24 G 
MR. WARREN ON THE SQUARE ROOTS 
This will appear from the expansion of l x ; 
One of the values of T is 1, 
This value is not a function of x, 
But 1 1 has other values, which are functions of x ; 
I 
For example, let x = §, and let l 7 = y, 
then if —1=0, 
an equation, whose roots are 1, 
— 1 + — 3 
O 5 
- 1 - v -S 
2 
Next, let x — and let 1* = y, 
then if — 1 — 0, 
an equation whose roots are 1, — 1, + — 1, — >/— 1 ; 
In like manner it will appear, if other values be given to x, that T will have 
values which are dependent upon the values of x ; 
that is, T is a function of x ; 
T may be expressed in the form A + B x + C x 2 -j- &c. 
where A, B, C, See. are constant quantities independent of the value of x ; 
First, to find the value of A, Let x = 0, then 1° = A, 
But 1° = 1, A = 1 ; T = 1 + B x + C x 2 4- Sec. ; 
Next, to find the law of the series, 1* . f = 1 a+ ’ / , 
B 2 x 1 B 3 x 2 
.’. the series is of the form, 1 + B x + y-y + l 2 "3 " 1 “ & c - 5 
■ng 2 2 
Next, to find the value of B, l nx = 1 + B n ,x -J — y* + &c. ; 
Let 
i n - 1 
l w + l 
= vi, then 1 
n 
1 + m 
1 — m’ 
X 
,m (1 + m ) /■, 1 x 
.-.l =L ^- = (1 + m) . (1 - m) ; 
(1 - m) 
Let M = m — \ vi 2 + ^ m 3 + &c. 
M' = — vi — %m 2 — ^ m 3 — \ m 4 — &c. 
„ M 2 .r 2 
Then (1 + in ) = 1 + M<r + 1 y + &c. 
r 2 
(1 — m) = 1 — M 7 x + y y — &c. 
l nI = (l + M*+^ + &c.).(l — M'* + ^-&c.) 
