844 



z— 2a- — = =^ h— i-, • • (29) 



l/2.-T;rl/y(: 1/2^ \/l\/k V V 



Therefore, putting .?•/? = r and considering, that iV k =^ l\/c 



-\-UR\'k (eq. 26), 



a['k= h . 0,924 y^^Tiry/ke'-y'^ (30) 



^ 9. According to (21) (f cannot become zero; on the other hand 2 

 can become zero at a point M (fig. 3) where (p goes through a 

 smallest value. The continuation of the curve QNSM to the left 

 again consists of a series of elongated «S-shaped curves, as repre- 

 sented diagramniaticallj in fig. 3'). 



The width of these curves is again small compared to / (abscissa 



of point M) and the following equation is found for them 



8 

 /:^» =4»m' ir/^ - r^,«' ± — -(1— cW ^y). . . (31) 



olyk 



(the positive sign referring to the part, where c>0, the negative 

 sign to the rest); 2 becomes zero for 



(n = y„, = or for (f = 2.T — ^/m , • . • (32) 



\/Sl[/k 



according to whether the order n of the point .]/„, where the n 



curve intersects the axis, is odd or even *). 



Introducing a new angle if' such that *) 



cot i rr / 2j/n\ 1 



sin ^ = = 1 f -— - co.'^ ^ rf = - cos ^ (f, . . (33) 



cos ^ (fm \ olyk J k 



we have in first approximation, as in the two-dimensional problem, 

 ^l/A;= ± 2 cos i|? ±u\/k—\ ^ 2 I Atprfifj, . (34) 



u being measured from M. The total width 2yj == ^q — -^l is therefore 

 given bj the same expression as in ^ 6 ; hence 



r/^,„= 1,082 «-/'i/i , (35) 



1) Nielsen, loc. cit. 



') Gomp. WiNKELMANN, loc. cit., p. 1138, fig. 400a. 



») In the first case the curve rises with increasing values of x, and <p changes 

 from TT to TT through <pm, in the other case the curve falls and c changes from 

 TT to T through 2t — <Pm- 



*) Cf. WiNKELMANN, loc. Cit., p. 1137. VVc shall take ^ in such a direction, that 

 ^=i ± ^ {tt—'s) + y. (infinitely small), with + for odd curves (cc < t), and — lor 

 even curves (f > n"). 



