144 Proceedings of the Royal Irish Academy. 



Section 8.— The surface $ and the two other surfaces which are similarly 

 related to the second and third correspondences are the limiting surfaces 

 (surfaces limitees) of a system of surfaces of the eighth order which have 

 been studied by de la Gournerie* under the name of quadricuspidal 

 surfaces. Tliese again, as will be shown, are the limiting surfaces of a system 

 of the sixteenth order. These systems may be derived as follows : — 



If a plane ax + bij + cz + dw = touch the three quadrics (X), (/.i), (") 

 its coordinates satisfy the three conditions 



Hence 



Aa iUa Va 



„ „ XaflaVa A/S/J^l'/S XyJ^I'y ^S^lSVS q^ 



a-b :c-:dr = j^- : y,^^^ : ^,^^ : ^,^^, ■ 



If {2Kfrsti(,) be the coordinates of a generator of (r) then (§ 2) 



s = ^eyVffy, t = {1 - Q'')y^a, n = ^^l + B~)yi^. 



If the generator lies in the plane, as + it -t cu = 0, or 



a^« ,_/i te; /'^^''^x.71 _fi2> /VtI 



Hence the values of B are independent of v, being, in fact, the parameters of 

 the generators of {fx) which meet in a point A on V For of the generators of (A) 

 which meet in a point f.i, as can be seen at once from (7), § 2. 



The equation of the surface generated by these generators when A and fi 

 are fixed and i- is variable, is given by eliminating v between the equations 



ax + ly + C2 + cliv = and v ^ -r^. Hence, from 1 1), it is 



y 



jjrJX.fx.KV- aU) ±jj^JXsf,p{r-(5U)±jj:^JXyfxy {V-jU) = O (3) 

 which, when rationalised, gives 







_ 64=^a;VV«-.{F- i.aT){V - i^V){V --^IT) [V -W). (4) 



as the surface generated by the generators of quadrics of the system 

 Xv - V, which lie in the tangent planes to the developable circumscribing 

 (A) and (ju). 



* De la Gournerie, " Reckerches sur les surfaces reglees-tetraedrales symefcriques. " 

 Paris, 1867 Compare also Matthews, P.L.M.S. 



