^16 
Transactions of the Boyal Society of So2ith Africa. 
we have 
p^i^,Pi-,{px -q)- qx{p_i_,p,_, +p_,_,Pi_,) + q^p_i_,Pi_, = 1. 
. • . p. p^i., .p,_, = q{p-i-,Pi-. +P-i-zPi-.) (1) 
and 
qp^i-,Pi^, = q^p^i^4h-2 - 1 
(2) 
Since 
qPi^2=Pp (1) reduces to p>_i_,pi = qp_i_,Pi_^ ; 
And 
PiPi^, -PU = (m-x - m-2)Vi-2 -Pi-Mnh-2 - m-3) 
§ 2. Now the remainder of the division is 
ao{Pn-iX — qPn-i) + (^liPn-zX — qi^i-s) • • • + «n-i^ + 
= x{aoPn-.i + + . . . + " q((^oVn-2 + <^ii?«-3 + • • • + <^,i-2^»o + (^nP-z)- 
Therefore the conditions for no remainder are 
Employing the relation — we can obtain from these two 
conditions any number of equations of the form 
aoPn-v + a^iPn-v-i" ' + <^n-iP-v + (^nP-v-i = 0- 
The notation will be easier to follow if we continue this investigation 
for the special case of 7^ = 8 : we shall write 0, 1, 2, etc. for a^, a^, a^, etc. 
Using j9_i = — we get for 1^ = 2, 3, ... the equations 
Hence 
P-i-.= - 
Pi-, 
q' 
and 
