151 



so that 



v=(— k^ =d( (a(kd= + ki)) (t + 1 2(a)' nog((a)> =t— 1) ((a) ', ^t^l) + (5 



(28) 



Eliminating "t" between (25) and (28) we hare the the geodesic lines for 



r = d given by 



— (u^k + k,)' = (kd2 + k,)i 2 d(u-'k + k,)i -'-uikd-' + k , ' ^ _^ r5 



V := log 



u.d 2d2 d(u2k + kj» = + u(kd = + k,) ', ^ 



(The above equation is equation 29.) 



When r — d (26) gives rise to an elliptic integral for the reduction of 



which we recall from the general theory of elliptic integrals. (See Note 1. ) 



R(x) = Ax^ + 4Bx3 + 6Cx- + 4B'x + A' 



g2 = AA' — 4BB^ + 302 



g3 = ACA' + 2BCB' — A'B^ AB'-' — C^ 

 In this case we have. 



R(t) = abt^ — (a+b)t- + 1 



g, = ab + (a+b)= 12 



g3 = (_ab(a+b)) 6 + (a+b):* 216 

 We also have 



R'(t) :=:4abt3- 2 a + b t 



R''(t) = 12abt= — 2(a + b) 

 Substituting in (26) 



t = .^ + (1 4R'(f)i (pu — 1 24R"(0) (See Note 2) (30) 



Wliere - is one of the roots of R(t) ;= 0. In this ease take f ^ 1 (a) '/^ 

 then, R'(l (a)' =) ^ (2(b— a) i (a) ^ ^ 



W (1 (a)i -) = 2( b — a) 

 So that (30) may be written, 



t = l (a)' - + ((b-a) (2(a)"-)) (pu-pv) 

 when pv ^ (1 12) (5b — a) and therefore 



abt= = b + (b(b— a)) (pu — pvi + (1 4)((b(,b -a) =) (pu — pv)- 

 Recalling now that, 



(p'v) = = 4p3v - g^pv — g3 (.31) 



p^'v = 6pv — 1 2 g, (32) 



and also, 



(p'v) 2 ' ( pu — pv ) 2 4- ( pu— p"v) / ( pv) = 



p(u + V) +p(u— V) -2pv (.33) 



Note 1.— Klein, Modiilai", Fuiictioueii, Vol. I, p. 15. 

 XoTE 2.— Enueper, Elliptisclie Functionen, p. 30. 



