320 
Mr. Robertson’s new Demonstration 
18. If in the series, which concludes the last article, n be 
equal to r, the whole series becomes equal to i-\-x. For in 
n i 
this case ~ = 1, and therefore = o, and consequently 
every term in the series, after the second, becomes equal to o, 
or vanishes. 
Hence it is evident that the rth root of 1 -\-x, or, which is 
i i 
the same thing, that l-j-xl r =i-{- 
I i 1 I X I 2 * 
— . jj’4- — . . . — &c. for this se- 
z 3 ' r 2 3 4 • 
ties being raised to the rth power becomes equal to l 
As by the general principles of involution the wth power of 
l n 
i -\-x\ r is l-j-.rl r , it therefore follows, from the last observa- 
n 
tion and the preceding article, that l-f-al r =i+ 
&c. 
19. By the general principles of involution a — b\ n =a n x 
1 — — =fl”xi — x\ n , by putting x= — . By article 14, n being 
a whole number, 1 — 3:1”= 1 — nx-\-n . - — - x * — n . . ~~ 2 — x* 
~Y n • T" • — • a &c. and by the same article, m being 
a whole number, 1 — x\ m — 1 — mx + m . 2 3 tyi . 2 
. — J 3 - a: 4 - — &c. But by the general 
principles of involution, and article 14,1 — x\" x 1- — <rf = 1 + m 
