[ 49 J 
III. On some Bemarkccble Belations which obtain among the Boots of the Fowr Squares 
into which a Number may be divided, as comjgared with the corres^ponding Boots of 
certain other Numbers. By the Bight Hon. Sir Frederick; Pollock, F.B.S., Lord 
Chief Baron. 
Received AprO. 26, — Read May 20, 1858. 
The property of numbers, which is the subject of this paper, first presented itself to my 
attention in the case of the odd squares 1, 9, 25, 49, &:c. (2?^+l)^ any two adjoining 
odd squares may be divided (each of them) into 4 square numbers, whose roots will 
have this remarkable relation to each other : two of them will be identically the same ; 
and as to the other two, one of them wiU be 2 less, and the other will be 2 more than 
the roots of the preceding or subsequent odd square ; for example, 25 and 49 may be 
dmded into squares, the roots of which being placed below, will appear thus : — 
25 49 so 49 81 
-2, 1,4, 2 —4,1, 4,4 0,2, 3, 6 -2,2, 3,8 
or thus 0, 0, 3, 4 — 2, 0, 3, 6. 
In comparing the roots of the adjoining odd squares, 2 roots (placed in the middle) are 
the same ; of the others, one is 2 more, the other 2 less than the corresponding roots of 
the other. 
The following Table presents the result of a comparison of the roots of all odd squares 
up to 27^=729:— 
1 
0, 0,1,0 
9 
0 , 1 , 2, 2 
25 
-2, 1,4, 2 
0, 0, 3, 4 
49 
0, 2, 3, 6 
81 
0 , 1 , 4 , 8 
-2,4, 5,6 
-4, 0,7, 4 
9 
- 2 , 0 , 1,2 
25 
-2, 1,2, 4 
49 
-4, 1,4, 4 
-2, 0, 3,6 
81 
-2,2, 3,8 
121 
-2,1,4,10 
-4, 4,5,8 
-6,0, 7,6 
121 
-2, 2, 7, 8 
169 
0, 3, 4, 12 
-2,4, 7,10 
-4,5, 8,8 
-6,4, 9,6 
225 
-4, 3,10,10 
-6, 5,10,8 
289 
-2, 5, 8,14 
-6, 3, 12,10 
169 
-4,2, 7,10 
225 
-2,3, 4,14 
-4, 4, 7,12 
-6,5, 8,10 
-8, 4, 9,8 
289 
-6, 3,10,12 
-8, 5,10,10 
361 
-4, 5, 8,16 
-8, 3, 12, 12 
MDCCCLIX. 
H 
