362 



B'C'D' with (VVO points of inflexion B' and D' for ^ <; ; 



the dotted line (two circular arcs) represents 

 the transition between the two cases for 

 ^ = 0. 



In that region the equation to the surface 

 may he written in the form 



^+^=^-<^''+^^^ 



(7') 



Fig. 3. 



where hc = h -\- ye, ye being the ordinate 

 of C (or C), and »i = // — ye- In the region 

 under consideiation, howevei', i] is small 

 compared to Z^, so tliat in second approxi- 

 mation 1} may be neglected with respect 

 to he and therefore with the same degree of accuracy to which hitherto 

 the deviation from the circular shape was calculated we may write : 



~~i--~-=khc constant (14) 



In third a|)proximation BCD is thus a part of the curve which 



was called uodoid by Platkau, B'C'D' a part of an onduloïd. 



The equations of these curves are known *) ; but in our case they 



2 . . 



may be materially simplified. Putting /Jif = ■^) (he first integral of 



(14) in the case of the nodoid {sin tp = for ,/■ = .v^) will be 



r ^x sin fp z:=. x"^ — xg* ....... (15) 



If .<;, and ,i\ [= .V() are the maximum- and minimum-values of .t 

 corresponding to ,sin (p = 1 and sin <p = — i we have approximately 

 since ./'/ƒ is very small with respect to i\ (see eq. 12') 



.^•,=r„ = i^/) 



X, of X( 



xjj 2 



= — kR„ 



r, 3 " 



(16) 



Further it follows from (15), as long as ,v is small with respect to r^ ^) 



, X + l/;»"— V 1 ^~— 



=h >^ = .r, log ^^-p^ w y x^ — - ir, . 



2R, 



(17) 



1) See for instance Winkelmann, loc. cit., p 1150. 



•) In first approximation ?-y = i?o; '" second approximation W^r =/,;(/? -f 2/?y) = 



2 2 



= — + 2A;Eo = ^ (1 + kR,), so that r,, = i^o (i - kRo"-). 



8) Here Xi belongs to the nodoid and has thus not the same meaning as xa 

 in § B. 



■*) Since in that case 



di] x^ — X* XB* — x' 



dx 



R^x,—x^ 



l/r.'.c" - {xjf -xy Vi\\x'' -XB* ^» V'x*— .^•,' 



