492 



REPOTCT — 1878. 



surface in A' ,B' ,C ,D' ,E' . If these five points be joined with the former five in 

 pairs, the points of intersection lie on the curve and are four, Q,R,S,T ; and six 

 other points are constituted on the curve by the intersections of AB', A'B ; 

 AC, A'C ; AD', A'D ; BO', B'C ; BD', B'D : and CD', CD. Call these six points 

 (a,b), («,c), (a,d), (b,c), (b,d), (c,d). 



It will be found that these 21 points lie on 40 chords, viz., A,B,C,D,E on 8 

 chords each, and the other 16 points on 5 chords each, according to the following 

 table : — 



P,A,A' 



P.B.B' 

 P,C,C 

 P,D,D' 

 P,E,E' 



Q.A.E' 



Q,C,(4,d) 

 Q,D,(6,c) 

 Q,E,A' 



E,A,(c/Z) 



E,B,E' 



R,C,(a,d) 



B,D,(a,c) 



E,E,B' 



S,A,(M) 



S,B,M) 



S.C.E' 



S,D,(a,6) 



S,E,C 



T,A,(6,c) 



T,B,(«,c) 



T,C,(a,6) 



T.D.E' 



T,E,D' 



~B',A,(a,b) 

 C',B.(6,c) 

 D',C,(e,d) 

 B',D,(M) 

 {a,b),E,(c,d) 



C\A,(a,c) 



A',B,(«,6) 



B',C,(6,c) 



A',B,(a,d) 



(a,c),E,(M) 



D',A,(a,d) 

 D',B,(b,d) 

 A',C,(a,c) 

 C.D.M) 

 (6,c),E,(a,rf) 



4. Let the quadro-quadric curve be denoted by the equations to two conicoids 

 of the system : — 



/ja 2 + mfi 2 + ttj-y 2 + ?\b- = o. 

 Z 2 a 2 + mJP + n 2 y 2 + r,8 2 = o. 



Let the equation to a transversal through a fixed point {f,g,h,k) be :— 

 a - f 3 - ff __ y - h 8 - k 



\ H v p 



For the segments of its distances from the conicoids : — 



= B. 



l^f+ XR) 2 + m t (ff + /xR) 2 + n^h + vll) 2 + r x (k + pR)' = o ■■ 

 l 2 (f+ XR) 2 -H» 2 (# + /xR) 2 + » S (A + vU), + rjk + pliy = o -- 



u. 



V. 



The following equation denotes the foci of these lines in involution: if — be 

 written for R, and finally a ■■ 



= o. 



1. 



dU dV_dU dV 

 dli da- da dR 

 This function yields on development : 



S^m,-^™,) (Xg-rf) (/+XR) + M R) = o. 

 Or, 2(l in i 2 -l 2 ? ni ) (ag-Pf) a(3 = o. . 



This may be reduced to the form — 



(Zja 2 + Wj/3 2 + ntf 2 + r x 8 2 ) (l..af+ m 2 f3g + n*yh + r„8k). 

 = (ha? + wi 2 /3 2 + n 2 y 2 + r 2 8 2 ) (l 1 af+ mfig + n x yh + ?\ok). 



The dual of this theorem (§ 1) may be noted. 



If a system of conicoids be inscribed in a quadro-quadric torse, and if from each 

 line of a fixed plane tangent planes are drawn to the conicoids, the envelop of the 

 focal planes in involution of the system is a cubic class-surface. 



5. The above equation may be obtained as the eliminant of a conicoid of the 

 system, and the pole-plane of a fixed point. 



(Zj - X?„)a 2 + («ij - Xw„)/3 2 + (n, - XM o y + (»-j - Xr„)5 2 = o. 

 2(Z 1 -X/ 2 )a/=o* 



This is Dr. Salmon's method (§1), which seems capable of generating surfaces of 

 any order or class from a surface of the next lower order or class. 



G. In this investigation, f=g = h = k or E is the fixed point, without loss of 

 generality. 



Professor Cayley's notation is adopted for the minors of the determinant : — 





J, UV 2 , 7. 2 , 



Thus 12 denotes 



L, m n 



12 + 21 = o. 



The following relation subsists between the minors, as has been pointed out. 

 (' Quarterly Journal of Mathematics,' vol. xv.) 



23.41 + 31.42 + 12.43 = o. 



