58 SIE F. POLLOCK ON THE EELATION BETWEEN EOOTS OF 
It is obvious, that what is done with these numbers may be done with any other 
odd numbers, and it would be supertuous to give an algebraic proof. 
This relation of all odd numbers to each other has not (as far as I am awme) 
remarked before; but it has occurred to me that 
‘‘ mysteries of numbers” alluded to by Feejiat in that “ ' 
theorem of the polygonal numbers was announced in a note to an edition of Diop . 
published aftei his death, p, 180*. The mysterious properties of numbers 'o 
by Feemat must (of course) be connected with the theorem he was announcing , mde 
he expressly refers to them as the source of his demonstration. 
PosTSCEiPT. — May 20, 1858. 
Since this paper was sent to the Society, some other theorems of a simdar kind have 
occurred to me, in which the terms of a series (not arithmetical of mavTe 
a similar relation with regard to the roots of the 4 squaies in o w 
divided that is. those which are equidistant from the middle (if the numbel of tei 
iTmri or from the middle term (if the number of terms be odd), have the mid^ 
the same and the exterior roots have an arithmetical relation to each othei, la ■; o 
t': raic“from the centre, rir. the one being less and the other greater by the same 
‘’“Thu‘s!'if any number of terms (exceeding 3) of either of the 2 series above-mentio^d 
viz 1 3 9, &c. (2»’+l), or 1, 6 , 13, 25, &c. (2rf+2«+l), and, beginning with 
the I’st term, the successive differences of the terms be added “ tnverso oi dme, a new 
series will be obtained possessing the property in question; thus the ^t 
of the 1st series are «!, ’3, -9, ‘W, «33, ■•51, .=73 ; the differences ^ UO 14. IK 
02 - if the differences be added “imerso ordim,” beginning with 22 instead of 
ie: becomes 1, 23, 41, 56, 65, 71, 73, each term of which may be divided mto 4 squares. 
whose roots will be as follows : — 
«55 ^65 
-3, 1,6, 3 -2, 3, 4 , 6 
1 , 2 , 5, 5 0 , 0 , 1 , 8 
+ 1 , 1 , 2 , 7 
Here obviously the result is as stated above ; 55 is the middle term , the terms eqmdivtan 
from it have tL same middle roots, and the difference between the other roots rrrcreases 
according to the distance from the middle term being 2, 4, 6 . 
there is nomiddle term; the result is similar.but the successive dlffeieiices. . , ■ 
The other series, 1, 5, 13, 25, &c., gives a similar result. 
The reason of these results is that the equidistant terms are always equal to 
ginal corresponding term in the series, increased by the same number; thus, 
41 and 66 = 9 + 32 and 33 + 32 respectively. 
* See Legendee, Theorie des Nombres, 1st edit. p. 187. 
01 
0, 0,1,0 
^23 
+1, 2, 3, 3 
441 
0, 3, 4, 4 
+ 2 , 0 , 1 , 6 
1071 
-3, 2, 3, 7 
1.273 
6 , 0 , 1 . 6 
