of Edinburgh, Session 1883-84. 
587 
so that the possible cycles are : — 
a, 2f, c, 1, 2c+l, 
/, 
2a , 
{ }. 
a,2/+l,2, 2, 1, 
/, 
2a, 
{ }> 
a, 1 , 1 , d , 1 , 
2cZ + 2, 
2a + 1 , { }, 
cc j 1 l , 2d +2,1, d , 
1, 
2a+l, { }, 
cif , 1 , 2 , 2 , 2 , 
2, 
2a + 1 , { }, 
where { } indicates the middle of the symmetric portion of the cycle. 
All of them are shown by the tree except the fourth, which is got 
from third by the theorem of § 8. 
The following are the like results for cycles of less than 16 
elements : — 
Elements 
in cycle. Cycle itself (up to middle element). 
2. 
{ },.... 
4. 
impossible. 
6. 
a, 2a, { }, 
8. 
a, 1, 2a+l, { 
. 
10. 
a , 1 , 2 , 2a + 1 , 
{ }>••• 
a, 26, 6, 2, 
{ }.••• 
12. 
a, 26 + 1 , 1 , 6, 
2a, 
{ },... 
a, 1, 2, 2, 
2a + 1 , 
{ 
14. 
a, 26, c, 2c, 
2a + 1 , 
{ 
},... 
a , 26 +1,2, 1 , 
2a + 1 , 
{ 
},... 
a, 1 , 26 + 1 , 6, 
1, 
2a + 1, 
{ 
a, 1, 2, 2, 
2, 
2a + 1 , 
{ 
},... 
a , 1 , 1 , 6 , 
26 + 1, 
2a + 1 , 
{ 
},... 
Numbers whose square roots have these cycles: 
2. a2 + 2. 
4. 
6. -j(2a2 + l)M + aj- ^ + 4<2M + 2. 
8. |(2a2 + 4a + l)M-(a + l)P + 4(a+l)M-2 . 
10. { (6«2 + 8a + 3 )m + (9a + 6) + i{Za + 2)m + 18 . 
I + iah + 2a^ + 1)m - (2«6^ + a + b){2b^ + 1) f ^ 
- 4(2a62 + a + j)m + 2(262 + 1)2. 
&c. &c. 
