152 



We get in the present case, 



(p'v)2 = ((b(b-a)2)/4 



p^^v = b(b— a) 



Equation (26) may be written, 



v= ((-kV^r )/((ab(kr2+k,)'/2)(kd''' + k, ■, -) . 



j'(b = p(u+v) + p(u + V) + 2pv)bu +r5 

 and so 



V = K((l/6)(b— a)u + (t'/ct)(u + v) + K'Vt)(u + v)) + -« (34) 



where K = ( -(k) '/2)/(d(ab) 7=) 



The geodesic lines on S are then given by means of t, 

 u^k + ki = (kit').(t2 — 1) 



V = K<t>[t) + (1 



where </>(t) is given in (34) and u = p-'((b— a) (2(a) ' ^t _ 2) + pv) 



V = p-'((6/12)b — (a/12J) 

 If (24) be put in the form 



(du dv) = (u-/r) ((d= — u2)(r= — u-))' - (u-(d- — 2Cd) + C-d-)V - 

 it is seen at once tliat the equation is satisfied by tlie values u — constant. 

 Bat from the geometric consideration it is evident that, in general, the 

 circles u =: constant are not geodesic lines since the normals to a geodesic 

 line must also be normal to the surface. And from figures V and VI it is 

 seen at once that this is only true for the circle u = d , where d , C , and 

 for the trivial case u ^ no matter what the value of d. 



The geodesic lines on the surfaces S i may be studied if the surfaces 



are divided into classes according as d = 2(J . 



In the case d ;= 20 the general integral (26) takes the form, 



V = (((d=r),(2n)) ((du)/((d= — u = )(r- — u-)) 

 which when u = 1/t may be written as 



V = — (— d-r)/2 f(t=dt)/((d-t- - l)(r-t= — 1) 



Here R(t) = d-r'^ t^ — (d^ + r-jt- + 1. It is evident tliat this is exactly 

 the same as the R(t) of the general case if we replace d ' by a and r- by b. 

 Taking note of this we may write the geodesic lines in terms of t 



u = 1/t 



V = (— l/2r) (1,6 (r-— d-)u -f (a',a,(u + v) + (n' f7)(u-v)) + '^ 

 where u = p-'((r=— d-)/(2dt— 2) + pv^and v + p-'(6 -'— d=)/(12). In tliis 

 case tlie geodesies are real for all values of r. 



