142 Proceedings of the Royal Irish Academy. 



Eliminating A-, /, the equation of the surface generated by tlie connectors 

 in the first correspondence is 



<!> = ; 7 - a; z^.t^ + 03 - S) 7f-i!fi + (/3 - a) a^/ + (y - g) s%-2 = 0. (2) 



This surface is of the tjpe described 'Salmon, Aiialytic Geometry of Tliree 

 Dime)tsioTis), being generated by the lines which meet two lines and a quartie 

 curve. 



The lines xw, ys, which are the directing lines, are double lines on the 

 surface, since they are met at every point by two intersecting generators. 



Taking, for the moment, Xl'ZTF as running coordinates, the tangent plane 



at the point (j-yzir) is 



X<I>i -^ Y^, + Z<P^ + TF$. = ; 

 or 



{ (y - a] z^' + ((5- a) f^\ xX ^ \(i5 - B) f- ^ {y - S) ^\wW 



+ { (/3 - 8) vf- + ()3 - a] ^\yY+ \{y - a)«2 -i- (y _ h)v^]tZ = 0. 

 Putting in the values x = (k -^ l)oi^, &c., the equation becomes 



{k - IfQc + llWy^' + 7^* - « d/^ + z,^, ] .rX+ \liy,^ + 7.^,^ - ^Cy^^-f ^2)j«;JF] 

 + {k + lf(k - r)[!/:i(a^^ + w^) - „;r,2 - lv:^\yY + (7 (x^-^ w^-) - a.r{~ - lw^\ %Z\. 

 Substituting from TJ^ = 0, Fj = 0, 



&y^ + 7^ - " (^i" + si^. = (a - S) w,^ &c. 

 Dividing by k^ - P, 



(k -I) {a- S) x^w^ {w^X -x.JT)- {k . (/3 - 7) (%F - 3,,^ = 0. (.3) 



is the equation of the tangent plane to <l> at the point {k, T on the connector 

 11, 1'}- 



Since ic^^X - x^W, z^Y - y^Z are the planes joining the connector to the 

 double lines, the tangent plane at any point on a connector passes through 

 the connector, and the tangent planes through a connector are homographic 

 ■with their points of contact, as is the case with all ruled surfaces. 



If we identify (.3) with the plane b: + my -^ nz + dv: = 0, we get the 

 tangential equation of $, 



(7 - a) m^cP ^(j3-8)Pn^ + (fi- a)n^(P + (7 - S) Pm^ = 0. 

 If the tangent planes to a quadric at points 1, 2 on a generator are 

 Pi and P^ the tangent plane at the point kj-^ + Ix^ is iP^ 4 IP.^. The 

 tangent planes at the points 1, 1' through the connector {1, I'j to tlie 

 quadric of which the connector is a generator are 



I'l (.fJiX - x^w) = s^ [z^y - y^z) = P^ and p^ {tc^x - XjW) + s^ (z^y - y^z) ^ P^, 

 where p^, &c., are the coordinates of the tangent to PF at the point 1. For 

 these planes pass thiough the connector !1, 1'}, and are evidently satisfied 



