260 



Proceedings of the Royal Irish Academy. 



If P = be the osculating plane at ti and ^ = 0, the osculating 

 plane at U, the " points " in P + kQ ■= Q and P - kQ = ^ are given by 



{t-Uf + K{t-u:f = 

 and (t - fif - K{t- to)^ = 0, respectively. 



Hence, " If the parameters of the ' points ' in any plane are the roots of a 

 cubic, the parameters of the ' points ' in the ' conjoint ' plane are the roots of 

 the cubico variant ; and the roots of the Hessian are the parameters of the 

 points, the osculating planes at which meet in the given plane." 



(Since this paper was written I have found that Mr. W. E. W. Eoberts 

 has given this geometrical interpretation of the Hessian and cubieovariant in 

 vol. xiii., Proc. Lond. Math. Soc.) 



The " points " in the plane of centres are therefore given by 



P/' + W,f~ + 3A^ + A = 0, 

 where D,f + ?>P,t' + ?,P,t + A 



is the cubieovariant of dj,^ + 'Mip + Md + d^, 



and the " points " wi, w^ are given by 



[d,d^ - d,'] f 4 [cl,d^ - d,ch) t + [d.d, - dJ) = 0. 



21. The direction ratios of the chord joining wi, wo are 



a^di - Sciido + 3«2f?i - Ozdg, 

 bf^ds - obidi + 35./Z1 - h-idf^, 

 c^^dz - 3ciC?2 + oc^di - c^dg. 

 The coordinates of the point are 



a^P^ - SaiPz + Sa^Pi - asl^u \ 



h,Ps - 3hP, + 3b,P, - hP, [ i- d,P^ - Sd.Ps + 3d,P, - d,P, 

 CqPs - 0C1P2 + ScoPi - C3P0 j 

 The equation of the plane of centres is 



X 



y 



z 



1 







«o 



K 



Co 



do 



A 



a, 



I. 



Ci 



d. 



P. 



az 



h 



C2 



d-i 



P. 



Ch 



b. 



C3 



dz 



A 



The parameters of corresponding points are connected by the relation 



2 [d,d, - d,') t,U + [d^d^ - d,dz) [U + t,) + 2 {d,d, - d^^) = 0. 



