3 1 6 
Mr. Robertson’s new Demonstration 
each equal to a, each of the members in any coefficient will 
become a power of a ; and, therefore, as the exponent of x in 
the first term is equal to n, it follows from the sixth and last 
articles that x ^-a\ n =x n -\-nax H *4 -n .- — -a?x n z -\-n. 
71 — [ 
2 
71— i n — z 
" * 3 
3 n—x 
ax J 
-f -n 
7 i— i n — 2 n — 3 
ax 
71 — i 
+ &C. 
14. If equations be generated from x — a . x—b .x—c .x—d y 
See. the coefficients will be the same, excepting the signs, as 
those which result from x-\-a . x~\-b . x+c . x-\-d, Sic. in the 
fifth article ; and as minus multiplied into minus gives plus, 
but minus multiplied into minus multiplied into minus gives 
minus, the coefficients in equations generated from x — a . 
x — b.x — c.x — d, Sic. whose members have each an even 
number of the quantities a, b, c, See. will have the sign 
but coefficients whose members have each an odd number of 
the quantities a, b, c, &c. will have the sign . And hence it 
Is evident that x — a\”==x n — nax~ x 
4 -n 
71 — I a 71 — 2 71 — I 
a x — n . 
>1—2 , 71— 2 . 71 — 1 71— Z 71 — 3 4 71 — A. Q ^ 
n ^ . . — i a 7 . r &c. 
a x 
15. By the general principles of involution a-\-b\ n =a n x 
14- ~| =a n x 14-vcf, by putting x= By article i4-.rl 
* 4 + 
, , 71 — I „ , 71 — I 71 — 2 3 . 71 — I 71 — 2 71— X 
: 1 4 -nx-$ r n. x*-\-n . . x-\-n . — 
Sic. and by the same article i-\-x\ m =zi-\-mx-\-m . 
■x Z ’\-m. 
3 4 
771 — I 
x*-\ -m . 
m-x m—z 3 , _ &c> But by the ge- 
neral principles of involution, and article 13, i4"‘ ri x 
1 ", tl -4- 1YI 0 » B B 
14-xl = 1 4~ n 4" m x 1 m . 
71 -f 771 — I 
n+ffi-i 
2 ±ZL=± , n + m rJ . . j? 4- & c . when n 
X • 2.3 4 1 
and rn are whole numbers. 
tl “I - “■ 2 3 _ . 
x -\-n-\-m 
