SQIJAEES INTO WHICH ODD NUMBEES MAY BE DIVIDED. 
53 
The middle roots are all the same ; the exterior roots increase on the right and decrease 
on the left, by 1, at each step. 
The other series will appear thus : 
11 35 5;|_3 725 ®41 &c. . . 
0,0, 0,1 -1,0, 0,2 -2, 0,0, 3 -3, 0,0, 4 -4, 0,0, 5 -w, 0, 0, (?^^-l) 
Here also the middle roots are the same, and the exterior roots increase and decrease in 
the same manner, by 1, at each step ; but if any odd number have two of its roots equal, 
it may be the first term of a series of the first kind ; or if any odd number have two of 
its roots difiering by 1, it may be the first term of a series of the second kind ; any odd 
number may therefore be the first term of a series of either kind ; thus — 
°23 
+ 3, 1,2, 3 
^23 
+1,3, 3, 2 
^25 
+ 2, 1,2, 4 
^27 
0, 3, 3, 3 
431 641 
+ 1 , 1 , 2, 5 0 , 1 , 2 , 6 
^35 ^47 
-1,3, 3, 4 -2, 3, 3, 5 
&c. . . . ^”(2^^^+23) 
-(w-3)l,2, (w+3) 
&c. . . . ^“+'(2^^"+2?^+23) 
[n — 1), 3, 3, (w+2) 
The middle roots are the same; the exterior roots increase and decrease in the same way, 
and the terms of the two series increase by the same numbers, as if 1 were the first term 
of both ; the difference between any two adjoining terms is the sum of their indices ; if 
not adjoining, the difference between any two terms is the sum of their indices, plus 
twice the sum of the intermediate indices ; the roots which compose any odd number to 
a given index may therefore be found thus: — Let 117 be the given number, and 12 the 
required index; then "a, V, % % '“/, 'H17 will represent the series whose seventh 
term is 'H17 : 0 + 12+2 x (2 + 4+6 + 8 + 10) = 72 is the difference between a and 117 ; 
a therefore =117 — 72 = 45 ; and the series with its roots will be 
045 2417 453 603 siytjr 1095 
+2, 1,6,2 +1,1,6, 3 0,1, 6, 4 -1,1, 6, 5 -2, 1,6, 6 -3, 1,6, 7 -3, 1,6, 8 
In the same manner ’^117 may be discovered by the other series, and the result will be 
133 '37 '45 ’57 "73 ^'93 ^'117 
+2, 2, 4, 3 +1,2, 4, 4 0, 2, 4, 5 -1,2, 4, 6 -2, 2, 4, 7 -3, 2, 4, 8 -4, 2, 4, 9 
What is stated above of 117 and the index 12, is obviously applicable to any other odd 
number and any other index ; and it follows that if every odd number has two of its 
roots equal, and also two of its roots differing by 1, it will have other roots differing by 
2, 3, 4, &c., as far as its magnitude will enable it. 
There are various other modes of proving the same result. If it be required to find 
the roots which make ®117, begin with *117, H21, ®129, ^141, ®157 ; 157 is 40 more 
than 117 ; deduct 40 from every term, and the series becomes 
177 '89 ’101 "117 
+ 1,6, 6,2 —3, 6, 6, 6 
