300 REV. T. P. KIRKMAN ON THE THEORY OF 



i lias all values < N, 



p has all values < N - i, >o, 



c has all values < N - i^ >'o. 



There are, of necessity, ^.^_^w^_ . - i groups 



eGe-' 6'iG^r' o^oe^^-- &c. (Art. 9), 



of which each is either identical with G or equivalent to it. 



Every equivalent group will have, as G has, (N-2«N-i) 

 substitutions, which have N in its natural place; and if 

 N be erased, we shall have a group of ISf - 2 • N - i sub- 

 stitutions made with N - 1 elements. We have proved 

 (theorem D) (14), that there are IT(N-3) equivalent 

 groups of N - 2 • N - 1 substitutions, when N - 1 is a 

 prime number. It follows that there are iT(N-3) equi- 

 valent groups of the order N N - 1 • N - 2, of which G 

 above constructed is one. 



Wherefore we have the theorem following — 



Theorem P. If 'N - 1 be any prime number, there are 

 IT(N - 3) equivalent groups of the order N • N - 1 N - 2. 



It is a property of these groups, that any one of them, 

 or any derived group of any one of them, gives a solution 

 of this problem : 



To seat N persons, N - 1 being any prime number, 

 N»(N-i)(N-2) times in N chairs, so that no three 

 persons shall twice occupy the same three chairs. 



There are (II(N-3))^ different ways of solving this 

 tactical problem. 



§ 6. 

 Ch'oups of the form G + RG, where R is composed of square 

 roots of unity, 

 24. Let 



{PlP^Pz' ■Pri){qaiiqa-,2' ' ?«+*) (n.+&+l »^a+5-3 ' '• ) &C, 



