241 



Conditions for the Deformation of Surfaces Referred to a 

 Conjugate System of Lines. 



Burke Smith. 



When a surface is subjected to a series of deformations, each form that 

 it assumes during the deformation may be thought of as a separate, dis- 

 tinct surface. We may thus regard a deformation of a surface as a 

 continuous system of surfaces, each representing some form into which 

 the original surface may be deformed. In this paper we consider the 

 problem of determining those surfaces which may be deformed so that a 

 conjugate system of lines will still remain a conjugate system after the 

 deformation is carried out. 



We sliall suppose that the equations of the surfaces that we consider 

 are given in the form, 



X = f 1 (//, V), y = fj ifi, i), z = f 3 ifJ, "). 



and that the first and second fundamental magnitudes are E, F, G and 

 D, D', D", respectively. 



If S2 represents the form that Sj takes when deformed so that a conju- 

 gate system remains a conjugate system, then Sj is applicable on Sj. But 

 the necessary and sufficient condition that two surfaces should be appli- 

 cable on each other is that tlicy shnll liave the same lineal element and the 

 same total curvature. 



If the parametric lines, ," = const. , 1-= const., on Sj and Sj form a 

 conjugate system, then D'= o for both Sj and S,. Since Sj and Sj must 

 have the same lineal element and the same curvature, we have from the 

 relation, 



K ">">" 



EG— F2 



that D2 = > Di and D^'^ =^ / D/^ where the subscripts refer to Sj and S2 

 respectively, and / is a function of f and v. To determine ?i we make use 

 of the fact that Codazzi's equations must be satisfied for both S^ and S^- 

 Bianchi* lias thus shown that X must satisfy the equations, 



r/ii:) — \2 1 It j 



*" Vorlesungen iiber Differential-Ueometrie," p. 336. 



16— A. OF SCIKNCE, '04. 



