242 



(1) 



Tv ^^) --\ 1 , I ^ -tJ 



f 12 ) ^ ( 12 ) ^ 



where ^ 1 ^ and , « ^^^ tl^^ symbols of Christoffel formed with 



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



Oil Op dv Ofi 



as the condition of integrability, 



[.5r "I 2 I ^\ 1 / \ 2 )■ J - |^,V' I 1 J ~ I 1 I' \ 2 / J 



Having given tlie 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 tliat may occur under (2). Suppose, 

 first, that the surface Si is such that 



/TN ' _^ f 12) ^_ „ f 121 ^ f 12) '_ ^ f 12\^ 



^ ' <U'\ 2 )' — ^ (. 1 / ■( 2 r — rW/'l 2 ) 



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

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

 surfaces S2 which are applicable on Si and sucli that tlieir 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 



±n2] '_^ fi2] ' n2\' i. "^ 12 1 ' 9 f 12 1 ' ( 12 \ ^ 



rfrl 2 J" — ^ "I 1 I \ 2 i ,h'\ 2 y ^^ \ 1 |- \ 2 j 



(II) or, 



(J f 121 ' „ ( 12) ^ f 121 ^ '5fl2)' „fl2l'fl2 



rf f 12) ' 2 (12^ ri2\^ '5fl2l'_„fl2l 



2 



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

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

 Suppose, finally, that 



^ f 12) ^^9 ri2) ' ri2l ' 



(III) 



'^ (12) '9 f 12) ' ( 121 ' 

 6u\ 1 j" ^"^ \ 1 i l 2 )■ 



We have in this case one, and only one, value for 'A-. If the surface Sj 

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



