of the Machinery Jor Calculating Tables, 125 
Another series of a similar kind, but more simple in its form, 
is derived from the following equation : 
Aw = units figure of \ 
If the constant or first term is equal to 2, then we may ex- 
press \ thus, 
« = 206 + 
where a is any of the numbers 1, 2, 3, 4, which, taken from 
leaves the remainder divisible by 4, and b is the quotient of 
that division : the series is, 
1... 2 
48 
Example : Let z = 18 
4 
56 
1 being subtracted, 1 
8 
62 
— 
16 
64 
12 which, 
.5... 22 
15... 68 
divided by 4, gives 3, hence. 
24 
76 
a = 1 b=8 
28 
82 
u = 20.3 -J- 2 s= 62 
36 
84 
13 
42 
88 
10.. .44 
20... 90 
Innumerable other series might be formed by the same en- 
gine, the differences of any order depending on the value of the 
figure which might occur in the units, or the tens, or the hun- 
dreds place, or in any one or more determinate places of the 
same, or the next, or preceding terms. Other laws might be 
observed by the same engine, of which the following is an ex- 
ample. A series of cube numbers might be formed, subject to 
this condition, that whenever the number 2 occurred in the tens’ 
place, that and all the succeeding cubes should be increased 
by ten. In such a series, of course, the second figure would 
never be a 2, because the addition of ten would convert it ixy 
to 8. 
