[SULLIVAN] A CERTAIN PROJECTIVE CONFIGURATION 155 



The invariance of the following functions under transformations 

 of the type (8) can be readily verified either by direct substitution or 

 by infinitesimal transformations: 



(9) a', h, 



6" = [b\ -2bb^ - T'b(a'u + g)], 



ou ov du ov 



These Invariants possess a geometrical significance of the utmost 

 importance in the succeeding discussion. 



Geometrical considerations. 



Through every point Py of the surface 5 two asymptotic curves 

 r' and r" pass; the tangents to V and T" at Py are the lines Py Pg 

 and Py Pp respectively. As the point Py moves over the surface S 

 these lines generate two congruences (C) and (G") ; and both of these 

 congruences have S as focal surface. If v remains constant and u 

 varies, the point Py moves along the curve V, the tangent Py Pz 

 describes a developable of the congruence (G'), and the tangent 

 Py Pp describes a ruled surface R"{v) of the congruence {G"). In 

 like manner the tangent Py Pp describes a developable of {G)" and 

 PyPz describes a ruled surface R'{u) of {G') as Py moves along T" . 



The lines of {G') and {G") can be assembled into ruled surfaces 

 according to any arbitrary law 



i/'(m, v) = Const. 



However, the surfaces that specially concern us in the present dis- 

 cussion are R' and R" tangent to S along r' and r" respectively. 



The linear complex 



C'{v) : — bv C034 — bojii + bui23 = 0, 

 where uij (i, j = l, 2, 3, 4) are the Pliickerian line coordinates referred 

 to the moving tetrahedron Py P^ PpPa, will contain the tangents to 

 the curve T'(v), if and only if the Invariant W vanishes. f Also (in 

 the same notation and coordinate system) the linear complex 



c"(u) : — a'u C042 + a'wu + a'co23 = 



will_ contain the tangents to the curve T"{u) when the Invariant fi" 

 vanishes.! 



The writer has shewn (section 3 of the paper cited above) that 

 when the Invariants 12', fi" vanish, the surfaces R'{v), R"{u) are con- 

 tained in line congruences each of which is the intersection of two 



*S, pp. 175, 180. 

 t5, pp. 175, 182. 

 XS, pp. 175, 182. 



