Section III., 1915 [I5i] Trans. R.S.C. 



A Certain Projective Configuration and the Integration of its 

 Normal Equations. 



By Charles T. Sullivan, B.A., M.Sc, Ph.D. 



Presented by J. Harkness, F.R.S.C. 



(Read May meeting 1915.) 



Introduction 



The investigation of geometrical problems frequently leads to 

 the solution of differential equations of the utmost importance in 

 other branches of pure and applied mathematics. Unfortunately, 

 however, many of these important formulae become known only to 

 those acquainted with highly specialized methods of geometrical 

 analysis. It is primarily with a view to emphasizing certain 

 results arising from geometrical considerations that this investi- 

 gation has been undertaken. 



In a recent paper* based on Wilczynski's Projective Theory of 

 Curved Surfaces! and devoted to a study of surfaces whose asymp- 

 totic curves belong to linear complexes, the writer has shewn that 

 these surfaces are organically related to a certain quadric, called 

 the Directrix Quadric. When the Directrix Quadric consists of two 

 coincident planes the Normal System of differential equations of the 

 surfaces reduces to a form which has since been integrated by Wil- 

 czynski (Directrix Curves, Math. Annalen, Dec, 1914). But the 

 geometrical basis of Wilczynski's investigation and the idea of a 

 Directrix Quadric are points of view entirely unrelated. It therefore 

 seems that the integration of the Normal Equations from this latter 

 point of view is of sufficient interest to merit a separate discussion. 



The integration of the Normal System can be effected by a more 

 direct analytical method than that employed here, J but the present 

 treatment is essential to a complete analysis of the geometrical con- 

 figurations involved. Moreover, it enables us to utilize Wilczynski's 

 formulae for the resolution of the general solution into a fundamental 

 set of solutions. 



*Transactions of the American Mathematical Society, Vol. 15 (1914). References 

 to this paper will be made under the symbol 5. 



tThe projective theory of curved surfaces has been developed by E. J. Wilczynski, 

 in a series of five memoirs published in the Transactions of the American Mathe- 

 matical Society, Vols. (8-10), (1907-1909). These memoirs will be cited as Mi, 

 Ml, etc. We shall also have occasion to refer to Wilczynski's memoir on One- 

 parameter Families and Nets of Curves (Trans. Am. Math. Soc, Vol. 12 (1911)); 

 we shall hereafter refer to this memoir under the symbol N. 



JFrom purely analytical considerations the form 



. f [ du dv 



J J 9 (u,v) 



of the general integral of the Normal System has been known for some time to the 

 writer. 



