to movement in three dimensions 159 



parameters p, q. There will thus be a rational equation con- 

 necting d and p, of the second order in p, equally satisfied by 6 

 and q. Through the transversal p can be drawn, beside the plane 

 {6, p), another tangent plane to S, which, as it contains p, will 

 contain an axis of Sq, say (f)^; thus the previous equation con- 

 necting {d, p) is of the second order in 6, and is equally satisfied 

 by 01 and p. Similarly through q can be drawn, beside {6, q), 

 another tangent plane to S, which, containing q, will contain an 

 axis of Sq, say (f)2, and the relation connecting {6, q) is equally 

 satisfied by (f>2 and q. There is thus a single relation of the second 

 order in each of 6, p which is equally satisfied by {9, q), (0^, p), 

 ((/>2, q). As in other cases it may happen that 0^ = ^3 for all values 

 of 6, provided a certain relation hold between the quadrics. 



To find a sufiicient condition for this, we may consider the 

 particular case when the axis 6 coincides with the identical axis 

 (f>i = 02' or the transversal p touches S; then, this line p, being a 

 generator of Sq, is a chord of the curve {S, Sq), and therefore, in 

 the particular case considered, is a tangent line of the curve. Thus 

 a sufficient condition is that the tangent planes of S at two of the 

 four points referred to above, on the curve {S, Sq), associated 

 with the transversals of Sq, should intersect on an axis of Sq. 



More generally, if Sq be x^ + y^ + z^ + t^ = 0, its generators of 

 the two systems being of the respective forms 



X + iy = iO {z + it)) x + iy = ip (2 — it)\ 

 X — iy = iO-^ (z — it)] ' x — iy = ip~^ {z -f- it)\ 

 the plane containing these, 



X + iy + dp {x — iy) — 6i {z + it) — pi {z — it) = 0, 

 touches S, say ax^ + hy^ + cz^ + dt^ = 0, provided 

 ^y ((^-1 _ 5-1) + 02 ((^-1 _ c-i) + f ((^-1 _ c-i) 



+ 2dp (a-i + 6-1 - c-i - d-^) + a-i - 6-1 = 0. 



Hence, for the porismatic relation in question, by considering this 

 equation, we see that we require that the discriminantal equation 

 I ^0 — A^S I = 0, which with our coordinates is 



(1 - Xa) (1 - A6) (1 - Ac) (1 - Xd) - 0, 



should have the sum of two of its roots equal to the sum of the 

 other two. Writing then as usual (Salmon, Solid Geometry, 1882, 

 p. 173) 



I /So - A5 I = Ao - A0O + A20 - A30 + A*A, 



the condition is 



©o^ - 4O0A0O -1- 8Ao20 = 0. 



(Cf. Purser, Quart. J. of Math., viii, 149, and Salmon, loc. cit., p. 181.) 



