[SULLIVAN] A CERTAIN PROJECTIVE CONFIGURATION 159 



Equations of the directrix plane and associated formulae, integrals of the 



Normal System, etc. 



It can be shewn that, when the Invariants d' , 6" , 12', 12" vanish, the 

 equations of the double directrices ô', 5" of the surfaces R' , R" are 



du 



dît 

 (14) _ 



2. ^^^ alogV _ X3 + 2*^2^4 = 0, 



dv 



dv 

 the moving tetrahedron PyPzPpPc being the system of reference.* 

 These equations shew that 5' and 5" intersect in the point 



( — , — aiog— ^ ôiog— ^ 



alogVc/) alogVc/) ,20^: ^: ^:1; 

 du dv / dv du 

 that is, in the pointf 



(/ — / — ^ d\o%—j= aiog— ^ 

 L<^- )y + .z-\ . p+a- 

 du dv / dv du 



To prove that this point is fixed in space, and therefore the same for 

 all surfaces R'{v), R"{u), it is sufficient to shew that 



dir , / \ dir , / \ 



~^= Mu,v)^, -^=Uu,v).. 



In virtue of equations (6) and (12), the first derivatives of the semi- 

 covariants can be written as follows: 



yu = z, Vv = p, Zu = — fy — 2(f)p, Zv = (T, Pu = (T, Pv = — gy — 2^2, 



<Tu =< 





] 



Z+4:(f>'^p. 



*S, p. 182, Equations 28; Equations 14 can be obtained from these by putting 



d'=d"=o. 



tCf., S, p. 195. 



Sec. Ill, 1915—12 



