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 t 



(A) ^ ?/" + 2pii?/' + 2pi2 2;' + gn?/ + gi2 2;=0, 



^ ^2" + 2pn y' + 2p'22 z' + q-n y + q22Z = 0, 



where pik and ^iu are functions of the independent variable x. 



The most general transformations leaving (A) unchanged in form 



are given by the equations 



(1) •< _ _ A = «ll'^22 — «12«21 5^ 0, 



( 2 = "21 ?/+ '^22 Z, 



(2) ^' = Hx), 



where ai^ and c 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 covanaw^. Semi-co variants 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 publica'tion 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) 



