112 On Hypocychids, Epicycloids, and Polyliedra. [June 15 T 



a 2 + v~ for_p, 2av for a, S lf £ 2 2 , . . . 5 rt 2 f or p lt p 2 , . . . p n , 

 W+*)W+*) • • • (W + ^ + ^ + vZ + xy-^na^iaZ + vZ + xy-* 



— . o 



+ . . fc 



4(??i-t-l)cu> 



6. Discussion of the first surface of the group. 



ft^aPW^^-jjCPi +P-2 +Ps +2> 4 ) 4 - 



It satisfies both the required conditions of § 2 ; and no other surface 

 formed from the regular tetrahedron satisfies the tests. Its quadri- 

 planar equivalent in point-coordinates is of the tenth order. 



The orthogonal projection on any face from its quadrantal pole, 

 whose equation is 



P±=i Cft +Pz +1H + A) = 3 Oi + Pi + Ps) » 

 gives the tricuspid hypocycloid (see § 4 of Memoir I) 



JPiP2Ps= +Pz +Ps) S =P*- 

 &2 6 2 _ 4 ac 3 _4 & 3d + igaM - 2 7a 2 i 2 = 0, 



where 



a,= etpy8 9 b = fri8 + et"{8+ . . . , C = a/3 + «7 + . . . , £Z=a + /3 + 7 + S. 



To ascertain its form two sections have been taken, (1) by a face, 

 (2) by a plane through an edge and a centre. 



From (1), when 8=0, &2(c 2 — 4M)=0, 



that is, <*W{( + ✓(*«) + y/("P)}=0. 



The first factors denote the three edges, which are conjugate double 

 lines, the last a tricuspid hypocycloid. 



(2) Let 7=5, or the surface be intersected by a plane AOB, which 

 passes through the edge AB, and bisects the edge CD perpendicularly. 

 This would give the sections of greatest' and least curvature ; another 

 such section superimposed vertically would give a clear conception of 

 the surface. 



The surface consists of six lobes, which are arranged in pairs, each 

 pair being touched by the same asymptotic cone. 



The edges of the asymptotic tetrahedron are conjugate lines, as is 

 also the great circle at infinity. 



7. In the same way the other surfaces are discussed and exhibited. 

 The property (B) of Memoir I has its analogues on this group of sur- 

 faces and their duals. 



