O'SuiJJVAN — Points on the Curve of Intersection of two Quadrics. 143 



by the coordinates of any point on the tangents to f7F at 1 and 1' respectively. 

 But (§ 2 (8)) p-^ = (a - S) x^n\, Sj = (/3 - 7) y-^z.,. Hence the tangent plane 

 at the point {k', I') on the connector [1, I'j, to the quadric of which it is a 

 generator, is 



(// + I') (a - S) x^w^ {w^x - x^w) - {¥ - V) (/3 - 7) 2/1^1 {z^y - y,z) = 0. (4) 



If we identify this plane with (3) above, we find 



{li + I] [k' + I') = (k - I) {k' - /'), or kl' + k'l = 0. 



Hence the points of contact of any plane through a connector with the 

 quadric of which it is a generator, and with <P, are harmonic with the points 

 in which the connector meets UV. 



The equation of a plane through a connector {1, I'J and the point 2 is 



Ihi ("'r^ - -h'v) - Sj., {z^y - y-^z) = F. 

 where 



P12 = Vih - 1/ih' ^12 = ^i'<'2 - '^"vh- 



The plane through tlie corresponding connector {!", 1'") and the point 2 

 is derived from this by changing the signs of x^, \. It is 



l^'i2 i'>^i^ + »'i'^) - «'i2 ihl/ + Vi^) = -P'> 

 where 



IHi = y-ih + Vih' «'i2 = *j«'2 + ^i^"v 

 If P, V meet any other connector j3, 3'j in the points (k, I), ik\ I') 



(k + /) PioS,3 = (k - I) SnVu ; [k' + /lo/is = {k' - I') sV./i3. 

 But, writing 1 for 4 in § 5 (1), 



PnP\2 • S,3-s'i3 = Pisp'n . SnS'n ■'■ [k + I) (k' + I') = {k - I) {k' - I'), OV 



kl' + k'l = 0. 



Hence the planes joining any pair of corresponding connectors jl, 1'}, 

 {1", 1'") to any point on UV meet any connector in a pair of points which 

 are harmonic with the points in which the connector meets UV. 



In particular, if four connectors aa'a"a!" be the coUinear axes of a quad- 

 rangle Pr'LL' (see fig. 3), the plane PLL' cuts the four connectors in the 

 points in which the planes joining them to P touch the quadrics (A,) (A'), of 

 which act a" a'" are generators. Hence the plane PL"L"' cuts them in the 

 points of contact of these planes with <I>. Hence, if four such connectors be 

 joined by planes to a point P on UV the points of contact with <I> of these 

 planes lie in a plane pas.sing through P, and also through the corresponding 

 connector L"L"' to LL', where PV'LL' form a quadrangle of which the four 

 connectors are the coUinear axes. In consequence of this property, four such 

 connectors may be called cotangential connectors. 



