172 THE ROYAL SOCIETY OF CANADA 



V' = EG-F'', X, Y, Z =- I ^' ^' ^' 



V II X2 y-i 22 



FT =m G-n F, V^A =n E-m F, 

 VV =m'G-n'F, V'à' =n' E-m' F, 

 V'T" = m"G-n"F, V^A" = n"E-m"F. 

 The Gauss differential equations of a surface are 



(1) A-ii= L.Y+XiF +.r2A, 

 yn = MY+y,T'-^y2A', 

 S22= iVZ+Zir" + Z2A", 



with similar equations arising from these by permuting x, y, z. 

 The integrability conditions of (1) are: 

 The Gauss equation 



T'=-h(E,,-2Fr,+ Gn) + (E,F,G){T',A'y-(E,F,G){T,A){T\à"), 



(2) and the Mainardi-Codazzi equations 



al©-4(f) = -^-— ---). 



If the surface defined by the above equations be non-developable, 

 then a curve C(p, q) traced upon the surface will be a straight line, 

 provided that this curv^e is both an asymptotic line and a geodesic line 

 on the surface (G. p. 202). These conditions are expressed by the 

 equations: 



(3) (i) A~Lp"' + 2MT'p'q' + Nq'' = Q, 

 (ii) Dx = Vp'^-{-2V'p'q'+V"q'^+p" = 0, 

 (iii) D2 = Ap'^ + 2A'p'q'-^AY^-hp" = 0, 



where , dp , da 



ds ds 



If now the asymptotic curves be chosen as parametric curves, and 

 if also the curves /> = const, be straight lines, we have the relations 



and from (3 — (ii)). 



dq dq dp 



These conditions lead to great simplification in the equations with 

 which we have to deal (G. p. 73). The quantities (r, T', V"; A, A', A") 

 are now expressible in the following convenient forms: 



