73 



I =^^2 l^E du2 + 2F du dv + Gdv^ .dt 



Denote t/E u'^ + 2 F u' v' + G v'^ by F. Then the first condition for 



d 

 a minimum of I is Fv — -sr Fv' = o|| 



d 

 Now, in this case Fv = o, so that -tt Fv' =: o 



Hence Fv'=:'', or substituting tlie values E, F and G this becomes 



u2 vi 



= =6 



J^^^ +u3y- 



Wlien '5 — o, v' = o, hence v:= constant, i. e., tlie meridians are geodesic 

 lines. 



When (^ ^ o 



r 'i d- ui 



( 1 ) V = s — i — - + (V 



'' -^ 2uV(d2— u^) (u^ — (55) ^ 



Making the substitution u^l t, (1) becomes 



r—ScV t2 dt 



(2) V— L, ^_ + 6' 



•^^v/(t2 d2 — 1) (1 — d2 t») 



We have for the reduction of the general elliptic integral 

 *R(x) = A x^ + 4 B x3 + 6 C x2 -f 4 B' X + A' 



g2=AA' — 4BB'4-3C2 



gg = AcA' 4- 2 BcB' — A'B2 — AB'^ — c^. 



These become in the present case 



R(t) = (t^d^ — 1)(1 — <?2t- ) = — JMn* + (d2 4-<52 )t2 — 1 

 (d2 + (52) 



ga — 6" I 6 



We get also 



Il'(t) = — 4 f52dn-3_^2(d2 4-,?2) t 

 R'(t) = — 12 (52d2t2 + 2 (d2 + rf2) 

 Making the substitution 

 i- R'(a) 



Where a is one of the roots of R(t), say 1 d, we get 



pu — pv where pv:=yV(cl^ — 5'^'*) 



II Kneser, Variationsrechnung. Fv denotes function v. 

 " Klein, Ellip. Mod. Funetionen, Vol. I, p. 15. 

 t Enneper, Ellip. Funetionen, 1890, p. 30. 



