TRANSACTIONS OF SECTION A. 493 



It will be convenient to denote by a, 6, c, d, the sums of certain minors : 



rt = 12 + 13 + 14: 6 = 21 + 23 + 24. 

 c = 31 + 32 + 34: d = 41+42 + 43. 



Then it will be seen that a+ 6 + c+ d = o. 



23a + 316 + 12c =o. 



346 + 42c + 23d = o. 



34« + 41e + 13« 7 = o. 



24a + 416 + l2d = o. 



Hence the cubic under discussion may be written : 



23/3y (/3— y)+31ya (y-a) + 12a/3 (a-/3)+41«S (8 -a) + 42/38 (8-/3) 



+ 34y8 (y - 8) = 0. 



7. The equations to the tangent planes at the vertices of the tetrahedron and at 

 E the centre, which constitute a pentad of points, are — 



12/3 + 13y + 148 = o. 



21a 23y + 24S = o. 



31a +32/3 +34S = o. 



41a + 42/3 + 43y =o. 



aa + 6/3 + cy + d8 = o. 



These are fine triple tangent planes ; and any two determine the line PQ on the 

 cubic, through which they are drawn. 



8. The equations to the pole-conicoids of the points of the pentad are — 



12/3 2 + 13y 2 + 14S 2 - 2a(12/3 + 13y + 128) = o. 



21a 2 + 23y 2 + 24S 2 - 2/3(21a + 23y + 248) = o. 



31a 2 + 32/3 2 + 34S 2 - 2y(31a + 32/3 + 348) = o. 



41a 2 + 42/3 2 + 43y 2 - 28(41a + 42a + 43y) = o. 



a« 2 + 6/3 2 + cy 2 + dS 2 =o. 



The four cone3 (12/3 2 + 12y 2 + 128 2 =o) and the like, belong to the same quadro- 

 quadric, and the fifth pole-conicoid is a hyperboloid of one sheet whose asymptotic 

 cone is inscribed in an orthogonal trihedral angle. 



8. To determine the ten lines on the cubic which intersect in PQ. 



The two lines AP, AQ, are obtained by the intersection of the tangent-plane 

 and cone 



12/3 + 13y +148 =o. 

 12/3 2 + 13y 2 +148 2 = o. 

 Their equations are — 



/3:y: 81:14 + ^:14 + ^: -(12 + 13), 



if u* + 12 . 13 . 14 (12 + 13 + 14) = o. 



Similar equations denote the other four pairs of lines. 



9. To determine the eight triple tangent planes, four through AP, and four 

 through AQ, other than PAQ ; and the sixteen lines on the surface, eight of 

 which intersect in AP, and eight in AQ, other than PQ and AQ or AP 

 respectively. 



Write «?: h:k::U + ~ : 14 + ^: -(12 + 13): 

 y -12 13 



so that (h$—gy = o) denotes the plane APD or AQD. Then, if p denote a para- 

 meter, the equation 



(3(12 + P h) + y (13 - P g) + 148 = o 



denotes any plane through AP or AQ. 



For brevity, write the coefficients — p =j) : ° ^ = q ; 



14 14 



the equation becomes p/3 + gy + 8 = o. 



