242 

 (1) 





where -^ , - and , ^C' - are the symbols of Christoflfel formed with 

 respect to the Gauss sphere. Since now 1^ — r ^ 1 — t- we have from (1), 



r)// ()!• or Oft 



as the condition of integrability, 



to, y2\ '^ '121' „fl2X' ( 121 '] _ [^ f 12) ' 0(12^(12)^1 



•''^ [rfr ( 2 )• — ^\ 1 J \ 2 r J - ^<h, { 1 ; ^\ 1 r 1 2 ; j 



Having given the surface Si, then to every value of > which satisfies (1) 

 and (2) there corresponds a surface S2 of the desired type. 



There are three possible cases that may occur under (2). Suppose, 

 first, that the surface Si is such that 



In this case the condition of integrability (2) is satisfied for every value of 

 >., and since equations ( 1 ) are of the first order, there are in this case 00^ 

 surfaces S2 which are applicable on Si and such that their parametric lines 

 form a conjugate system. We thus have in this case a continuous system 

 of surfaces, and the above equations are the necessary and sufficient condi- 

 tion that a surface may belong to such a system. 

 Suppose, next, that Si is such that 



'^ I 12 ) ^_ o ri2 1 ' / 12 -) ^ ■ i_ * 12 ) ' . „ f 12 ) ' f 12 \ ^ 



Ji'X 2 r - \ 1 r I 2 i r5r I 2 J^ ^ ^ \ 1 |- \ 2 ) 



(II) or, 



rf f 121 ' ( 121 ' ri2-i ^ '^ f 12) '_„ ri2) ' n2\' 



jjix 1 ( ^ ^ \ I i' \ 2 I ruM i -'^ \i i \2 i 



In this case > vanishes or is undefined, and the condition of integrability 

 is not satisfied. Consequently there exists no surface S2 in this case. 

 Suppose, finally, that 



(in) 



(5 f 12) ' „ f 12) ' f 12) ^ 



6v\ 2 / ^^ \ 1 r \ 2 ) 



'5 ri2) ' ^ f 12\' ) 12) ' 



,5,/ \ 1 1' ^ "= \ 1 ) \ 2 r 



We have in this case one, and only one, value for >.^. If the surface Sj 

 is such that in addition to (III) being satisfied, (1) are also satisfied, then 



