246 MR. WARREN ON THE SQUARE ROOTS 



This will appear from the expansion of l' ; 



One of the values of 1* is 1, 



This value is not a function of x. 



But 1* has other values, which are functions of x ; 



1 

 For example, let .r = §, and let 1^ = i/, 



then «/3 — 1 = 0, 



> ^ , — 1 + a/ —3 — 1 — ^/~^3 

 an equation, whose roots are 1, , ; 



Next, let a? = 5, and let 1* = y, 



then 3/4 — 1 rs 0, 



an equation whose roots are 1, — 1, + v — 1, — v — 1 ; 



In like manner it will appear, if other values be given to x, that l' will have 

 values which are dependent upon the values of <r ; 

 that is, l' is a function of ^r ; 



.-. 1* may be expressed in the form A + B a;- + C ^r^ + &c. 

 where A, B, C, &c. 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 ; .-. f = 1 + B 0? + C j:^ + &c. ; 



Next, to find the law of the series, 1.1=1, 



B* ■r' B* x" 



.-. the series is of the form, 1 + B «■ + y^g- + j^^Ts "^ ^^' ' 



Next, to find the value of B, T'' = 1 + B «a? + -:—- + &c. ; 



... i-=ii±4=(i+mr.(i-mr; 



{l-m) 



Let M = wi — ^ m2 + I w^ — I m* + &c. 

 M' = — m — § m^ — ^ m^ — I m* — &c. 



Then (1 + m)' = 1 + Mo^ + ^ + &c. 



(1 - m)-' = 1 - M'^ + Trf - &c. 

 .-. 1"^= (i+M^+^+&c.).(l-M'a: + ^-&c.) 



