132 Professor Booth on a new Theorem in the 



or developing 

 1 



2. - L. _1 



Hence the development of x <i +x <i in powers of (x + x ) 

 or z, will necessarily contain negative powers of z, which has 

 been just shown to be impossible. 



To determine the coefficients A, B, C, &c. in the develop- 

 ment (VI.). 



As z = x + x , we find by the binomial theorem, 



«, «-2 , n.n — lyn-4, . n.n-l.n-2 n ~G ~ 

 z —x +nx H — — 1 TY3 



« 9 « 2 v n-4 (tt — 2) (» — 3) m-6 •__ 



* w ~ 2 = »«-*+(«_ 8)*" 4 + V J-| 'x ,&C 



^- 4 = ^'- 4 + (n-4) tf w ~°, &c. 



>-* = ^ W - G , &c. 



Substituting for the powers of ^ in (VI.) their developments 

 here given, and bringing over the terms on the left-hand side 

 of the equation, we get 



+ . . . D <x ...j" 



Now the coefficients of the powers of x in this development 

 are separately equal to cypher; and it is easy to show that the 



coefficient of x"~~ ' or of the general term, in this develop- 

 ment, is 



A.n. n — I . w — 2 .. . (n — 1+\) B . w — 2 . n— 3 . . . («— / 

 1 23...^ + 1 2 3...(t — 1) 



C.w — 4.« — B...(n — t—l) D.n-6...(n-t-Q) 

 + ~ 12 3... (£-2) + 12... (#-3) 



But this coefficient of x may be thus written, 



