110 Mr. H. M. Jeffeiy. [June 15, 



(A.) Let there be a regular polyhedron of (n) faces, inscribed in a 

 sphere of radius (.a). If from the summits and centre there be drawn 

 VmPz-> • • ■ Pn> V perpendiculars on any plane (exterior to the solid, if 

 (m) is odd), the sum of the (7^)th powers of the perpendiculars from 

 the summits is a function of the perpendicular from the centre. 



KPr) m ~ " {(p + ~ (p-4"+ 1 }. 



2(m+ l)a 



This formula is applicable to all five Platonic bodies, if m be 1, 2, 3 ; 

 if m be 4, 5, and not larger, it is restricted to the dodecahedron and 

 icosahedron. 



(B.) Under the same conditions as in (A), if there be taken any 

 point, whose distance from the centre is (v), the sum of the (2m)th 

 powers of the distances of this point from all the summits will be a 

 function of its distance from the centre. 



2(d,) 2; "= — {(v + ay>'i + *-(v-ay>» + Z}. 



2(m+l)av 



2. Following the analogy of plane geometry, I propose to consider 

 a group of five surfaces, whose orthogonal projections are the tricuspid 

 and quadricuspid hypocycloids, and which have the property, that the 

 product of the perpendiculars drawn on any tangent plane from all 

 the summits of one of three regular polyhedra (which are cuspidal 

 points on those surfaces), is a function of that perpendicular only 

 which is drawn on the same tangent plane from the centre of the 

 sphere circumscribed about the polyhedron. 



These three surfaces are defined by tangential polyhedral coordi- 

 nates referred to the three first of the regular polyhedra. 



(!) PiPrthPi =P*- 



(2) p^o. ■ - p 6 =p 6 -aY- 



(3) p^o . . . p 6 =jo 6 . 



(4) pip 9r .,p s =p*—ia?p*. 



(5) Pl p 2 . . . p 8 =p 8 . 



3. By generalising the results of examination in each case of the 

 regular polyhedra, it is found that the continued product of the per- 

 pendiculars drawn from all their summits on any plane may be thus 

 expressed in terms of that drawn from the centre — 



P1P2 • • -Pn=P' 



Subsequent terms would involve the inclinations of lines and planes. 

 But the following scale is found to exist : — 



