TRANSACTIONS OF SECTION A. 563 



have been assigned. So the order of the frroiip is the product of three numbers, 

 viz., the number of vertices, the number of edi,'e3 coutainiug a given vertex, and 

 the number of faces passing through a given edge. 



4. For the six regular cells of S' the results are : 



Five-cell . . . (C,) ... T, x 4x3= 60 



Eight-cell . . . (Cj ... IG x 4x3= 192 



Sixteen-cell . . . (a,,;) ... 8x 6x4= 192 



Twenty-four-cell . . (C.J ... 24 x 8x3= 676 



Hundred and twenty-cell ((V,„) ... 600 x 4x3 = 7,200 



Six hundred-cell . . (C,,o„) . • . 120 :■: 12 x 5 = 7,200 



5. Semarlcs. — (a) A deeper study proves the group of the five-cell to be 

 bolohedrically isomorph with that of the icosahedron. 



(A) In the pairs of cases (Cg, C,„) and (C|„,„ C„„,) the results are equal. This 

 is due to the fact that these pairs of regular cells are reciprocal polars of each 

 other with respect to a hypersphere. 



(c) The order of the group is equal to 2;- times the number of faces, ;• represent- 

 ing the number of vertices situated in any face. 



Five-dimensional Space (S''). 



6. General Principle extended. — The order of the group is the product of four 

 numbers, viz., the number of vertices, the number of edges through a given vertex, 

 the number of faces through a given edge, and the number of limiting bodies 

 adjacent at a given face. 



7. Results: 



Six-being . . . (B,.) . . . 6x5x4x3= 360 

 Ten-being . . . (B,^) . . . 32 x 5 x 4 x 3 = 1,920 

 Thirty-two-being . . (B3.,) . . . 10x8x6x4 = 1,920 



8. Remarks.— (a) The cases (B,,,) and (Bj,) are reciprocal polars of each 

 other, &c. 



(6) The order of the gi'oup is equal to fir times the number of limiting bodies, 

 ^' representing the number of vertices situated in any limiting body. 



Space of n-Dimensionfs (S"). 



9. The extension of the principle is evident. The results are : 



M + 1-being (B„+i) . . (n + 1) w(w-l) . . . x4 x 3 = it(M + l) ! 

 2M-being (B..„) . . . 2" . n (w-l) .... x 4x3 = 2"-' . n\ 

 2"-being (B>) . . . 2w.2(h-1) .... x6x4 = 2n-i.M! 



10. Remarks. — («) The cases (B,„) and (B._,") are reciprocal polars of each 

 other, &c. 



(6) The order of the group is equal to (w,-2) ! r times the number of limitino- 

 beings of n - 2 dimensions, r representing the number of vertices situated in each o'f 

 these. 



8. On Mersenne's Numbers. By Lieut. -Colonel Allan Cunningham, E.E. 

 Fellow of King's College, London. 



These are numbers of form N = 2''-l, where p is prime. Lucas has shown 

 that N is composite, and contains the factor (2^; + 1) when p and (2^ + 1) are both 

 prime, and jj is of form (4;; + 3). 



Such numbers N may for shortness be called Lucasia^is. The hio-hest 

 Lucasians, determinable by the existing tables of primes (extending to 9 000 000) 

 are given by 



;j = 4,499,591 and 4,499,783, 



2 



