56 
SIE r. POLLOCK ON THE EELATION BETWEEN BOOTS OP 
-4, 0,4, 7 
-5, 2, 4, 6 
81 
-4, 0,1, 8 
121 
-4,1,2,10 
®25 
0, 0, 4, 3 
-1,2, 4, 2 
so the odd squares taken every third term, as 
9 
•4-2, 0,1, 2 
25 
4-2, 1,2, 4 
present a difference of 6 for the exterior roots. , Km- in the case 
The theorem affords a solution of all these and every other ^ 
of any arithmetical progression of odd numbers having an odd number 
terms Tf wLh have been indexed as directed (by making the common drfference th 
index of the middle term, &c.), it will be found that all 
from the middle term are derived from 2 terms of one, or other, of the - 
^I^i^^d by adding the ~ ^ 
flprived from terms in the series 1, o, y, iy, <x . , 
deriveu irom te „ ^ terms in the series may immediateh 
derived from the series 1, 5, 13, Zb, ecc., anu i 
be found, as they are the terms having the same indices as the pan 
arithmetical progression. Let 
«5 '9 ns, n7, "21, '25, ®29, ^33, ®3/ 
be an arithmetic series (with a common difference 4 and 21 the ^ " 
according to the theorem; place under each term that term m either of the two sene, 
which has the index of the term in the arithmetical senes ; thus 
«5 '9 ns n7 "21 '25 ®29 '33 ®37 
o;iL n 23 ®5 "9 ns n9 ^25 ®33 
4 8 10 12 12 12 10 8 4 
Deducting the one from the other, it is obvious that the terms equidistant 21 “e 
derivedfrom terms of the two series, by adding to them the ^ 
in the 2 series be adjacent, the difference of the extenor roots W 1 be 1 , ^ 
and so on ; if the terms in the 2 series be » places distant the -teiioi 
roots will be m : this may be shown generally m an algebraic form, . 
Let the middle term of an arithmetic series be n and t le common i er 
even number 2m, the series with its indices will be 
-i(»_ 6 m), — (»-4m), -‘(n-2m), »«(«+2m), -“(«+4>»). ' 
and the terms of the 2 series to be placed under each term will be 
&c. ( 2 m‘- 4 m+ 3 ), ( 2 m>- 2 m+l), ( 2 m’+l), ( 2 m“+ 2 m+l), (-« + ™+^)' 
deducting the one from the other, the remainders will be 
m-(2m“+5), (m-(2m'‘+3)), (n-(2m’+l)), «-(2ro=+l), »-(2m-+l). +=)■ ' 
