THE KANSAS UNIVERSITY 

 SCIENCE BULLETIN. 



Vol. XIII.] MAY, 1920. [No. 5. 



A Calculation of the Invariants and Covariants for 

 Ruled Surfaces.* 



BY E. B. STOUFFER. 



IN Wilczynski's Projective Differential Geometry of Curves 

 and Ruled Surfaces f it is shown that the projective differ- 

 ential properties of a non-developable ruled surface may be 

 studied by means of a system of differential equations of the 

 form % 



(A) ^ ?/" + 2?>n?/' + 2pi2 2' + giii/ + gi2 2;=0, 

 '\ z" -^2pny' + 2pnz' -{-qny + q2-2Z = 0, 

 where Pik and qi^ are functions of the independent variable x. 

 The most general transformations leaving (A) unchanged in form 

 are given by the equations 



,,, \ 1/ = «11 ?/ + «12 2, 



(1) "j _ _ A = «ll«2-2 — «12'^21 ?^ 0, 



[ z = any-{- a<n z, 



(2) ^=nx), 



where a-i^ and I are arbitrary functions of x . 



A function of the coefficients of ( A ) and their derivatives and 

 of the dependent variables and their derivatives which remains 

 unchanged in value by the transformation ( 1 ) is called a semi-cova- 

 riani and if it remains unchanged in value also by the transforma- 

 tion (2) it is called a comn'aw^. Semi-covariants or covariants 

 which do not involve the dependent variables or their derivatives 

 are called seminvariants or invariants, respectively. The invari- 

 ants and covariants of system (A) are used in the study of the 



* Received for publication May 10. 

 t Teubner, Leipzig, 1906. 



t Wilczynski writes his system without the factor 2 in the coefficients of y' and z'. Its 

 introduction makes some of the results appear in simpler form. 



(59) 



