Section III, 1918 (171] Trans. R.S.C. 



Concerning the Integrals of Lelieuvre. 



By Charles T. Sullivan, Ph.D., D.Sc. F.R.S.C. 

 (Read May Meeting, 1918). 



INTRODUCTION. 



It is proposed in this paper to discuss the form assumed by the 

 Integrals of Lelieuvre when they fulfil the condition that one family 

 of the parametric curves consists of straight lines. In their final 

 form the expressions obtained are not wholly free from quadratures, 

 but they are entirely general and sufficiently simple to be readily 

 applicable and serviceable in the study of scrolls. The procedure 

 employed in this discussion is suggested by a comparison of the 

 equations of Lelieuvre with those of Gauss for the representation of 

 the coordinates of a point on a surface in terms of two independent 

 parameters p and q. The notations and formulae of the general 

 theory of surfaces which we require in this work are drawn largely 

 from Forsyth's Differential Geometry; and this treatise will be cited 

 hereafter, for brevity, as (G). In the concluding section of this 

 paper, we shall apply the formulae developed in the earlier sections 

 to a study of Scrolls whose asymptotic tangents are contained in 

 linear complexes. 



The Gauss Equations. 



It will be convenient to have before us a number of expressions 

 from the general theory of surfaces, and these we shall now write down. 



The fundamental magnitudes of the first and second orders are 

 (£, F, G) and (L, M, N) respectively. The symbols {X, Y, Z; r. r', 

 r''. A^ ^'^ ^". V; T; K) have the following definitions: 



Let m, m', in", n, n', n" be defined by the equations 

 w = SxiXn, m' = i:xiXi2, m" = i:xiX22, 

 n = 20:2X11 , n' = "^xoXii, n " = 'LXiX-n, 



where the summation extends over the three axes and the subscripts 

 denote differentiation ; thus 



dx dx d'^x 



The quantities given above are then defined by the equations 



Sec. III. Sig. 13 



