Calculus of Finite Differences. 1 3 1 



tains its value, changing its sign. Hence all the powers of z 

 in the second member of (VI.) are odd, or the index n is 

 diminished by even numbers. 



Again, when n is an even number, and x is changed into 

 — x, z is changed into— z; and the first member of (VI.) re- 

 maining unaltered, the second can contain only even powers 

 of z, or n is diminished by even numbers. 



The development of x 11 + x~ n , in powers q/Qx + x~ J or z, 

 can contain no negative powers of z. 



For if it were possible that the development should contain 

 such powers, put x — v— T, then x = 0, and the first member 

 of (VI.) becomes + 2 when n is divisible by 4, — 2 when n 

 is divisible by 2, and not by 4, and zero when n is odd, while 

 the second member becomes infinite, as having positive powers 

 of zero in the denominator. 



Hence it follows that when n is even, the coefficient of z° 

 is + 2 ; for by putting z = 0, all the terms of the development 

 vanish, except the one containing z°, while the first member 

 of the equation (VI.) becomes + 2. 



Wlien n is a fraction = — , the development (VI.) becomes 



I _I 



impossible. To show this, put xi=-y, x i=y"~ ; then 



—l — — — p —p 



putting y + y — u, x q + x i = y + y 



= P/ + Q« 2, - 2 +R« p - 4 ,&c 

 by (VI.), since p is a positive integer number, and P, Q, R 

 are functions of p, as will be shown further on. 



We must now substitute for the powers of «, in the last de- 

 velopment, their values in terms of z, 



Hence 



or 



now z = y q + y q . 



q z + Vz 2 -4> „,- g _ s — vV-4 



y- o 1 y q = 9 » 





K2 



