60 THE UNIVERSITY SCIENCE BULLETIN. 



properties of ruled surfaces. Their calculation as given by Wil- 

 czynski involves the solution of several rather complicated sys- 

 tems of partial differential equations. It is the purpose of this 

 paper to obtain the same results by much shorter methods. 



In 1915 Green published a paper* in which he obtains the in- 

 variants and covariants of the general form of the system of 

 partial differential equations associated with curved surfaces 

 from the invariants and covariants of a canonical form of these 

 equations. Green points out that his general method is of wide 

 application. This scheme of making the calculations first for a 

 simplified system and then transforming to the coefficients of the 

 original system is used in the present paper. 



The results in this paper carry the same label as do the corre- 

 sponding results in Wilczynski's book but there are differences in 

 numerical coefficients and in signs because of the introduction of 

 the binomial coefficients in equations (A) and because of a 

 change of sign in the defining expression for wik. 



1. The Semi-Canonical Form. 



Let us make the transformation (1) upon the system (A). 

 There immediately results the system 



'«ll2/"+ «12 2"+ 2 ("'u + Pll«ll + Pl2«2l)?/' + 



2 («'l2+ Pll«12 + Pl2«22) Z' 



+ ( «"n + 2 pii « 'ii + 2 pi2 « '21 + qw'w + gi2 «2i ) y 



+ {""n + 2 pu « '12 + 2 P12 « '22 + q\\ «i2 + 912 o-ii ) 2 = , 



«2i y" + «22 2:"+ 2 ( « '21 + P21 «ii + P22 «2i) y ' + 



2 ( «'22 + P21 «i2 4- P22 "22) z ' 

 -t- ( »-"i\ + 2 P21 « '11 + 2 P22 « '21 + Q21 «ii + Q22 «2i ) y 



^ + ( «"22 + 2 P2I « '12 + 2 P22 « '22 + Q2I «12 + 922 «22)2; = 0. 



If ajt are so chosen that 



2 



(4) «'ik= - - Pij «jk, (^^ = 1,2), 

 i=i _ _ 



the coefficients of y' and z' in (3) vanish. Such a solution for 



ajk is always possible since it is equivalent merely to choosing 



(«ii, «2i) and («i2, «22) as two distinct pairs of solutions of the 



system of differential equations 



(>' = — ( Pll /' + Pl2 <^ ) 

 '^ ' = — ( P2I /' + 2522 <^ ) . 



r 



(3) J 



*G. M. Green, On the Theory of Curved Surfaces, and Canonical Systems in Projective Differen- 

 tial Geometry. Transactions of the American Mathematical Society, Vol. 16 (1915), pp. 1-12. 



