73 



I ==/t2 i/E du2 -I- 2F du dv + Gd.v2.dt 



Denote i/E u'^ _[- 2 F u' v' + G v^^ by p. Then the first condition for 



d 

 a minimum of I is Fv — ^ Fv' = o|l 



d 

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



Hence Fv' = i\ or substituting the values E, F and G this becomes 

 u2 v^ 



J^^^ +U2V- 



\4u2(d'^— U2) ^ 



When (^ = o, \^=zo, hence v = constant, i. e., tJie meridians are geodesic 

 lines. 



When (5 = 



r s d^ ui 



^^^ ^^-^ 2uV(d2— u2) (u"2^^^^d2y + ' 

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



r—dd'' t2 dt 

 (2) v= s — -- — 4- J^ 



^ ' '^2,/(t2 d2— 1) (1—^2 t2) ^ 



We have for the reduction of tlie general elliptic integral 

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



g2==AA' — 4 BB' + 3 C- 



g3=AcA' + 2 BcB' — A'B2— AB'2_c'5. 



These become in the present case 



R(t) = (t2d2— 1)(1— rS2t2)— _rf2(i2t4-^ (Cl2_^j2^t;2 — 1 



(d2+(52) 

 g3=<12d2+^-^r 



f52d2(d2 + J2) rd2 + J2-] 3 



-] 



^3— 6 16 



We get also 



R'(t) = — 4 f52d2t3 + 2(d2+<S2) t 



R'(t) = — 12 d2d2t2 + 2 (d2 + fS2 ) 



Making the substitution 



_ 1 R'(a) , 



(3) t-a + ^^^^_^i^ j^,,(^^^t 



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



i(d2-,;2) 



p u — p V where j^ v := ^V ( d2 — 5^2 ) 



II Kneser, Variationsreehnung. Fv denotes function v. 

 •'■' Klein, Ellip. Mod. Functionen, Vol. I, p. 15. 

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



