225 



The first has two marginal plane triangles and a + b submargins ; 

 the second has one marginal plane triangle ; the third has none. 

 The fourth has a-\-c plane submargins and a solid submarc/ins. 

 We see that polyedra are removeable by the external primary efface- 

 ables in the third, which have a (3 + l)-gcmal and a (4 + l)-gonal 

 face; and that the fourth has been constructed by charging the 

 plane marginal triangles of a subject penesolid, one with a polyedron 

 having a 3-f 1-gonal face, and the other either with a polyedron 

 having an (A+l)-gonal face, or with a full penesolid having the 

 solid margin A- The edge that crowns the first is the intersection 

 of an ( + 3)-gon and a (6 + 3)-gon. The edge that crowns the 

 third may be the intersection of an ( + 3)-gon and a (c+6)-gon, 

 or of an (a + 4)-gon and a (c + o)-gon, &c. 



The polar summits being known for P-edra Q-acra and Q-edra 

 P-acra, their reciprocal polar faces are of course known also. 



This may suffice as an account of the data (C, D, G), except that 

 we shall return to the construction of the plane and of the full reti- 

 culations. 



We consider next the data B. 



Analysis of the polar j anal summits p'p / of a f-edron Q-acron W. 



Let p' and p, be w-aces terminating a janal axis. The ordinary 

 effaceables of either summit, which are what we have already consi- 

 dered about a polar p-o.ce, can be restored in one way only. The 

 janal symmetry remains, though modified by such restoration. 



There are in faces about p 1 a certain number of triaces collateral 

 vtiihp t . These are (r^0) rhombo famous triaces. Let each be made 

 collateral with p', and let the or similar triaces in faces about p t be 

 made collateral with p 1 . They are now all rhombotomous tessaraccs, 

 sucb that quadrilaterals p' p p t p t are drawn through pairs pp t of the 

 tessaraces. We have now a pair of janal ?r'-aces perfect both in their 

 ordinary and their peripolar ejfaccables, which is registered in our 

 table of perfect janal poles. 



A peripolar ejfaceable is any ray drawn from a principal jaiial 

 pole to a rhombotomous tessarace, which is always collateral with 

 either pole, and may lose either of its effaceables. 



