TRANSACTIONS OF THE SECTIONS. 5 



order. These roots can be enumerated by a simple general method for 6 of any 

 order, made with n letters. 



The fundamental theorem is the following : — 



If «=Aa-f Bb + Cc+ , the number of different groups of the order K, which 



is the least common multiple of ABC . . . ., of the form 



\,6,6\.. .6*-\ 

 wheretfhasa circular factors of the order A,& of the order B, &c, is (7rn=l .2.3 .. . n)» 



R K A a B*C c . .TrajriTTC. . . 



R K being the number of integers, unity included, which are less than K and prime 



to it. 

 The partition n = 9 = 3.3=Ao 



& ives „ n9 =8.7.5.4 



R 3 33.7r3 



groups, 166' 2 , (G) 



of the third order, which is that of 6 and of d 2 . 

 The partition 



n = 9 = 6 . 1 + 3. l=Aa+B6 



6 ives « 9 , =8.7.6.5.3.2 



Re . 6 . 3 



groups, 1$<£ 2 • • • $''» (H) 



of the 6th order. Every group (H) contains a group (G), namely, 



and (f> of the 6th order is the square root of (j> 2 of the 3rd order, and the fourth root 

 of <£' of the 3rd order. Also <j)' of the 6th order is the square root of <p\ and the 

 fourth root of (ft 2 . 



The number of groups (II) being nine times that of the group (G), the group ]60 2 

 will be comprised in nine different groups (H) ; that is, 6 has nine square roots of 

 the 6th order, and nine fourth roots of the same order. 



The partition 



w = 9 = 9. l=Aa 



giveS J* =8.7.5.4.3.2 



Rg9 



groups, l^^P • • • V' H ' ^ 



of the 9th order. This comprises the group (G), 



where \^ 3 has the cube roots \j/ \jr l \^ 7 of the 9th order, and the sixth roots yfr* yjr' ^ 

 of the same order. There are six times as many groups (J) as groups (G). Therefore 



\66 2 

 will be found in six groups (J), and either 6 or 6 2 has IS cube roots, and 18 sixth 

 roots all of the 9th order. 



In the same manner it is easily proved that the substitution of the 2nd ordcr(w=8), 



ffl _ 341278 56 

 12345678* 

 which has four circular factors of the 2nd order, has twelve square roots all of the 

 4th order. These form with unity and 8' the two groups following, 



12345678 12345678 



34127856 34127856 



58763214 23418567 



76581432 41236785 



23416785 78561234 



41238567 56783412 



87652143 85674123 



65874321 67852341 



