54 SIE E. POLLOCK OK THE EELATIOK BETTVEEK EOOTS OF 
‘77 is +1 6 6 2, and therefore »117 is - 3, 6, 6, 6. Obnonsly it is easy to thiw the proof 
into a general algebraic form; the general property of odd numbers above stated mat 
thcroforG b© consid-Grod. 8.s cstfiblishGd. v • ■ r. Tt-if-Vi 
The theorem alluded to is this :-If any odd number of 
13 (or any other number), be in arithmetical progiession m a co 
4 [or any other number] (for the purpose of this example let the ^ 
25 29 33, 37, with a common diiference of 4), then if the comm { 
:jum:d a; th; index of the difference of roots to the mMe term m (4+1) 
higher tei-ms beyond the middle have as mdices of the chffeiences o ( ^ 
(4+2), (4+3), &c. in succession, and the lower terms have as indices - ■ 
i4_3) &c the series with the indices will be ‘13, ®17, “21, -o, , o- , o, , 
having the differences indicated by the respective indices [which may be done b. 
propGrty of odd numbors just proved], thus, 
^13 ^17 
1 , 2 , 2, 2 
-1,0, 4, 2 -2, 1,4, 2 
2,0,0,3 +1,0,2,4 0, 0,3,4 
then the terms greater than the middle term will have this 
than the middle term, the two terms next to the micldle eim w 
roots,-one, less by 1, the other, greater by 1, than those 
next but one wiU have thmr "/;2rbt:« &oni tlm centre; and 
:u re (enuidJant from the middle term) will respectively have the same 
middle roots; thus 
-2,1, 4,2 0,0, 2, 5 -2,2,3, 4 -2, 2, 2, 5 
- 1 , 0 , 0,6 
Comparing them, the result is as stated above; the difference between the exterior roots 
o, ,£ ,4— . I.- x»-" — 
The algebraic proof of this theorem is 1^10 series with the indices 
with « as a middle term and y) as a common difference , then 
of thG difforoncGS of roots will bo 
7.-2(^„2p), &c. ; 
rrr’; r r. .« J ... x 
in ordm’that the difference of the roots may correspond 
(®-»i); then the corresponding roots of «+myi wall be, accoi „ 
(*->»), r, s, a+p. If the first set of roots be squared, the sum is 
„‘-+r“+s“+(«+i‘-m)“=«’+»-’+«’+(“+I''’+"'"~'^"“"''^^"'’ 
