494 report — 1878. 



Let this equation be combined with that to the cubic (§ 6.) 



(8 2 y (23 + 42? + 43p 2 + 2 . 42pq) + /3y 2 ( - 23 + 43/r» + 42? 2 + 2 . 43p?) 

 + y 2 a (31 + 41? 2 ) + ya 2 ( - 31 + 41?) + a 2 /3(12 + 41p) 

 + a/3 2 ( - 12 + 41p 2 ) + 2a/3y . 41pq + /3 3 42(p 2 ->■ p) + y 3 43(? 2 + q) = o. 



After rejecting the known factor fih — yg, the conic of intersection is thus denoted. 



-a-p + ^42 (p*+p) - ^ 43(? 2 + ?) + -^(-12 + 41;j 2 ) 

 h g h 



"V/qt At 2n o /43b 2 42? 2 , 43.12 42.13 \ 

 ~j (31+41? )+/3 H-^-T- + T4T-14FJ = - 



This may assume the more tractable form : — 



+ -£ (41a 2 + 42/3 2 + 43y 2 ). 



This ternary quadric may be resolved into linear factors if 



42.43(2 P + 1 -V±^- 1 i£!±in 2 

 I h g J 



= (41+42 + 43) {i?(14« 2 + 12) 2 + ^(14? 2 + 13) 2 ) 

 L h- g' J 



where 14/? = 12 + ph, 14? = 13 — pg. 



This quartic yields four values of the parameter p, so that the equations are 

 determined to eight triple tangent planes through A, since g, h have each two 

 values dependent on the values of u. 



The two preceding linear factors denote the traces on the co-ordinate plane 

 ABC, or rather the planes through D and those traces of a pair of lines which 

 intersect in AP or AQ. The four values of p yield sixteen in all — eight which 

 meet in AP, and eight in AQ. 



To complete the investigation of the equations to the lines on the cubic, it 

 would be necessary to combine another form of a tangent plane : — 



12/3 + y (13 + ph) + 8(14 + ph) = o. 



By proceeding as above, the rejection of the known factor Jcy + hd, leads to]the 

 conic of intersection: — 



<2w 23y+ 2«){^ + ;-f) + *(,,. + y)} 



+ ^(21a 2 +23 y 2 + 248 2 ) = o, 

 where for brevity p. = — — ^- : v = -£- . 



If this quadric be resoluble into linear factors 



23.24<[2p + i(l2 M 2 + 13) + I(l2v 2 + 14)} 2 



= (21 + 23 + 24) {|f (l2„ 2 + 14) 2 + ^ (l2 M 2 + 13)'}. 



The actual solutions of these quartics has not been attempted, since the auxi- 

 liary cubic is cumbrous ; although we may infer from the circumstance, that the 

 same sixteen lines may be determined indifferently from A, B, C, D, or E, that the 

 expressions would be explicit. This quartic yields four values of p, which substi- 

 tuted in the preceding quadric, determine the projections on the co-ordinate plane 

 ACD of four pairs of lines on the cubic which intersect in AP and four nairs 



