252 MR. A. CAYLEY ON THE A FACED POLYACRONS IN 



acrons of the first class are obtained by the first process ; 

 the second process is only required for finding the {n+i)- 

 acrons of the second class ; and these being all obtained 

 by means of it, the third process is only required for find- 

 ing the {n+ i)-acrons of the third class. Hence the second 

 process need only be made use of when the w-acron has no 

 tripleural summit, or when it has only one tripleural sum- 

 mit, or when, having two tripleural summits, they are the 

 opposite summits of two adjacent faces. In the last-men- 

 tioned two cases respectively it is only necessary to con- 

 sider the basic quadrangles which pass through the single 

 tripleural summit and the basic quadrangle which passes 

 through the two tripleural summits; for with any other 

 basic quadrangle the derived (w+i)-acron would retain a 

 tripleural summit, and would consequently be of the first 

 class. The condition is more simply expressed as follows, 

 viz. : The second process need only be employed when 

 there is on the w-acron a basic 

 quadrangle the summits of which 

 are at least of the number of 

 edges shown in the annexed 

 figure, and all the other summits 

 are at least 4-pleural. Again, 

 by the third process (as already mentioned) we seek only 

 to obtain the {n+ i) -acrons of the third class ; the process 

 need only be applied to the w-acrons for 

 which there exists a basic pentagon the 

 summits of which are at least of the 

 number of edges shown in the annexed 

 figure, all the other summits being at 

 least 5-pleural ; for it is only in this ^ ^ 



case that the derived (w+i)-acron will be of the third 

 class. The condition just referred to obviously implies 

 that the w-acron is of the second or third class. It is 

 to be noticed that in applying the foregoing principles 



