MORENO. — ON RULED LOCI IN W-FOLD SPACE. 1-11 



a restricted system equivalent to two independent equations. The 

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

 equations 



9W 9U> 



9 x 9 y' 



9 V 9 V 



9x' dy' 



and 



L W , = AU , + x (^ dxl + J^L i W + . ..) + .. .. = <>, 



(9' 2 V 9 2 V \ 



A V" = A V + x ~ dx> + ^-p^ dy' + ....+.... = 



\9 x'~ 9 x' d y' I 



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' = 

 V =0 



U"= U' + ^jdx' + = 0, 



d x 



9 V 

 V" = V + %^dx> + = 0, 



dx 



From these three sets of equations we derive 



(9 2 U' 9' 2 U' \ 



x [jx^ dx ' + 9*9y-< dy ' + ' ' ' 'J + * * • ' = °' 



(d 2 V 9 2 V \ 



x \9^ dx ' + d*w? d * + • • • ■) +■ • • • = °' 



9 J^dx> + .... = 0, 



9x' 



9 V 



? rT rf a r' + .... = 0. 



dx' 



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

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

 eliminate the other four. We have 



