841 



found in the same manner as equation (4). Representing by z^ the 

 minimum of z (ordinate of the lowest point F, fig. 2) and putting 

 (f=zO for z=^z^ we find 



8 



(the positive and negative signs corresponding to y > and <^ <I 

 respectively) '). Since the curves have to join oii to each other, it 

 is easily found, that for the n'^^ curve 



4 \/n 



where the abscissa of F may conveniently be taken for /. 



Putting At}> = y^\ — ;.' sin* if>, where /* = and ip=: h {n-\-(f), 



4 + ^z," 



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



zp/^ = 2Z\if' M 1/ '^ = I — — 2 I Aif> <iif?, . . (16) 



u being the abcissa of an arbitrary point P of the curve relatively 

 to point F, i.e. u=zxp — xp; the total width Ip of the curve is 

 therefore such that 



2pi/k = {.,H — ^r:B)\/k^^2{K-2E), ... (17) 



K and E being the complete elliptical integrals [ j ) of the 1*' 



and 2"^ kind respectively, with modulus ;.. Now A is but little smal- 

 ler than \, viz. 



2n 

 A=l— i^z' = l ; (18) 



K is therefore very large, namely in a first approximation ') 



4 2w 



K = log — = i /09 8 — i loq . . . (19) 



•V2(l-A) ^ ^^l\/k ' ^ ^ 



E being finite, viz. =r 1 ; it follows that 



z,\/k= \,0S2e-J>^J' (20) 



M The change of sign at ^ = is due to the change of sign of sw ^ 9, whence 



for ^ > y'kiz^—ZQ^) = -{-2sin^<p and for ip < = — 2 sin ^ tp. 



*) See for instance Winkelmann, loc. cit,, p. 1134. 



*) See for instance 0. Schlömilch, Compendium der höheren Analysis. 3e Aufl. 

 1879, II, p. 322. 



