TO Proceedings of the Royal Irish AcoAemy. 



It is easy to see that (35) represents the same curve as (34). F ow the 



equations (35) were deduced from the relation f 0' = 1, which was deduced 



■fr'om the hypothesis that a quadrilateral of the kind in question could be 



inscribed in a rational quintic. It follows that (35) is the only rational 



quintic with this property. 



69. Again, if an B,-, has a bitangent, we take the points of contact as 

 and oc. The 2, 2 relation, 



F= at-Q'' -=- m{} -I- 0) + c(t- + 0=) -f dtQ -r e{t -i- Q) + /= 0, 



has double roots ^ = 0, cc, corresponding to the values 6 = cc, 0. Hence 

 c = c = b = 0, and V can be written in the form V = (atO + h) (ctO + d). 

 Hence propertj' (a) involves (6), and it is easy to see that the converse holds. 



Supposing (or) — or (b) — to hold, then by putting f = kt, we can write the 

 2, 2 relation (tO - I) {id - a) = 0. 



The equations of the corresponding curve, after the analogy of equations 

 (35), are 



{t-iy{t^) " «5 = ^ = {t + lf^t^a)' 



The condition that the tangents should belong to a linear complex is easily 

 found to be (a + 1)' = 0, so that the cui-ve is identical \vith (35) or (34) ; 

 hence no other R^ possesses properties («) or (6). 



70. The three pairs (±1), (i i], (0, ocj are harmonically conjugate with 

 respect to one another, and the vertices of any one of the quadrilaterals 

 (f, r', —t,- 1'^) are harmonically conjugate by twos \vith respect to these 

 three pairs. 



71. We may sum up these results as follows : — 



If in a rational quiiitic curve onu quadrilateral can he viiscrvbed such thai tlie 

 plane of any two adjacent sides is the oscv.lating plaim at their vertex, then 

 (1) ayi infinity of such qvMdrilaterals can be inscribed; (2) the tangents to the 

 curve belong to a linear complejr which contains the sides of the quadrilate7-als ; 

 (3) the curve has a bitangent ; (4) the secants of the curve v:hich belong to the 

 complex (Le. the sides of the quadrilaterals) determine two involutions on the 

 tv/nce, whose foci are the two pairs into which the four inflexions fall, o.nd whose 

 comynon pair of conjugate points are the points of contact of the bitangent i 

 (5) if the vertices of one of the quadrilaterals are determined, by the quxirtic 

 f(t) = 0, the sextic covariant of f gives the four inflexions and the points of 

 contad, of the bitangent, and all the other quadrilateraJs are determined^ by 

 eqvMtions of the form f-rXh = 0, where h is the ITessian off; (6) the curve will 

 be a Tiomographic transformation of that given by the eguations (35); (7) no 

 other Hi possesses either of the properties (3) or (4). 



