204 
Proceedings of Eoyal Society of Edinburgh. [sess. 
[This means that if ( q , n)a = b, then a — 
Remembering the meaning of equation (11), we see that 
(q,n)q-”( )q~ n = - (?, - n) .... (17). 
The last property we propose to prove is that when n is a positive 
integer 
(q,n) = ( )q n ~ l + q( )q n ~ 2 + q 2 ( )q n ~ 3 + . . . + q n ~\ ) (18). 
Calling, for brevity, the linear function on the right Q, we see that 
• a' and q are coplanar 
Qa' - nq n ~ l a'. 
Again 
2Q a" = Q Y~ 1 q(qa - ^[equation (7)] = V" l qQ(qa - aq) 
Ya n 
= Y~ 1 q(q n a - aq n ) = 2^-a" [equation (7)] . 
Qa = Qa' + Q a" = nq n ~ x a' + 
\q 
a " = (q, n)a , 
which proves the proposition. [Notice that equation (16) combined 
with (18) solves the equation aq n ~ x + qaq n ~ 2 + . . . + ^ 7l_1 a = 5]. 
We can now prove that for all scalar values of n 
d.q n = (q,n)dq (19). 
Equation (18) proves this for n a positive integer. Next suppose 
n = ljm where l and m are positive integers. Let 
q n = r , i.e. q l — r m . 
Differentiating, and using the first case, 
( q , l)dq = (r, m)dr 
d. q n = dr = (r, m) ~ 1 (g, T)dq 
= ( ) '“ k) t,dq [ e i uation e 1 6 )] 
= (?> n)dq 
[equation (15)] . 
