of the Machinery for Calculating Tables. 125 
should calculate tables of other species, whose analytical laws 
were unknown. On this suggestion, I proceeded to write down 
a table which might have been made, had such an engine exist- 
ed ; and finding that there were no known methods of express- 
ing its wth term, I thought the analytical difficulty which was 
thus brought to light, was itself worthy of examination. The 
following are the first thirty terms of a series of this kind : 
0... 2 
11. ..222 
22... 924 
1 ... 2 
264 
1010 
2... 4 
310 
1096 
3... 10 
356 
25... 1188 
4... 16 
15.. .408 
1288 
5... 28 
468 
1396 
48 
536 
1510 
76 
610 
1624 
110 
684 
30.. .1742 
144 
20... 762 
1862 
10.. .182 
842 
1984 
The law of formation of which is, that the first term is 2, its 
first difference 2, and its second difference equal to the units fi- 
gure of the second term ; and generally, the second difference 
corresponding to any term, is always equal to the units figure of 
the next succeeding term. This engine, when once set, would 
continue to produce term after term of this series without end, 
and without any alteration ; but we are not in possession of me- 
thods of determining its wth term, without passing through all 
the previous ones. If u n represent any term, then «« must be 
determined from the equation 
l\u — the units figure of u , , . 
an equation of differences of a species which I have never met 
with in treatises on that subject. 
If we push the inquiry one step farther, it is possible to ex- 
press the units figure of any number in an analytical form. 
Thus, let $ represent the sum of the v\h powers of the tenth 
roots of unity, then will 
