243 



there is one, and only oue, surface S2 which represents the result of deform- 

 ing Sj so that a conjugate system remains a conjugate system after the 

 deformation. 



There are two cases which may occur under (III). Suppose that 



Then ? = d= 1 and the surface S2 is such that its second fundamental 

 magnitudes D2 and D2" are either equal to the corresponding magnitudes 

 Di and Di" of Si or they are the negatives of Di and Di". 



But from the equations (*) 



,\^ — eg — f 2 I ^ fS,u "^ * 77 J 



(5 11 eg — f^ I (^ fj. '5i' ) 



Where e, f, g are the fundamental magnitudes of the Gauss' sphere, it 

 is seen that a change in the sign of D and D'' corresponds to a change of 

 sign in the co-ordinates x, y, z of the surface, and therefore the surface S2 

 is either identical with Si or it is symmetrical to Si with respect to a plane 

 or to the origin of co-ordinates. 



Suppose next that 



In this case there is a "inique value of ^2 =#=1. Si may therefore be de- 

 formed so that after the deformation is carried out the lines fi = const. , 

 iz= const., form a conjugate system, although they do not form a conju- 

 gate system at any time during the deformation. Now, by a theorem of 

 Dini, (**) from relation (Illb) no surface So exists, the spherical images of 

 whose asymptotic lines are tlie same as the spherical images of a conjugate 

 system of lines on Si. But from the definition of associate surfaces, there 

 is then no surface to which Si is associated, and thus we have the result 

 that when (Illb) is true for any surface Si referred to a conjugate system, 

 there exists no surface So to which Si is associated. 



( * ) Bianchi, 1. c. p. 134. 

 ( *■■ ) Bianchi. 1. c. p. 125. 



