively equal to — , n and — the coordinates of 

 iV, Q and R are found as follows: 



(24) 

 I 0,372i 



u^^V^- = |%(l/2-l) -f l/2|-^^|3M^/2-l)+^/2-lj=0,532+-j^| 



ili^ = 7x z(^l/A=2-— ^ WQ 1/^ = . . . (25) 



3/1 1 0,609 I 



r. =— , r i/A:=v/2_—— (2v/2-l)= 1,414- 



\ (26) 

 1 0,0391 



"^l/^=%(|/2+l)-l/2-^3%(p/24-l)-(|/2+l)!=-0,532H-^| 



^ 8. The outside portion of the curve /^*S J', corresponding to values 

 of (f , which differ but infinitely little from 2 jt and extending to 

 infinity, is again determined by (9). Since, however, in this case z 

 approaches zero at infinity, the solution will now be 



zz=:aiH^^\iz\/k). (27) 



where H^^' is Hankei/s function of order zero and of the first kind. 

 The integration-constant n is found by joining on to equations 

 (22) and (23j putting y = 2 .t — tf?, where t|> is infinitely small. In 

 the same manner as in § 5 we then find 



logz\/k=z~u\/k — 0,QU—^, ..... (28) 



whereas for large vahies of x differing little from / equation (27) 

 gives 



^) This is a known formula, except for a small reduction (cf. for instance Win- 

 KELMANN, loc. cit., p. 1148). See also A. Ferguson, Phil. Mag., (6). 24, (1912) p. 837. 



