152 



We get in the present case, 



(pV)2z=: ((b(b— a)=) 4 



p'^v = b(b— a) 

 Equation (26) may be written, 



v= ((-k^ =r ) ((ab(kr-+k,)V2)(kd=+k, ^ -) . 



("(b = p(u+v) + p(u + V) + 2pv)bu +'S 

 and so 



v = K((l 6)(b— a)u + (<7' (T)(u + v) + (g' a)(n + v)) + fi (34) 



where K= ( — (k) i.'2)/(d(ab) V/ =) 



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

 u-k 4- kj = (kjt^) (t= — 1) 



V = K o t) + £5 



where o(t) is given in (34) and n = p-M(b — a) (2(a) ' -t — 2) + pv) 



V = p-M (5 12)b — (a, 12J) 

 If (24) be put in the form 



(du dv) = (u^ r) ((d- — u2)(r= — u = ))' ■ (n=(d= — 2Cd) + C=d=)' - 

 it is seen at once that the equation is satisfied by the valnes 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 Sj may be studied if the siufaces 



are divided into classes according as d =: 2C . 



In the case d = 2C the general integral (26) takes the form, 



V = I ((d=ri (2a)) ((dn)'((d-' — u = )(r= — u-)) 

 which when u = 1/t may be written as 



V = — (— d2r)/2 f (t-dt)/((d2t= — 1)(tH'- — 1) 



Here R(t) = d-r- t* — (d- + r=)t- + 1. It is evident that this is exactly 

 the same as the Il(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 = l;t 



v = (— 1 2r) (1 6 (r-— d2)u + (crc.(u+\) + (c' o)(u-v)) + A 

 where u = p-H(r-— d2)/(2dt— 2) + pviand v + p-'(5 --d-)/(12). In this 

 case the geodesies are real for all values of r. 



