[SULLIVAN] CONCERNING THE INTEGR.\LS OF LELIEUVRE 173 



(5) 



In the case of a real surface, F does not vanish for real and inde- 

 pendent parameters p, q. Also for a non-developable surface having 

 its asymptotic curves parametric, M is difïerent from zero. The 

 expression 



MV=(xiy2Zi2) 



is therefore difïerent from zero and may be used as a divisor without 

 further discussion. 



TJie Integrals of Lelieuvre. 

 It is known that a non-developable surface referred to its asymp- 

 totic curves as parametric curves can be represented by the Integrals 

 of Lelieuvre (Eisenhardt — Diflferential Geometry, p. 194) 



J \ dp dp/ \ do day 



dp dp/ \ dq dq 



dp dp/ \ dq dq 



J V dp dp/ \ dq dq/ 



dp dp/ V dq dq 



where 6i, 62, dz are three independent particular solutions of the Laplace 

 equation with equal invariants: 



dpdq 

 Throughout the subsequent discussion, in writing down deter- 

 minants whose elements are built up from ^1, Q2, O3 and their derivatives, 

 we shall use the familiar notation employed to express MF in terms of 

 X, y, z above. Thus from (6) we find at once 



\dq dp/ \ dpdq/ \dq dp/ 

 by virtue of (A), and similar expressions for y, z. 



