Dr Burnside, Convex Solids in Higher Space 439 



which is associated in a similar way with the symbol 



{12} {34} {56} {78}. 



There is no difficulty in verifying that for three symbols and for 

 four symbols, all possible separations arise from these by permuta- 

 tions of the single symbols. It will be shown here that if the scheme 

 is general for 7 it is general for 8. If the proof is examined it will 

 be quite clear that the same method may be used for any two 

 consecutive numbers. Assuming that the separation of 21 points 

 is given by the scheme (i), the question to be settled is how the 

 seven points 18, 28, . . . . , 78 fit into it and how they are separated 

 among themselves. 



Suppose for instance that 48 is the first of the new symbols 

 that occurs to the left of the bar. Then 48 is opposite to 45, 46 and 

 47; so that 45 and 58, 46 and 68, and 47 and 78 are adjacent. It 

 follows that 68 occurs to the left of the bar and 58, 78 to the right. 

 Since 18, 28 are both adjacent to 12, 28 is opposite to 18. So 23, 28 

 and 23, 38 being adjacent, 28 and 38 are opposite: 24, 28 and 24, 48 

 being adjacent, 28 and 48 are opposite. In a similar way 28 is 

 shown to be opposite to 58, 68, 78. The scheme that thus arises 

 for the 28 points is 



Comparing this with scheme (ii) it is changed into the latter by the 

 permutation (846), the other single letters remaining unchanged. 

 Similarly if 38 is the first to occur to the left of the bar the scheme 

 that arises is changed in (ii) by the permutation (8357). 



