ALLTJDED TO BY FEEMAT. 
633 
4, 5, 5, 5 
= 
91 
4, 4, 5, 6 
= 
93 
3, 5, 5, 6 
= 
95 
3, 4, 6, 6 
= 
97 
3, 4, 5, 7 
= 
99 
2, 5, 6, 6 
= 
101 
2, 5, 5, 7 
= 
103 
2, 4, 6, 7 
= 
105 
3, 3, 5, 8 
= 
107 
2, 4, 5, 8 
= 
109 
1, 5, 6, 7 
= 
111: 
2, 3, 6, 8 
= 
113 
1, 4, 7, 7 
= 
115 
1, 4, 6, 8 
= 
117 
2, 3, 5, 9 
= 
119 
2, 2, 7, 8 
= 
121 
1, 3, 7, 8 
= 
123 
2, 2, 6, 9 
= 
125 
1, 3, 6, 9 
= 
127 
0, 4, 7, 8 
= 
129 
0, 5, 5, 9 
= 
131 
0, 4, 6, 9 
= 
133 
1, 2, 7, 9 
= 
135 
0, 3, 8, 8 
= 
137 
0, 3, 7, 9 
= 
139 
1, 2, 6, 10 = 
141. 
So much for the series 1, 3, 7, 13, 21, &c. ; but The Square has three series. The 
other two (when the first term is 1) are 1, 5, 13, 25, 41, 61, &c., and 1, 3, 9, 19, 33, 
51, 71, &c. ; and each of these affords a similar facility of passing from one term to the 
next ; but these last series assist each other, and the condition is, not the sum of all the 
roots being the same, but the difference of two of them, and any term of the second 
being halfway between two of the first; thus 25 + 8 = 33, and 33 + 8 = 41, Avhich is 
true throughout both series. 
