MORENO. 



ON RULED LOCI IN Jl-FOLD SPACE. 141 



a restricted system equivalent to two inde^Dendent equations. Tlie 

 tangent planes at P' and P", any two consecutive points, have for their 

 equations 



9U> 9U' 

 d x' 3 y 



9V' 9 V 

 AV' = x'^+y%^ + .... = 0, 

 9 x' d y' 



and 



/52 U' 9'^ U' \ 



Let us take the first two equations in each set to be independent, then 

 the rest are superfluous. Since P' and P" are points of the surface, 



U' =0 



F' =0 



9U' 

 c' x' 



9V' 

 V" = V + ^dx' + = 0, 



d x' 



From these three sets of equations we derive 



"" \9x'-' ^ 9x'9v'^ +....; + u, 



X 



fd'V , , 5-F' , , \ 



9 x' 9 y 



-^r—r dx' + . . . . = 0, 



9x' 



9V' 

 dx' 



These four equations are homogeneous in the five differentials dx', 

 d >/', . . . We may take one of these differentials to be zero and 

 eliminate the other four. We have 



