1 30 Professor Booth on a new Theorem in the 



ft"-/+«(2»-l)/- 1 +».w-l./" 2 =» 2 /-w(w-l) 2 /" 1 



+ — YJ— (n ~ 2) 9 z . 

 Performing the same operations m times successively, we find 



Ax n + nBx n ~ l +n.n-l Cx n ~ 2 ...-Vn {n- \)...{n-m -+- 2)1 

 p jr »-«+ 1 + n (ft - 1) {n - 2) {n - m + 1) x n ~ m \ 



m „ , .%■ ■—I , n.n— I . . m n -2 f (TV 



= n z n —n (n—1) z H — — (n — 2) z 



W . W — l.n— 2 . -.m n -3 



— — — (n — 3) z .... 



12 3 v ; 



A B C...P being functions of n. 



Now let z = 1, or x = 0, and the first member of the last 

 equation becomes a series of positive powers of 0, multiplied 

 by constant finite coefficients, and therefore each term of the 

 first member is equal to zero. Hence the second member 

 becomes also zero, or 



= n —n {n — 1) -J — — (n — 2) , &c. 



Let m = n, then all the terms of the first member of equation 

 (IV.) vanish, as before, except the last, and it becomes w(w— 1) 

 (n - 2). ..2.1 . 0°, but 0° m 1. Hence 



123{n-l)n = n -n{n- l) w + n '^~ * (n-2)" 



From these principles it follows that if n be an improper frac- 

 tion greater than the integer m, that the series (III.) continued 

 to infinity is equal to zero. 



In the genera] theorem (I.) let a = w — 1, b — n— 2, c — n 

 — 3 1 — (n—t + 1), h = 1, and n — t, then 



(n-1) {n-2)...{n-t+\)-t {{n-2) (*-8)...(n -t)}y 



tit - i) K v * ) 



+ 12 {(»-3)(«-4)...(w-*-l)}-,&c.= J 

 This theorem will be found useful further on. 



To develope x n + x~ n in descending powers of{x+x~ )> 

 Put x + x~ =z, and assume the series 



^ + ^- n =A^ + B^- 2 + C/- 4 + D/ i - 6 ...&c.&c. (VI.) 



The following considerations will show that the assumption 

 of the development in this form is correct; for, in the first 

 place, if n is an odd number, and x is changed into — x, z 

 will be changed into — z j and the first member of (VI.) re- 



