306 Transactiojis of the Boyal Society of South Africa. 
8n^ 
48n3. 
-(2^ + D) 
-1 
-(5^• + 3D) 
-(2i + D) 
-1 
-(26+ 17D) 
-(10^ + 6D) 
-{2^ + D) 
-1 
&c. 
&c. 
(a) 
(b) 
(0) 
which is read as follows : — 
0= -(2i + D) n°-2n; &c. 
Subtract now 3(a) from {h), and generally subtract whole multiples of 
(a) (b)... from the following letter and the following table results : — 
2n:. 
sui. 
48n^ 
-(2i + D) 
-1 
i 
-2^; + 3-D 
-1 
2i 
-2i + 6-D 
-1 
&c. 
&c. 
(«)' 
(6)' 
(0)' 
The final results are presented in Table I. a, which contains precepts for 
its indefinite extension. The Ul\'^', will be found in Table I.e. It will alsa 
be seen that the property — 
n;;/ = n;'xn°^ 
falls naturally into the scheme of these tables. 
In practice we first compute — 
and — 
then — 
2n;, sui, 4:Sui ... &c. 
2n°:;, 8n°:^, 48n°„j ... &c. 
4ir,:;, I6n;^:;, 96ni:i ... &c. 
and so on, step by step, each individual step being simple enough, and 
natural checks on the results abound. This method requires, however, 
the quantities 0*^110 where g - 1 is equal to the highest power of e or e^,. 
which has to be included. Thus in actual work we must compute — 
2nj, 8n-;, 48ro ... &c. 
mu], mui, msm ... &c. 
D^2n;, D^8m, &c. L &c. 
